Publications mathématiques de Besançon

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1 Publications mathématiques de Besançon Algèbre et théorie des nombres 2018 Presses universitaires de Franche-Comté

2 Laboratoire de Mathématiques de Besançon (CNRS UMR 6623) Publications mathématiques de Besançon A l g è b r e e t t h é o r i e d e s n o m b r e s - F o n d A t e u r : g e o r g e s g r A s Comité de rédaction Directeur de la revue : le directeur du laboratoire Éditeur en chef : Christophe Delaunay Comité scientifique Bruno Anglès, Université de Caen bruno.angles@math.unicaen.fr Éva Bayer, École Polytechnique Fédérale de Lausanne (Suisse) eva.bayer@epfl.ch Jean-Robert Belliard, Université de Franche-Comté jean-robert.belliard@univ-fcomte.fr Jean-Marc Couveignes, Université Bordeaux 1 jean-marc.couveignes@math.u-bordeaux1.fr Vincent Fleckinger, Université de Franche-Comté vincent.fleckinger@univ-fcomte.fr Farshid Hajir, University of Massachusetts, Amherst (USA) hajir@math.umass.edu Nicolas Jacon, Université de Reims Champagne-Ardenne nicolas.jacon@univ-reims.fr Jean-François Jaulent, Université Bordeaux 1 jean-francois.jaulent@math.u-bordeaux1.fr Henri Lombardi, Université de Franche-Comté henri.lombardi@univ-fcomte.fr Christian Maire, Université de Franche-Comté christian.maire@univ-fcomte.fr Ariane Mézard, Université Paris 6 mezard@math.jussieu.fr Thong Nguyen Quang Do, Université de Franche-Comté tnguyenq@univ-fcomte.fr Hassan Oukhaba, Université de Franche-Comté hassan.oukhaba@univ-fcomte.fr Manabu Ozaki, Waseda University (Japon) ozaki@waseda.jp Emmanuel Royer, Université Blaise-Pascal Clermont-Ferrand 2 emmanuel.royer@math.univ-bpclermont.fr Publications mathématiques de Besançon Laboratoire de Mathématiques de Besançon - UFR Sciences et Techniques - 16, route de Gray - F Besançon Cedex ISSN

3 Publications mathématiques de Besançon A l g è b r e e t t h é o r i e d e s n o m b r e s 2018

4 Laboratoire de Mathématiques de Besançon (CNRS UMR 6623) Directeur de la revue : le directeur du laboratoire Éditeur en chef : Christophe Delaunay Presses universitaires de Franche-Comté, Université de Franche-Comté, 2018

5 Publications mathématiques de Besançon A l g è b r e e t t h é o r i e d e s n o m b r e s 2018 Presses universitaires de Franche-Comté

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7 Sommaire H. Chen Sur la comparaison entre les minima et les pentes D. Roberts Hurwitz Belyi maps D. Roberts A three-parameter clan of Hurwitz Belyi maps F. Ulpat Rovetta A strategy and a new operator to generate covariants in small characteristic A. Sedunova A partial Bombieri Vinogradov theorem with explicit constants M. Watkins Jacobi sums and Grössencharacters

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9 , 5-23 SUR LA COMPARAISON ENTRE LES MINIMA ET LES PENTES par Huayi Chen Résumé. On compare les minima et les pentes successifs d un fibré vectoriel hermitien sur une courbe arithmétique et on démontre un encadrement uniforme de leurs différences. La preuve repose sur un principe général de comparaison des R-filtrations et un lien entre la géométrie des fibrés vectoriels hermitiens et le problème de transfert en géométrie des nombres. Ce résultat s applique à l étude des invariants birationnels des fibrés en droites hermitiens sur une variété arithmétique projective. Abstract. (On comparaison between minima and slopes) One compares the successive minima and successive slopes of a Hermitian vector bundle over an arithmetic curve and establishes a uniform bound for their differences. The proof relies on a general comparison principle of R-filtrations and a link between the geometry of Hermitian vector bundles and the transference problem in geometry of numbers. This result can be applied to study the birational invariants of Hermitian line bundles on an arithmetic projective variety. 1. Introduction Les minima et les pentes successifs sont des invariants naturels en géométrie des fibrés vectoriels hermitiens. Les minima avaient été déjà étudiés dans la théorie classique de Minkowski dans le cadre des réseaux euclidiens puis ont été généralisés dans le cadre de géométrie des nombres algébriques par différent auteurs ; les pentes ont été introduites par Bost [3], en s inspirant de la géométrie des fibrés vectoriels sur une courbe projective (notamment la théorie de Harder Narasimhan [17]), ainsi que les travaux de Stuhler [22] et Grayson [16] dans le cadre arithmétique. La comparaison entre ces invariants a été étudiée par plusieurs auteurs [2, 12, 21] dans divers contextes. Le meilleur résultat dans la littérature est dû à Borek (cf. l inégalité (1.1) plus bas), ce qui majore la différence entre la i ème pente et le i ème minima par i fois une fonction du rang du fibré vectoriel hermitien dont l ordre de grandeur Classification Mathématique (2010). 14G40, 11H50. Mots clefs. Hermitian vector bundle, slopes, successive minima.

10 6 Sur la comparaison entre les minima et les pentes est le logarithme du rang (où l entier i varie entre 1 et le rang du fibré vectoriel hermitien). Cette majoration, qui n est pas uniforme en i, est peu efficace lorsque i est grand. Dans ce travail on revisite le problème de comparaison entre les minima et les pentes en établissant une majoration uniforme de la différence entre les minima et les pentes successifs. L amélioration majeure consiste à enlever le facteur i devant le majorant, c est-à-dire que le nouveau majorant est d ordre logarithmique (du rang du fibré vectoriel hermitien) et est uniforme en i. Pour obtenir ce résultat, on adopte une stratégie complètement différente de celle de Borek (reposant sur le deuxième théorème de Minkowski), qui consiste à combiner l approche de R-filtration et le théorème de transfert de Banaszczyk. Cette nouvelle méthode révèle aussi le lien étroit entre la comparaison entre les minima et les pentes (notamment celle entre le dernier minima et la dernière pente) et le problème de transfert, et pourrait avoir des applications à l étude de ce dernier dans le futur. Soient K un corps de nombres et O K la fermeture intégrale de Z dans K. Par fibré vectoriel normé sur Spec O K on entend un O K -module projectif de type fini E muni d une famille de normes ( σ ) σ:k C paramétrée par l ensemble des plongements du corps K dans C, où σ est une norme sur E OK,σC. On demande en plus que la donnée des normes ( σ ) σ:k C soit invariante par la conjugaison complexe. Si de plus les normes σ sont toutes hermitiennes, on dit que le fibré vectoriel normé (E, ( σ ) σ:k C ) est un fibré vectoriel hermitien. Dans le cas où K = Q (et donc O K = Z), la donnée d un fibré vectoriel hermitien sur Spec Z est équivalente à celle d un réseau euclidien. En effet, il y a un unique plongement de Q dans C. La restriction de la norme Q C à E Z R donne une norme euclidienne sur ce dernier. Ainsi on peut considérer E comme un réseau euclidien dans (E Z R, Q C ). Pour tout O K -module projectif de type fini E, on désigne par rg OK (E) le rang de E sur O K, qui s identifie à la dimension de E OK K sur K. Étant donné un fibré vectoriel normé E = (E, ( σ ) σ:k C ) sur Spec O K, pour tout i {1,..., rg OK (E)}, le i ème minimum (de Bombieri Vaaler au sens de [15]) de E (noté λ i (E)) est défini comme l infimum de l ensemble des nombres positifs R tels que l espace vectoriel }) Vect K ({s E max s σ R σ:k C soit de rang i sur K. Dans le cas où K = Q, cela revient à la définition classique des minima successifs à la Minkowski. Pour faciliter la comparaison on introduit la version logarithmique des minima en posant ν i (E) := ln λ i (E). On a alors ν 1 (E) ν 2 (E)... ν r (E), r = rg OK (E). Les pentes successives d un fibré vectoriel hermitien sont construites d une façon similaire à la théorie de Harder Narasimhan en géométrie des fibrés vectoriels sur une courbe projective. À tout fibré vectoriel hermitien E = (E, ( σ ) σ:k C ) on associe un nombre réel deg(e), appelé degré d Arakelov (normalisé), qui est construit comme suit. On fixe une famille {s i } r i=1 d éléments de E qui forme une base de E OK K sur K (r est donc égal au rang de E sur O K ) et on définit ( 1 ( ) deg(e) := ln # Λ r E/O K (s 1 s r ) ln s 1 s r σ ). [K : Q] σ:k C Il s avère que cette définition ne dépend pas du choix de la famille {s i } r i=1.

11 H. Chen 7 Soit E un fibré vectoriel hermitien non nul sur Spec O K. On considère l ensemble des points de R 2 de la forme (rg OK (F ), deg(f )), où F parcourt l ensemble des sous-o K -modules de E, et dans la structure de fibré vectoriel hermitien de F on considère les normes induites. Le bord supérieur de l enveloppe convexe de cet ensemble est le graphe d une fonction concave P E sur [0, rg OK (E)], qui est affine sur chaque intervalle [i 1, i], i {1,..., rg OK (E)}. On désigne par µ i (E) la pente de la fonction P E sur [i 1, i] (qui est égale à P E (i) P E (i 1)), appelée i ème pente de E. Comme la fonction P E est concave, on a µ 1 (E) µ 2 (E)... µ r (E), r = rg OK (E). La comparaison entre les minima et les pentes consiste à majorer et minorer la différence µ i (E) ν i (E) par des termes qui ne dépendent que de K, rg OK (E) et i {1,..., rg OK (E)}. On dit qu un majorant ou un minorant est uniforme s il ne dépend pas de i. Par l inégalité de Hadamard, il est facile de montrer que la i ème pente µ i (E) est toujours minorée par ν i (E) (voir la proposition 3.4 et le théorème 3.7 pour les détails, cf. [2, théorème 1] pour une autre démonstration). En outre, l égalité µ i (E) = ν i (E) est atteinte pour tout i lorsque E est un fibré vectoriel hermitien trivial. La majoration de µ i (E) ν i (E) est cependant plus subtile. Le résultat de Borek donne (cf. [2, théorème 3]) i (1.1) µ i (E) ν i (E) rg OK (E) C(rg O K (E), K), où pour tout n N, avec C(n, K) := ( 1 n(r 1 + r 2 ) ln(2) + n ) [K : Q] 2 ln K r 1 ln(v n ) r 2 ln(v 2n ), r 1 et r 2 : les nombres des places réelles et des places complexes de K respectivement, K : le discriminant absolu de K, v m : la mesure de Lebesgue de la boule unité dans R m, m N, m 1. La stratégie de Borek repose sur le deuxième théorème de Minkowski, qui pourrait être considéré comme une comparaison entre la somme des minima (logarithmiques) successifs et le degré d Arakelov (qui s identifie à la somme des pentes successives). Dans le langage de la géométrie d Arakelov, le deuxième théorème de Minkowski s énonce comme (1.2) deg(e) rg OK (E) i=1 ν i (E) = rg OK (E) i=1 ( µi (E) ν i (E) ) C(rg OK (E), K). La méthode de Borek consiste à combiner cette inégalité avec la filtration de Harder Narasimhan. Elle conduit à un facteur i dans le terme à droite de l inégalité (1.1). Par la formule de Stirling, on peut montrer que C(n, K) = O(n ln(n)), n +. La somme de l inégalité (1.1) pour i {1,..., rg OK (E)} donne une majoration de la différence entre le degré d Arakelov et la somme des minima successifs qui est beaucoup moins bonne que (1.2), où on multiplie le majorant C(rg OK (E), K) par (rg OK (E) + 1)/2. Cela suggère que l inégalité (1.1) n est pas assez précise lorsque i est grand.

12 8 Sur la comparaison entre les minima et les pentes Dans cet article on établit une majoration de la différence entre la i ème pente et le i ème minima comme suit (cf. le théorème 3.7). Théorème 1.1. Soit E un fibré vectoriel hermitien non nul sur Spec O K. Pour tout i {1,..., rg OK (E)}, ( 3 ln K (1.3) µ i (E) ν i (E) ln(rg OK (E)) + min 2[K : Q], ln K [K : Q] + 1 ) 2 ln[k : Q]. La majoration obtenue dans le théorème est uniforme en i. Par ailleurs, le majorant dans (1.3) a le même ordre de grandeur que C(rg OK (E), K)/ rg OK (E) quand rg OK (E) +. Ainsi ce résultat améliore considérablement l inégalité (1.1) lorsque i est grand. En outre, la somme sur i des inégalités dans (1.3) donne une majoration de deg(e) rg O (E) K i=1 ν i (E) dont l ordre de grandeur est identique à celui du majorant dans le deuxième théorème de Minkowski lorsque rg OK (E) tend vers l infini. La nouveauté principale de la méthode de démonstration du théorème 1.1 est de découvrir que l on peut ramener le problème à comparer les R-filtrations par minima et de Harder Narasimhan. Cela permet de montrer que, si on désigne par δ(n, K) la borne supérieure de µ n (F ) ν n (F ), où F parcourt l ensemble des fibrés vectoriels hermitiens de rang n sur Spec O K, alors pour tout fibré vectoriel hermitien E de rang n sur Spec O K et tout i {1,..., n}, on a (1.4) µ i (E) ν i (E) δ(n, K). Cette inégalité est obtenue en s appuyant sur l approche de R-filtration introduite dans [10], voir aussi mon mémoire d habilitation [8, 1.2.4] pour la comparaison des R-filtrations. L idée de ramener la comparaison entre les minima (de Roy Thunder ou ceux des quotients) et les pentes à celle entre le dernier minimum et la dernière pente apparaît aussi (d une manière plus directe) dans les notes du cours de Gaudron [14] à l école d été 2017 de l institut Fourier. Le passage à la comparaison des R-filtrations donne cependant une interprétation conceptuelle de cette approche. Pour en déduire une majoration explicite de µ i ν i, on relie la comparaison entre la dernière pente et le dernier minimum au problème de transfert en géométrie des nombres. Pour tout entier r 1 on désigne par d(r, Q) la borne supérieure des (ν 1 (V ) + ν r (V )), où V parcourt l ensemble des réseaux euclidiens de rang r. On établit l inégalité suivante (cf. la proposition 4.7) (1.5) δ(n, K) d(n[k : Q], Q) + ln K [K : Q] pour tout entier n 1. Une version faible de l inégalité (1.3) s ensuit via le théorème de transfert à la Banaszczyk (cf. [1, Theorem 2.1]). Après avoir soumis l article pour publication, le rapporteur m a signalé l article [15] de Gaudron et Rémond et m indique une amélioration de la majoration (1.5) en utilisant les minima de Roy Thunder et des résultats dans cette référence. Cela permet d améliorer le terme constant dans la majoration obtenue dans la version précédente de l article (comparer le théorème 1.1 et le corollaire 4.8). Je tiens à lui exprimer mes remerciements. Soient π : X Spec O K un morphisme projectif et plat et L un fibré en droites hermitien sur X. Pour tout entier n 1, soit E n le O K -module projectif π (L n ) muni d une famille

13 H. Chen 9 ( n,σ ) σ:k C de normes hermitiennes telles que 1 (1.6) lim sup n + n ln 0 s E n OK,σC s n,σ s n,σ,sup = 0, où s n,σ,sup = sup x Xσ(C) s(x) σ. On désigne par η n et η n les mesures boréliennes sur R définies comme η n = 1 r n δ µi r (E, η n)/n n = 1 r n δ n r νi (E, n)/n n i=1 où r n est le rang de π (L n ) sur O K, et pour tout x R, δ x est le mesure de Dirac en x. Il est démontré dans [10] et dans [9] que, si L K est un faisceau inversible ample sur X, alors les suites de mesures (η n ) n 1 et ( η n ) n 1 convergent faiblement vers des mesures boréliennes η L et η L respectivement (qui ne dépendent pas du choix des normes hermitiennes n,σ vérifiant (1.6)). Ces deux mesures ont des interprétations géométriques différentes. La mesure η L est liée au comportement asymptotique des polygones de Harder Narasimhan du système linéaire gradué de L (cf. [10, théorème 3.1.8], voir aussi du loc. cit. pour le lien entre les polygones et les mesures boréliennes sur R). La mesure η L est liée aux propriétés birationnelles de L. En particulier, le volume arithmétique de L est égal à (dim(x) + 1) vol(l) max(x, 0) η L (dx). En outre, les convergences des suites (η n ) n 1 et ( η n ) n 1 ont été obtenues par des méthodes différentes. La convergence de (η n ) n 1 repose sur la construction d un couplage discret entre des répartitions uniformes sur les monômes ; tandis que celle de ( η n ) n 1 découle du théorème de Hilbert Samuel pour les systèmes linéaires gradués démontré par Lazarsfeld et Mustaţǎ [19]. Comme application du théorème 1.1, on démontre que les mesures limites η L et η L sont identiques (cf. le théorème 5.1 infra). La majoration uniforme (1.3) est le point clé dans la comparaison des deux suites de mesures (η n ) n 1 et ( η n ) n 1. Cela établit un lien entre le comportement asymptotique des polygones de Harder Narasimhan et la géométrie birationnelle de la variété arithmétique de X (voir la remarque 5.2). L article est organisé comme suit. Dans le deuxième paragraphe on rappelle la construction des R-filtrations par minima et de Harder Narasimhan d un fibré vectoriel hermitien sur une courbe arithmétique. Le troisième paragraphe est consacré à démontrer un principe général de comparaison des R-filtrations et à en déduire l inégalité (1.4). Dans le quatrième paragraphe, on propose une majoration de la fonction δ(, K) et on établit le théorème 1.1. Enfin, on conclut l article par des applications et quelques commentaires dans le cinquième paragraphe. Ce travail est partiellement soutenu par le fond de recherche ANR-14-CE Une partie de recherches qui ont conduit à cet article ont été réalisées à Beijing International Center for Mathematical Research (BICMR) lors de ma visite scientifique. Je voudrais remercier le centre pour son hospitalité. R i=1 2. R-filtrations associées à un fibré vectoriel hermitien Le but de ce paragraphe est de rappeler la construction des R-filtrations par minima et par pentes d un fibré vectoriel hermitien (voir la définition 3.1 pour la notion de R-filtration en général). Ces R-filtrations sont des outils pour comparer les minima et les pentes.

14 10 Sur la comparaison entre les minima et les pentes Soit E un fibré vectoriel normé non nul sur Spec O K. Pour tout t R, soit F t m(e) le sousespace K-vectoriel de E K := E OK K engendré par l ensemble { s E max σ:k C s σ e t}. On voit aussitôt que, si t 1 et t 2 sont deux nombres réels tels que t 1 t 2, alors on a F t 1 m (E) F t 2 m (E). Autrement dit, la famille (Fm(E)) t t R est décroissante. En outre, par définition, pour tout i {1,..., rg OK (E)}, on a ν i (E) = sup{u R rg K (F u m(e)) i}. En d autres termes, pour tout i {1,..., rg OK (E)} et tout t R, (2.1) rg K (F t m(e)) i ν i (E) t. Proposition 2.1. Soient E un fibré vectoriel normé non nul sur Spec O K et r le rang de E sur O K. a. Si t > ν 1 (E), alors F t m(e) = {0}. b. Si t ν r (E), alors F t m(e) = E K. c. Pour tout i {1,..., r 1}, les sous-espaces vectoriels de E K dans la famille (Fm(E)) t t ]νi+1 (E),ν i (E)] sont égaux. Démonstration. Ce sont des conséquences immédiates de la relation (2.1). Définition 2.2. Soit E un fibré vectoriel normé non nul. On désigne par ν min (E) le dernier minimum logarithmique de E. Par définition, si r est le rang de E sur O K, alors ν min (E) = ν r (E). La proposition suivante donne une construction de F m en utilisant ν min. Proposition 2.3. Soit E un fibré vectoriel normé non nul sur Spec O K. Pour tout t R, on a Fm(E) t = F K, 0 F E ν min (F ) t où F parcourt l ensemble des sous-o K -modules non nuls de E, et dans la structure de fibré vectoriel normé de F on considère des normes induites. Démonstration. Soit F un sous-o K -module non nul de E tel que ν min (F ) > t. Il existe alors une famille {s i } n i=1 d éléments de F linéairement indépendants sur K, qui vérifie i {1,..., n}, max s i σ e t. σ:k C L espace vectoriel F K est donc contenu dans F t m(e). Réciproquement, pour tout t R tel que F t m(e) {0}, il existe des éléments u 1,..., u k dans E qui engendrent F t m(e) comme espace vectoriel sur K, et tels que j {1,..., k}, max u j σ e t. σ:k C Soit F le sous-o K -module de E engendré par {u 1,..., u k }. Par définition le dernier minimum de F est t.

15 H. Chen 11 Les pentes successives peuvent aussi être décrites par une R-filtration en s appuyant sur la théorie de Harder Narasimhan. Cette construction (de R-filtration) a été introduite dans [10]. Dans la suite on rappelle la théorie de Harder Narasimhan pour les fibrés vectoriels hermitiens et la construction de R-filtration de Harder Narasimhan. Pour tout fibré vectoriel hermitien non nul E, on désigne par µ(e) le quotient deg(e)/ rg OK (E), appelé pente de E. Le fibré vectoriel hermitien E est dit semi-stable si, pour tout sous-o K -module non nul F de E, on a µ(f ) µ(e). Étant donné un fibré vectoriel hermitien E sur Spec O K, il existe un unique drapeau {0} = E 0 E 1... E n = E de sous-o K -modules de E (appelé drapeau de Harder Narasimhan) qui satisfait aux conditions suivantes (cf. [3, A.3]) : 1. pour tout j {1,..., n}, le O K -module quotient E j /E j 1 est projectif et, si on le munit des normes sous-quotients, le fibré vectoriel hermitien E j /E j 1 est semi-stable ; 2. les pentes des fibrés vectoriels hermitiens sous-quotients vérifient les inégalités µ(e 1 /E 0 ) >... > µ(e n /E n 1 ). Soit E l enveloppe convexe dans R 2 des points de la forme (rg OK (F ), deg(f )), où F parcourt l ensemble des sous-o K -module de E. Le bord supérieur de E est le graphe d une fonction P E (définie sur l intervalle [0, rg OK (E)]) qui est concave et affine par morceaux. La fonction P E est appelée polygone de Harder Narasimhan de E. Les abscisses où la fonction P E change sa pente sont précisément rg OK (E 1 ),..., rg OK (E n 1 ), et la valeur de la fonction P E en rg OK (E j ) est deg(e j ) pour tout j {0,..., n}. On désigne par µ min (E) la pente de E n /E n 1, qui est la dernière pente parmi les pentes successives de E (appelée pente minimale de E). Elle est aussi égale à la plus petite pente des quotients projectifs non nuls de E munis des normes quotients. On désigne par µ max (E) la pente de E 1, appelée pente maximale de E. Elle est égale à la plus grande pente des sous-o K -modules de E munis des normes induites. Soit r = rg OK (E). Pour tout i {1,..., r}, on désigne par µ i (E) la pente de la fonction P E sur l intervalle [i 1, i]. On a µ i (E) = µ(e j /E j 1 ) si et seulement si rg OK (E j 1 ) < i rg OK (E j ). En outre, on a µ max (E) = µ 1 (E) and µ min (E) = µ r (E). Soient E un fibré vectoriel hermitien non nul sur Spec O K, et {0} = E 0 E 1... E n son drapeau de Harder Narasimhan. Pour tout t R, soit {0}, si t > µ(e 1 /E 0 ), FHN(E) t := E i OK K, si µ(e i+1 /E i ) < t µ(e i /E i 1 ), i {1,..., n 1}, E OK K, si t µ(e n /E n 1 ). La proposition suivante, qui est parallèle à la proposition 2.1, résulte directement de la définition de F HN.

16 12 Sur la comparaison entre les minima et les pentes Proposition 2.4. Soient E un fibré vectoriel hermitien non nul sur Spec O K et r le rang de E sur O K. a. Si t > µ 1 (E), alors FHN t (E) = {0}. b. Si t µ min (E), alors F t HN (E) = E K. c. Pour tout i {1,..., r 1}, les sous-espaces vectoriels de E K dans la famille (Fm(E)) t t ] µi+1 (E), µ i (E)] sont égaux. d. Pour tout i {1,..., r}, on a µ i (E) = sup{t R rg K (FHN t (E)) i}. Le résultat suivant, qui est parallèle à la proposition 2.3, a été démontré dans [10]. On renvoie les lecteurs au corollaire du loc. cit. pour une démonstration. Proposition 2.5. Soit E un fibré vectoriel hermitien sur Spec O K. On a FHN(E) t = F K, {0} =F E µ min (F ) t où F parcourt l ensemble des sous-o K -modules non nuls de E, et dans la structure de fibré vectoriel hermitien de F on considère des normes induites. 3. Comparaison des filtrations Dans ce paragraphe, on considère le problème suivant. Étant données deux R-filtrations du même espace vectoriel de rang fini sur un corps, comment comparer les points de saut de ces deux filtrations? On démontre un principe formel et on l applique à la comparaison des filtrations par minima et de Harder Narasimhan. Définition 3.1. Soient K un corps et V un espace vectoriel de rang fini sur K. On appelle R-filtration de V toute famille F = (F t (V )) t R de sous-espaces K-vectoriels de V paramétrée par R, qui satisfait aux conditions suivantes : a. (décroissance) pour tout (t 1, t 2 ) R 2, t 1 > t 2, on a F t 1 (V ) F t 2 (V ) ; b. (séparation) pour tout t R assez positif, F t (V ) = {0} ; c. (exhaustivité) pour tout t R assez négatif, F t (V ) = V ; d. (continuité à gauche) la fonction t rg K (F t (V )) est continue à gauche (et donc est localement constante à gauche). Pour tout i {1,..., rg K (V )}, on définit Pour tout t R, on a Z F (i) := sup{t R rg K (F t (V )) i}. (3.1) Z F (i) t rg K (F t (V )) i.

17 H. Chen 13 Exemple 3.2. Soit E un fibré vectoriel hermitien non nul sur Spec O K. Les familles (F t m(e)) t R et (F t HN (E)) t R sont des R-filtrations de E K = E OK K, comme le montrent les propositions 2.1 et 2.4. Proposition 3.3. Soient K un corps et V un espace vectoriel de rang fini sur K. Soient F et G deux R-filtrations de V. On suppose que a est un nombre réel tel que, pour tout t R, on ait F t (V ) G t a (V ). Alors on a Z F Z G + a. Démonstration. Soit G(a) la R-filtration (G t a (V )) t R. Par définition on a Z G(a) = Z G +a. Il suffit de démontrer la proposition dans le cas particulier où a = 0. Par la relation (3.1), pour tout i {1,..., rg K (V )}, si Z F (i) t, alors rg K (F t (V )) i, qui implique que rg K (G t (V )) i et donc Z G (i) t. Ainsi on obtient Z F (i) Z G (i). Proposition 3.4. Soit K un corps de nombres. Pour tout fibré vectoriel hermitien non nul E sur Spec O K, on a µ min (E) ν min (E). Démonstration. Soit {s i } r i=1 une famille d éléments de E qui forme une base de l espace vectoriel E K et telle que i {1,..., r}, max s i σ e t, σ:k C où t est un nombre réel. Soit G un O K -module quotient de E qui est projectif et non nul. Pour tout i {1,..., r}, soit α i l image canonique de s i dans G. Sans perte de généralité, on suppose que {α 1,..., α n } forme une base de G K, où n {1,..., r}. Par définition on a ( 1 deg(g) = ln #(det(g)/o K (α 1 α n )) ) ln α 1 α n σ [K : Q] 1 [K : Q] 1 [K : Q] σ:k C n i=1 σ:k C ln α 1 α n σ 1 [K : Q] ln s i σ tn, σ:k C n i=1 σ:k C ln α i σ où la deuxième inégalité provient de l inégalité d Hadamard. On obtient donc µ(g) t. Comme G est arbitraire, on a µ min (E) ν min (E). Définition 3.5. Soit r un entier, r 1. Pour tout corps de nombres K, on désigne par δ(r, K) la borne supérieure des µ min (E) ν min (E), où E parcourt l ensemble des fibrés vectoriels hermitiens sur Spec O K qui sont de rang r. Remarque 3.6. Soient K un corps de nombres, et E et F deux fibrés vectoriels hermitiens sur Spec O K. On a µ min (E F ) = min( µ min (E), µ min (F )) et ν min (E F ) min(ν min (E), ν min (F )). Par conséquent, la fonction (r N 1 ) δ(r, K) est croissante. Théorème 3.7. Soit K un corps de nombres. Pour tout fibré vectoriel hermitien non nul E et tout i {1,..., rg OK (E)}, on a ν i (E) µ i (E) ν i (E) + δ(rg OK (E), K).

18 14 Sur la comparaison entre les minima et les pentes Démonstration. D après les propositions 3.4, 2.5 et 2.3, pour tout t R on a F t m(e) F t HN (E). En outre, pour tout sous-o K-module non nul F de E, on a (voir la remarque 3.6 pour la deuxième inégalité) Cela montre que µ min (F ) ν min (F ) + δ(rg OK (F ), K) ν min (F ) + δ(rg OK (E), K). t R, FHN(E) t F t δ(rg O (E),K) K m (E). Donc les inégalités annoncées proviennent de la proposition Comparaison entre la dernière pente et le dernier minimum Dans le paragraphe précédent, on ramène la comparaison entre les minima successifs et les pentes successives à celle entre le dernier minimum et la pente minimale. On présente ici une majoration de la fonction δ(, K) introduite dans la définition 3.5. Grâce au théorème 3.7, cela donne une comparaison explicite et uniforme entre les minima successifs et les pentes successives Lien avec le problème de transfert. On commence par relier la comparaison entre les minima et les pentes au problème de transfert en géométrie des nombres. Proposition 4.1. Soient K un corps de nombres et E un fibré vectoriel hermitien sur Spec O K. On a µ min (E) ν min (E) (ν 1 (E ) + ν min (E)) où E désigne le fibré vectoriel hermitien composé du module dual de E muni des normes duales. Démonstration. Rappelons que le degré d Arakelov du dual d un fibré vectoriel hermitien est l opposé du degré d Arakelov du fibré vectoriel hermitien. On en déduit µ max (E ) + µ min (E) = 0 (cf. [4, (4.2)]). En outre, par le théorème 3.7 on obtient ν 1 (E ) µ 1 (E ) = µ max (E ), d où µ min (E) ν min (E) = µ max (E ) ν min (E) ν 1 (E ) ν min (E). Définition 4.2. Soient n un entier, n 1, et K un corps de nombres. On désigne par d(n, K) la borne supérieure des ν 1 (E ) ν min (E), où E parcourt l ensemble des fibrés vectoriels de rang n sur Spec O K, et E désigne le O K -module dual de E muni des normes duales. La proposition 4.1 conduit naturellement à la comparaison suivante. Corollaire 4.3. Pour tout entier n 1 et tout corps de nombres K, on a δ(n, K) d(n, K). L estimation de l invariant d(n, K) est connue comme l un des problèmes de transfert dans la littérature. Le cas des réseaux euclidiens (c est-à-dire K = Q) a été étudié par Banaszczyk (cf. [1], Theorem 2.1) que l on rappelle comme suit. Théorème 4.4 (Banaszczyk). Pour tout n N 1, on a (4.1) d(n, Q) ln(n).

19 H. Chen 15 Remarque 4.5. On compare la borne supérieure (4.1) de d(n, Q) à la constante 1 nc(n, Q) introduite dans la majoration (1.1) de Borek. Rappelons que dans le cas où K = Q on a 1 n C(n, Q) = ln(2) 1 n ln(v n). Par la formule de Stirling, on obtient (cf. [5, 3.2]) 1 n C(n, Q) = 1 2 ln(n) 1 ( eπ ) 2 ln n ln(πn) + 1 6n 2 θ(n/2), où θ est une fonction dont les valeurs sont comprises entre 0 et 1. Asymptotiquement quand n +, 1 nc(n, Q) et ln(n) ont le même ordre de grandeur. Cependant la combinaison du théorème 3.7, du corollaire 4.3 et de la borne supérieure (4.1) montre que, pour tout fibré vectoriel hermitien non nul E sur Spec Z et tout i {1,..., rg Z (E)}, on a (4.2) µ i (E) ν i (E) ln(rg Z (E)), où le facteur i n apparaît pas dans le majorant. Cette inégalité est meilleure que l estimation de Borek dès que i 8. Dans la suite, on propose une variante de la proposition 4.1 qui permet de majorer δ(n, K) par 1 d(n[k : Q], Q) + [K : Q] ln K. On fixe un corps de nombres K et on désigne par π : Spec O K Spec Z le morphisme canonique. Si E est un fibré vectoriel hermitien, on désigne par π (E) le groupe abélien sous-jacent à E. Il s avère que π (E) Z C = E OK,σ C. σ:k C On munit cet espace vectoriel de la norme hermitienne telle que, pour tout s = (s σ ) σ:k C π (E) Z C, on ait s 2 = s σ 2 σ. σ:k C Soit ω OK /Z = Hom Z (O K, Z) le module canonique associé au corps de nombres K, qui est un O K -module projectif de rang 1. Il s avère que l application de trace Tr K/Q est un élément non nul de ω OK /Z. On munit ω OK /Z de la structure de fibré vectoriel hermitien sur Spec O K de sorte que Tr K/Q σ = 1 pour tout σ : K Q. On a (4.3) deg(ω OK /Z) = ln K [K : Q]. De plus, pour tout fibré vectoriel hermitien E sur Spec O K, on a un isomorphisme (4.4) π (E ω OK /Z) = π (E) de fibrés vectoriels hermitiens sur Spec O K (c est-à-dire un isomorphisme de O K -modules qui induit une isométrie pour tout σ : K C). On renvoie les lecteurs à [5, Proposition 3.2.2] pour une démonstration. Proposition 4.6. Soit E un fibré vectoriel hermitien non nul sur Spec O K. On a ν 1 (E) ν 1 (π (E)), ν min (E) ν min (π (E)).

20 16 Sur la comparaison entre les minima et les pentes Démonstration. Soit s un élément de E. Par définition, on a s 2 = s 2 σ max σ:k C s 2 σ. σ:k C Soit λ > 0. Si s est un élément non nul de E tel que s λ, alors pour tout σ : K C on a s σ λ. On obtient donc ν 1 (E) ν 1 (π (E)). De même, si {s 1,..., s n } est une base de π (E) sur Z telle que s i λ pour tout i {1,..., n}, alors on a s i σ λ pour tout i {1,..., n} et tout σ : K C. On peut donc soustraire une sous-famille (s i ) i I de {s 1,..., s n } qui forme une base de E K sur K telle que max σ:k C s i σ λ. On en déduit donc ν min (E) ν min (π (E)). Proposition 4.7. Pour tout entier n 1, on a (4.5) δ(n, K) d([k : Q]n, Q) + ln K [K : Q]. Démonstration. Soit E un fibré vectoriel hermitien de rang n sur Spec O K. D après la proposition 4.6, on a ν min (E) ν min (π (E)) ν 1 (π (E) ) + d(n[k : Q], Q), où la deuxième inégalité provient de la définition de la fonction d(, Q). Par (4.4) on obtient (4.6) ν 1 (π (E) ) = ν 1 (π (E ω OK /Z)) µ max (E ω OK /Z), où l inégalité provient de la proposition 4.6. D après [3, (A.2)], on a µ max (E ω OK /Z) = µ max (E ) + deg(ω OK /Z) = µ min (E) + ln K [K : Q], où la deuxième égalité provient de (4.3) et [4, (4.2)]. On en déduit La proposition est donc démontrée. µ min (E) ν min (E) d([k : Q]n, Q) + ln K [K : Q]. Corollaire 4.8. Pour tout entier n tel que n 1 et tout corps de nombres K, on a δ(n, K) ln(n[k : Q]) + ln K [K : Q]. Démonstration. C est une conséquence directe de (4.1) et (4.5) Minima de Roy Thunder et démonstration du théorème 1.1. Il existe d autres notions de minima successifs dans la littérature. Soient K un corps de nombres et E un fibré vectoriel hermitien sur Spec O K. Pour tout s E K, on définit deg(s) := 1 [K : Q] p Spm(O K ) ln s p + σ:k C ln s σ, où pour tout idéal maximal p de O K, la norme p sur E OK K p (K p étant le complété de K par rapport à la place p) est induite par la structure de O K -module de E. Pour tout i {1,..., rg OK (E)}, le i ème minimum de Roy Thunder de E (cf. [20]) est défini comme (4.7) νi RT (E) := sup{t R rg K (Vect K {s E K deg(s) t}) i}.

21 H. Chen 17 On peut aussi définir de façon similaire le degré d Arakelov pour les vecteurs non nuls dans E OK K a, où K a désigne la clôture algébrique de K. Pour tout i {1,..., rg OK (E)}, le i ème minimum absolu de E est défini comme νi abs (E) = sup{t R rg K a(vect K a{s E K a deg(s) t}) i}. Il n est pas difficile de voir que (cf. [15, 4] par exemple) ν i (E) ν RT i (E) ν abs (E) µ i (E) pour tout i. La méthode de comparaison des R-filtrations peut être appliquée à la comparaison de ces invariants. Il est aussi possible d étendre les résultats de l article dans le cadre adélique, suivant le chemin indiqué dans [12, 5.4]. Il faut cependant tenir en compte du défaut de pureté et du défaut de hermitianité dans la majoration (voir [13, 2] pour plus de détails). Comme mentionné dans l introduction, le rapporteur suggère une amélioration de la majoration (4.5) que je résume comme suit. Soit E un fibré vectoriel hermitien de rang n sur Spec O K comme dans la démonstration de la proposition 4.7. Le raisonnement est similaire sauf que l on identifie le premier minimum ν 1 (π (E) ) du réseau euclidien π (E) au premier minimum ν1 RT (π (E)) au sens de Roy Thunder (voir (4.7) pour la définition) en utilisant [15, proposition 4.8] (avec les notations du loc. cit. on a c 1 (Q) = 1). L intérêt de ce passage au premier minimum de Roy Thunder est de pouvoir appliquer le [15, corollaire 4.28] pour obtenir ν 1 (π (E) ) = ν1 RT (π (E ω OK /Z)) ν1 RT (E ω OK /Z) 1 ln[k : Q]. 2 Cela permet de gagner un terme 1 2 ln[k : Q] par rapport à (4.6) et donc conduire à la majoration améliorée δ(n, K) d([k : Q]n, Q) + ln K [K : Q] 1 ln[k : Q]. 2 On en déduit l inégalité suivante, via le théorème de transfert de Banasczcyk (4.8) δ(n, K) ln(n) + ln K [K : Q] + 1 ln[k : Q]. 2 Les minima de Roy Thunder sont aussi naturellement liés aux R-filtrations. En effet, si E est un fibré vectoriel hermitien non-nul sur Spec O K, pour tout t R, on désigne par FRT t (E) le sous-espace K-vectoriel de E K engendré par l ensemble {s E deg(s) t}. Ainsi F RT définit une R-filtration de E K et on a νi RT (E) = sup{u R rg K (FRT(E) u i)}. L avatar de la proposition 2.3 et du théorème 3.7 est encore valable pour les minima de Roy Thunder. En particulier, si, pour tout entier n 1 on désigne par δ RT (n, K) la borne supérieure des µ min (F ) ν n (F ), où F parcourt l ensemble des fibrés vectoriels hermitiens de rang n sur Spec O K, alors pour tout fibré vectoriel hermitien non nul E sur Spec O K et tout i {1,..., rg OK (E)} on a ν RT i (E) µ i (E) νi RT (E) + δ RT (rg OK (E), K). i

22 18 Sur la comparaison entre les minima et les pentes Soient n un entier, n 1, et F un fibré vectoriel hermitien de rang n sur Spec O K. On a µ min (F ) ν n (F ) = µ max (F ) ν n (F ) ν 1 (F ) ν n (F ). D après le théorème 36 de [14], on a Cela conduit à la majoration ν 1 (F ) ν n (F ) ln(n) + ln K [K : Q]. (4.9) δ RT (n, K) ln(n) + ln K [K : Q]. D après les propositions 4.8 et 5.1 de [15], pour tout fibré vectoriel hermitien F de rang n sur Spec O K, on a νn RT (F ) ν n (F ) + ln K 2[K : Q]. On en déduit (4.10) δ(n, K) δ RT (n, K) + ln K 2[K : Q] ln(n) + 3 ln K 2[K : Q], où la seconde inégalité provient de (4.9). La combinaison de (4.8) et (4.10) conduit au théorème 1.1, en s appuyant sur le théorème Applications et commentaires 5.1. Comparaison entre des mesures limites. Soit π : X Spec O K un morphisme projectif et plat d un schéma intègre X vers Spec O K. On appelle fibré en droites hermitien sur X tout couple L = (L, ϕ), où L est un O X -module inversible et ϕ = (ϕ σ ) σ:k C est une famille de métriques continues sur L qui est stable par la conjugaison complexe. On précise que ϕ σ = ( σ (x)) x Xσ(C) est une métrique continue sur L σ, la tirée en arrière de L sur X σ = X Spec OK,σ Spec C. L invariance par la conjugaison complexe signifie que, pour tout σ : K C et tout x X σ (C), l isomorphisme canonique entre x (L σ ) et x (L σ ) induit une isométrie entre σ (x) et σ (x). Théorème 5.1. Soit L un fibré en droites hermitien sur X. On suppose que la fibre générique de L est gros. En d autres termes, le volume de L K, défini comme vol(l K ) := lim sup n + dim K H 0 (X K, L n K ) n dim(x K) / dim(x K )!, est strictement positif. Pour tout n N, soit E n = (H 0 (X, L n ), ( n,σ ) σ:k C ) un fibré vectoriel hermitien dont le O K -module projectif sous-jacent E n est égal à H 0 (X, L n ). On suppose en outre que, pour tout σ : K C, 1 (5.1) lim n + n sup ln s n,σ ln sup s σ (x) = 0. s E n OK,σC x X σ(c) Pour tout entier n 1 tel que r n := rg OK (E n ) > 0, soient η n := 1 r n r n i=1 δ µi (E n)/n et η n := 1 r n r n i=1 δ νi (E n)/n,

23 H. Chen 19 où pour tout nombre réel t, δ t désigne la mesure de Dirac en t. Alors les suites de mesures (η n ) et ( η n ) convergent vers la même mesure de probabilité borélienne qui ne dépend que de L. Démonstration. Rappelons que la convergence faible d une suite (λ n ) n N de mesures boréliennes vers une mesure limite λ signifie que, pour toute fonction continue et bornée h, la suite d intégrales ( R h(x) λ n(dx)) n N converge vers R h(x) λ(dx). Pour tout n N, soit π (L n ) le O K -module projectif E n = H 0 (X, L n ) muni de la famille de normes ( n,σ,sup ), où s E n OK,σ C, s n,σ,sup := sup s σ (x). x X σ(c) C est un fibré vectoriel normé sur Spec O K. Si r n est strictement positif, soit λ n = 1 r n r n i=1 δ νi (π (L n ))/n. D après [9, théorème 2.11], les supports des mesures λ n sont uniformément borné supérieurement et la suite de mesures (λ n ) converge vaguement vers une mesure de probabilité borélienne λ. En d autres termes, pour toute fonction continue à support compact h sur R, on a lim h(x) λ n (dx) = h(x) λ(dx). n + R R En outre, d après [18, Theorem D], les supports des mesures λ n sont uniformément bornés inférieurement. Donc la suite de mesure (λ n ) converge en fait faiblement vers λ. Il reste à vérifier que les deux suites (η n ) et ( η n ) convergent faiblement vers λ. Le point clé repose sur les estimations suivantes : (5.2) lim max 1 n + i {1,...,r n} n ν i(e n ) ν i (π (L n )) = 0, (5.3) lim max 1 n + i {1,...,r n} n µ i(e n ) ν i (π (L n )) = 0, où (5.2) provient de l hypothèse (5.1), et (5.3) résulte de (5.2), du théorème 1.1 et du fait que r n = O(n dim(x) ) (on souligne que l inégalité (1.1) de Borek ne suffit pas pour déduire (5.3) de (5.2)). L estimation (5.1) montre que les supports des mesures η n sont uniformément bornés (puisque ceux des mesures λ n les sont). En particulier, pour montrer que la suite (η n ) converge faiblement vers λ, il suffit de vérifier que, pour toute fonction continue à support compact h : R R, on a (5.4) lim n + h(x) η n (dx) h(x) λ n (dx) = 0. R R Par définition on a h(x) η n (dx) R R h(x) λ n (dx) = 1 r n r n i=1 ( ) ( ) 1 1 h n ν i(e n ) h n ν i(π (L n )). Donc l estimation résulte de (5.2) et de la continuité uniforme de h. La convergence faible de ( η n ) vers λ découle de (5.3) par le même argument.

24 20 Sur la comparaison entre les minima et les pentes Remarque 5.2. On désigne par η L la mesure limite dans le théorème précédent. D après [10, proposition 1.2.9] la suite des polygones normalisés (t [0, 1] 1 nr n P En (r n t)) converge uniformément vers une fonction concave P L sur [0, 1] (le polygone associée η L avec la terminologie de [10, définition 1.2.8]). La mesure η L est un invariant birationnel, c est-à-dire que, pour tout morphisme f : X X projectif et birationnel, on a η f L = η L (cf. [11, proposition 5.2]). Cela implique que la limite P L des polygones normalisés est aussi un invariant birationnel du fibré en droites hermitien L Quelques commentaires. Soient K un corps de nombres et E le fibré vectoriel hermitien sur Spec O K. Soit r = rg OK (E). En général, un problème de transfert cherche à majorer (ν i (E) + ν r+1 i (E )) pour tout i {1,..., r}. Si 0 = E 0 E 1... E n 1 E n = E est le drapeau de Harder Narasimhan de E, alors 0 = (E/E n ) (E/E n 1 )... (E/E 1 ) (E/E 0 ) = E est le drapeau de Harder Narasimhan de E. En particulier, on a pour tout i {1,..., r}. On en déduit µ i (E) + µ r+1 i (E ) = 0 i {1,..., r}, (ν i (E) + ν r+1 i (E )) 2δ(r, K). En particulier, on a d(r, K) 2δ(r, K). De ce point de vue-là, il est plus raisonnable de conjecturer que δ(r, K) est majorée par une fonction d ordre 1 2 ln(r)+o K(1). Plus précisément, le rapporteur relève la question suivante : est-ce que l inégalité δ(r, K) 1 2 ln(r) + ln K 2[K : Q] est vraie? Une réponse positive à cette question conduira à la majoration uniforme i {1,..., rg OK (E)}, µ i (E) ν i (E) 1 2 ln(rg O K (E)) + ln K 2[K : Q] pour tout fibré vectoriel hermitien non nul E sur Spec O K. La somme de ces inégalités par rapport à i donne une amélioration importante du deuxième théorème de Minkowski dans le cas de fibré vectoriel hermitien (correspondant aux cas d ellipsoïdes dans le langage classique) qui est valable pour tout corps de nombres (cf. [15, Proposition 5.1]). En outre, la comparaison à la proposition 29 de [14] conduit naturellement à la question suivante. Soit r un entier, r 1. Pour tout i {1,..., r}, on désigne par δ i (r, K) la borne supérieure de µ i (E) ν i (E), où E parcourt l ensemble des fibrés vectoriels hermitiens de rang r sur Spec O K. Les inégalités sont-elles vraies? δ 1 (r, K)... δ r (r, K)

25 H. Chen Interprétation géométrique. Si on se contente de majorer la différence entre les pentes et les minima absolus, on peut transformer le problème dans un cadre géométrique. Soit E un fibré vectoriel hermitien sur Spec O K. Tout élément non nul s de E K a détermine une droite dans l espace vectoriel E K a, qui correspond à un point algébrique du schéma P(EK ) que l on note P s. En outre, on a deg(s) = h O(1) (P s ), où la hauteur (absolue) h O(1) est calculée par rapport au modèle entier tautologique (P(E ), O P(E )(1)) et les métriques de Fubini Study sur O P(E )(1). Rappelons que le minimum essentiel de la fonction de hauteur h O(1) est défini comme µ ess (E ) = sup inf h Z P(EK ) P (P(EK )\Z)(Ka ) O(1) où Z parcourt l ensemble des parties fermées Zariski strictes de P(EK ). Il s avère que ν abs min(e) + µ ess (E ) 0. On conjecture que le minimum essentiel µ ess (E ) est majoré par la pente maximale µ max (E ) plus 1 2 ln(rg O K (E)). Cette conjecture est vraie lorsque E est une somme directe orthogonale de fibrés en droites hermitiens (cf. [7]). Cela conduit à la majoration (P ), µ i (E) νi abs (E) 1 2 ln(rg O K (E)) pour tout i {1,..., r}, en utilisant la méthode proposée dans cet article (il suffit de remplacer la R-filtration par minima par la R-filtration par hauteur, cf. [6, 3.2]). Plus généralement, on peut considérer le problème suivant. Soient f : X Spec O K un morphisme projectif et plat d un schéma intègre X vers Spec O K. Soit L un fibré inversible hermitien sur X tel que L K soit ample, que L soit relativement nef et que les métriques de L soient semi-positives. La donnée de L permet de définir une fonction de hauteur (logarithmique et absolue) h L ( ) sur l ensemble des points algébrique de X K. Soit µ ess (L ) le minimum essentiel associé à cette fonction de hauteur, définie comme µ ess (L ) := sup inf h Z X K P (X K \Z)(K a ) L (P ) On conjecture que le minimum essentiel µ ess (L ) peut être majoré par la limite µ asy µ max (f (L n )) max(l ) := lim n + n plus une fonction du degré de L K. On revoie les lecteurs au 4.2 de [10] pour une étude de l invariant µ asy max( ).

26 22 Sur la comparaison entre les minima et les pentes Références [1] W. Banaszczyk, «New bounds in some transference theorems in the geometry of numbers», Math. Ann. 296 (1993), n o 4, p [2] T. Borek, «Successive minima and slopes of Hermitian vector bundles over number fields», J. Number Theory 113 (2005), n o 2, p [3] J.-B. Bost, «Périodes et isogénies des variétés abéliennes sur les corps de nombres (d après D. Masser et G. Wüstholz)», in Séminaire Bourbaki, Vol. 1994/95, Astérisque, vol. 237, Société Mathématique de France, 1996, p [4], «Algebraic leaves of algebraic foliations over number fields», Publ. Math., Inst. Hautes Étud. Sci. (2001), n o 93, p [5] J.-B. Bost & K. Künnemann, «Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers», Adv. Math. 223 (2010), n o 3, p [6] S. Boucksom & H. Chen, «Okounkov bodies of filtered linear series», Compos. Math. 147 (2011), n o 4, p [7] J. I. Burgos Gil, P. Philippon & M. Sombra, «Successive minima of toric height functions», Ann. Inst. Fourier 65 (2015), n o 5, p [8] H. Chen, «Géométrie d Arakelov : théorèmes de limite et comptage des points rationnels», Mémoire d habilitation à diriger des recherches, Université Paris Diderot, [9], «Arithmetic Fujita approximation», Ann. Sci. Éc. Norm. Supér. 43 (2010), n o 4, p [10], «Convergence des polygones de Harder-Narasimhan», Mém. Soc. Math. Fr., Nouv. Sér. (2010), n o 120, p [11], «Differentiability of the arithmetic volume function», J. Lond. Math. Soc. 84 (2011), n o 2, p [12] É. Gaudron, «Pentes des fibrés vectoriels adéliques sur un corps global», Rend. Semin. Mat. Univ. Padova 119 (2008), p [13], «Géométrie des nombres adélique et lemmes de Siegel généralisés», Manuscr. Math. 130 (2009), n o 2, p [14], «Minima and slopes of rigid adelic spaces», Notes de cours de l école d été «Géométrie d Arakelov et applications diophantiennes», Grenoble, [15] É. Gaudron & G. Rémond, «Corps de Siegel», J. Reine Angew. Math. 726 (2017), p [16] D. R. Grayson, «Reduction theory using semistability», Comment. Math. Helv. 59 (1984), n o 4, p [17] G. Harder & M. S. Narasimhan, «On the cohomology groups of moduli spaces of vector bundles on curves», Math. Ann. 212 (1974/75), p [18] H. Ikoma, «Boundedness of the successive minima on arithmetic varieties», J. Algebr. Geom. 22 (2013), n o 2, p [19] R. Lazarsfeld & M. Mustaţă, «Convex bodies associated to linear series», Ann. Sci. Éc. Norm. Supér. 42 (2009), n o 5, p [20] D. Roy & J. L. Thunder, «An absolute Siegel s lemma», J. Reine Angew. Math. 476 (1996), p. 1-26, addendum and erratum in ibid. 508 (1999), p

27 H. Chen 23 [21] C. Soulé, «Hermitian vector bundles on arithmetic varieties», in Algebraic geometry (Santa Cruz 1995), Proceedings of Symposia in Pure Mathematics, vol. 62, American Mathematical Society, 1997, p [22] U. Stuhler, «Eine Bemerkung zur Reduktionstheorie quadratischer Formen», Arch. Math. 27 (1976), n o 6, p Huayi Chen, Université Paris Diderot, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Bâtiment Sophie Germain, Boîte Courrier 7012, Paris Cedex 13, France huayi.chen@imj-prg.fr Url : webusers.imj-prg.fr/~huayi.chen

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29 , HURWITZ BELYI MAPS by David P. Roberts Abstract. The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties covering configuration varieties. Certain one-dimensional slices of these coverings are particularly interesting Belyi maps. We present systematic examples of such Hurwitz Belyi maps. Our examples illustrate a wide variety of theoretical phenomena and computational techniques. Résumé. (Applications d Hurwitz Belyi) L étude des modules de revêtements de la droite projective conduit à la théorie des variétés de Hurwitz comme revêtements des variétés de configurations. Certaines sections de dimension un des ces revêtements sont des applications de Belyi particulièrement intéressantes. Nous présentons des exemples de telles applications «d Hurwitz Belyi» qui illustrent une large variété de phénomènes théoriques et techniques de calculs. 1. Introduction The theory of Belyi maps sits at an attractive intersection in mathematics where group theory, algebraic geometry, and number theory all play fundamental roles. In this paper we first introduce a simply-indexed class of particularly interesting Belyi maps which arise in solutions of Hurwitz moduli problems. Our main focus is then the computation of sample Hurwitz Belyi maps and the explicit exhibition of their remarkable properties. We expect that our exploratory work here will support future more theoretical studies. We conclude this paper by speculating that as degrees become large, Hurwitz Belyi maps become extreme outliers among all Belyi maps. The rest of the introduction amplifies on this first paragraph Belyi maps. In the classical theory of smooth projective complex algebraic curves, ramified covering maps from a given curve Y to the projective line P 1 play a prominent role. If Y is connected with genus g, then any degree n map F : Y P 1 has 2n + 2g 2 critical points in Y, counting multiplicities. For generic F, these critical points y i are all distinct and 2010 Mathematics Subject Classification. 11G32, 14H57. Key words and phrases. Hurwitz variety, Belyi map, ramification.

30 26 Hurwitz Belyi maps moreover the critical values F (y i ) are also also distinct. A Belyi map by definition is a map Y P 1 having all critical values in {0, 1, }. One should think of Belyi maps as the maps which are as far from generic as possible, with their critical values being moreover normalized to a standard position. The recent paper [24] provides a computationally-focused survey of Belyi maps, with many references An example. The main focus of this paper is the explicit construction of Belyi maps with certain extreme properties. A Belyi map from [15] arises from outside the main context of this paper but still exhibits these extremes: (1.1) π : P 1 P 1, x (x + 2)9 x 18 ( x 2 2 ) 18 (x 2) (x + 1) 16 (x 3 3x + 1) 16. We use this map as an introductory example, because it represents a class of very extreme Belyi maps which provide some context for this paper, as discussed further in 1.3 and 11.1 below. The degree of π is 64 and the 126 critical points are easily identified as follows. From the numerator A(x), one has the critical points 2, 0, 2, 2, with total multiplicity = 59 and critical value 0. From the denominator C(x), one has critical points 1, x 2, x 3, x 4 with total multiplicity = 60 and critical value. Since both A(x) and C(x) are monic, one has π( ) = 1. The exact coefficients in (1.1) are chosen so that the degree of A(x) C(x) is only 56. This means that is a critical point of multiplicity = 7. As = 126, there can be no critical values outside {0, 1, } and so π is indeed a Belyi map. In general, a degree m Belyi map π has a monodromy group M π S m, a number field F π C of definition, and a finite set P π of bad primes. We call π full if M π {A m, S m }. Our example π is full because M π = S 64. It is defined over F π = Q because all the coefficients in (1.1) are in Q. It has bad reduction set P π = {2, 3} because numerator and denominator have a common factor in F p [x] exactly for p {2, 3}. In the sequel, we almost always drop the subscript π, as it is clear from context. To orient the reader, we remark that the great bulk of the explicit literature on Belyi maps concerns maps which are not full. Much of this literature, for example [13, Chapter II], focuses on Belyi maps with M a finite simple group different from A m. On the other hand, seeking Belyi maps defined over Q is a common focus in the literature. Similarly, preferring maps with small bad reduction sets P is a common viewpoint An inverse problem. To provide a framework for our computations, we pose the following inverse problem: given a finite set of primes P and a degree m, find all full degree m Belyi maps π defined over Q with bad reduction set within P. The finite set of full Belyi maps in a given degree m is parameterized in an elementary way by group-theoretic data. So, in principle at least, this problem is simply asking to extract those for which F π = Q and P π P. Our inverse problem is in the spirit of the classical inverse Galois problem [13]; however it focuses on constrained ramification, rather than unusual Galois groups. While the Belyi map (1.1) may look rather ordinary, it is already unusual for full Belyi maps to be defined over Q. It seems to be extremely rare that their bad reduction set is so small. In fact, we know of no full Belyi maps defined over Q with m 4 and P π 1. For P π = 2

31 D. Roberts 27 we know of only a very sparse collection of such maps [14], [15], as discussed further in our last section here. The largest degree of these with both primes less than seventeen is m = 64, coming from (1.1) Hurwitz Belyi maps. Suppose now that P contains the set P T of primes dividing the order of a finite nonabelian simple group T. The theoretical setting for this paper is a systematic method of constructing Belyi maps of arbitrarily large degree defined over Q and ramified within P. In brief, the method has two steps and goes as follows. First, from T one can build infinitely many natural covers from a Hurwitz variety to a configuration variety. In our notation, these covers are written π h : Hur h Conf ν, and the common dimension of both cover and base can be arbitrarily large. Second, there can be many non-trivial maps u from the thrice-punctured projective line P 1 {0, 1, } into Conf ν. Let X 0 be the preimage of u(p 1 {0, 1, }) in Hur h, and let X be its smooth completion. Then the corresponding Hurwitz Belyi map π h,u is the induced map from X to P 1. As we explain in our last section, we expect infinitely many of these π h,u to be full, and thus satisfy the remaining condition of our inverse problem Contents of this paper. Our viewpoint is that Hurwitz Belyi maps form a remarkable class of mathematical objects, and are worth studying in all their aspects. This paper focuses on presenting explicit defining equations for systematic collections of Hurwitz Belyi maps, and exhibits a number of theoretical structures in the process. The defining equations are obtained by two complementary methods. What we call the standard method centers on algebraic computations directly with the r-point Hurwitz source. The braid-triple method is an alternative method introduced in this paper. It uses the r-point Hurwitz source only to give necessary braid group information; its remaining computations are then the same ones used to compute general Belyi maps. We focus primarily on the case r = 4 which is the easiest case for computations for a given T. This case was studied in some generality by Lando and Zvonkin in [8, 5.5] under the term megamap. In the last two sections, we shift the focus to r 5, which is necessary to obtain the very large degrees m we are most interested in. The standard method is insensitive to genera of covering curves X, and so we could easily present examples of quite high genus. However, to give a uniform tidiness to our final equations, we present defining equations only in the case of genus zero. Thus the reader will find many explicit rational functions in Q(x) with properties similar to those of our initial example (1.1). All these rational functions and related information are available in the Mathematica file HBM.m on the author s homepage. Section 2 reviews the theory of Belyi maps. Section 3 reviews the theory of Hurwitz maps and explains how carefully chosen one-dimensional slices are Hurwitz Belyi maps. Of the many Belyi maps appearing in Section 2, two are unexpectedly defined over Q. These maps each appear again in Section 4, with now their rationality obvious from the beginning via the Hurwitz theory. Section 5 introduces the alternative braid-triple method for finding defining equations. We give general formulas for the preliminary braid computations in the setting r = 4. Passing from braid information to defining equations can then be much more computationally demanding than in our initial examples, and we find equations mainly by p-adic techniques. Section 6 then presents three examples for which both methods work, with these examples

32 28 Hurwitz Belyi maps having the added interest that lifting invariants force X to be disconnected. In each case, X in fact has two components, each of which is full over the base projective line. Sections 7, 8, and 9 consider a systematic collection of Hurwitz Belyi maps, with all final equations computed by the braid-triple method. They focus on the cases where P T 3. By the classification of finite simple groups, the possible P T have the form {2, 3, p} with p {5, 7, 13, 17}. Section 7 sets up our framework and presents one example each for p = 13 and p = 17. Sections 8 and 9 then give many examples for p = 5 and p = 7 respectively. Section 10 takes first computational steps into the setting r 5. Working just with T = A 5 and r = 5, we summarize braid computations which easily prove the existence of full Hurwitz Belyi maps with bad reduction set {2, 3, 5} and degrees into the thousands. We use the standard method to find equations of two such covers related to T = A 6, one in degree 96 and the other in degree 192. Section 11 concludes by tying the considerations of this paper very tightly to those of [22] and [20]. It conjectures a direct analog for Belyi maps of the main conjecture there for number fields. The Belyi map conjecture responds to the above inverse problem in the case that P contains the set of primes dividing the order of some finite nonabelian simple group. In particular, it says that there then should be full Belyi maps defined over Q and ramified within P of arbitrarily large degree Notation. Despite the arithmetic nature of our subject, we work almost exclusively over C. Following [22] and [20], we use a sans serif font for complex spaces as in Y, P 1, Hur h, Conf ν, or X above. The phenomenon that allows us to work mainly over C is that to a great extent geometry determines arithmetic. Thus an effort to find a function π(x) C(x) giving a full Belyi map π : P 1 P 1 involves choices of normalization. Typically, one can make these choices in a geometrically natural way, and then the coefficients of π(x) automatically span the field of definition. When this field is Q, and the normalization is sufficiently canonical, the primes of bad reduction can be similarly read off. Often there will be several projective lines under consideration at once. When clarifying, we distinguish them by subscripting by the coordinate we are using. We commonly present a Belyi map π : P 1 x P 1 v not as a rational function v = A(x)/C(x) but rather via the corresponding polynomial equation A(x) vc(x) = 0. This trivial change in perspective has several advantages, one being that it lets one see the three-point property and the primes of bad reduction simultaneously via discriminants. For example, the discriminant of A(x) vc(x) in our first example (1.1) is v 59 (v 1) Acknowledgements. This work was partially supported by the Simons Foundation through grant #209472, and, its last stages, by the National Science Foundation through grant DMS I thank Stefan Krämer, Kay Magaard, Hartmut Monien, Sam Schiavone, Akshay Venkatesh, and John Voight for helpful conversations. Finally I thank the anonymous referee for detailed comments which helped streamline this paper. 2. Two Belyi maps unexpectedly defined over Q This section presents twenty-eight Belyi maps as explicit rational functions in C(y), two of which are unexpectedly in Q(y). Via these examples, it provides a quick summary, adapted to

33 D. Roberts 29 this paper s needs, of the general theory of Belyi maps. We will revisit the two rational maps from a different point of view in Section 4. Our three-point computations here are providing models for later r-point computations. Accordingly, we use the letter y as a primary variable Partition triples. Let n be a positive integer. Let Λ = (λ 0, λ 1, λ ) be a triple of partitions of n, with the λ τ having all together n + 2 2g parts, with g Z 0. The two examples pursued in this section are (2.1) Λ = (322, 421, 511), Λ = (642, , 5322). So the degrees of the examples are n = 7 and n = 12, and both have g = 0. Consider Belyi maps F : Y P 1 with ramification numbers of the points in F 1 (τ) forming the partition λ τ, for each τ {0, 1, }. Up to isomorphism, there are only finitely many such maps. For some of these maps, Y may be disconnected, and we are not interested here in these degenerate cases. Accordingly, let X be the set of isomorphism classes of such Belyi maps with Y connected. One wants to explicitly identify X, and simultaneously get an algebraic expression for each corresponding Belyi map F x : Y x P 1. The Riemann-Hurwitz formula says that all these Y x have genus g. Note that in the previous paragraph we have finitely many Belyi maps F x : Y P 1 indexed by the finite set X. In the bulk of this paper, we will have infinitely many maps F x : Y x P 1, which are now ramified above more than three points. These less extreme covers will be continuously indexed by the covering curve X in a Belyi map π : X P 1. Our notations are chosen so that the computations of this section are in the same notation as the computations of the later sections, even though the position of Belyi maps in these computations is different. Computations in our current three-point setting can be put into a standard form when g = 0 and the partitions λ 0, λ 1, and λ have in total at least three singletons. Then one can pick an ordered triple of singletons and coordinatize Y by choosing y to take the values 0, 1, and in order at the three corresponding points. In our two examples, we do this via (2.2) Λ = (3 0 22, x 1, 5 11), Λ = ( x, , 5 322). Also we have chosen a fourth point in each case and subscripted it by x. This choice gives a canonical map from X into C, as will be illustrated in our two examples. When the map corresponding to such a marked triple Λ is injective, as it almost always seems to be, we say that Λ is a cleanly marked genus zero triple. When g = 0 and there is at least one singleton, computations can be done very similarly. All the explicit examples of this paper are in this setting. When g = 0 and there are no singletons, one often has to take extra steps, but the essence of the method remains very similar. When g > 0, computations are still possible, but they are very much more complicated The triple Λ and its associated 4 = splitting. The subscripted triple Λ in (2.2) requires us to consider rational functions and focus on the equation F (y) = 1 + c + d (1 + a + b) 2 y3 (y 2 + ay + b) 2 y 2 + cy + d (2.3) 5y 4 + 3(a + 2c)y 3 + (4ac + b + 7d)y 2 + (5ad + 2bc)y + 3bd = 5(y 1) 3 (y x).

34 30 Hurwitz Belyi maps The left side is a factor of the numerator of F 0 (y) and thus its roots are critical points. The right side gives the required locations and multiplicities of these critical points. Equating coefficients of y in (2.3) and using also F (x) = 1 gives five equations in five unknowns. There are four solutions, indexed by the roots of (2.4) fλ0 (x) = (x + 2) 16x3 248x2 77x 6. In general from a cleanly marked genus zero triple Λ, one gets a separable moduli polynomial fλ (x). The moduli algebra KΛ = Q[x]/fΛ (x) depends, as indicated by the notation, only on Λ and not on the marking. It is well-defined in the general case when the genus is arbitrary, even though we are not giving a procedure here to find a particular polynomial. While the computation just presented is typical, the final result is not. We give three independent conceptual explanations for the factorization in (2.4), two in 4.1 and one at the end of 6.3. For context, the splitting of the moduli polynomial is one of just four unexplained splittings on the fourteen-page table of moduli algebras in [11]. While here the degree 7 partition triple yields a moduli algebra splitting as 3 + 1, in the other examples the degrees are 8, 9, and 9, and the moduli algebras split as 7 + 1, 8 + 1, and Dessins. A Belyi map F : Y P1t can be visualized by its dessin as follows. Consider the interval [0, 1] in P1t as the bipartite graph. Then Y[0,1] := F 1 ([0, 1]) inherits the structure of a bipartite graph. This bipartite graph, considered always as inside the ambient real surface Y, is the dessin associated to F. A key property is that F is completely determined by the topology of the dessin. Figure 2.1. Dessins Yxi,[0,1] P1y corresponding to the points of XΛ0 = {x1, x2, x3, x4 } with Λ0 = (322, 421, 511) Returning to the example of the previous subsection, the roots indexing the four solutions are x1 = 2, x , x i, x i.

35 D. Roberts 31 The complete first solution is (2.5) F 1 (y) = y3 ( y 2 + 2y 5 ) 2 4(2y 1)(3y 4). The coefficients of the other F i are cubic irrationalities. The four corresponding dessins in Y i = P 1 y are drawn in Figure 2.1. The scales of the four dessins in terms of the common y-coordinate are quite different. Always the black triple point is at 0 and the white quadruple point is at 1. The white double point is then at x i Monodromy. The dessins visually capture the group theory which is central to the theory of Belyi maps but has not been mentioned so far. Given a degree n Belyi map F : Y P 1, consider the set Y of the edges of the dessin. Let g 0 and g 1 be the operators on Y given by rotating minimally counterclockwise about black and white vertices respectively. The choice of [0, 1] as the base graph is asymmetric with respect to the three critical values 0, 1, and. Orbits of g 0 and g 1 correspond to black vertices and white vertices respectively. In our first example, the original partitions λ 0 = 322 and λ 1 = 421 can be recovered from each of the four dessins from the valencies of these vertices. On the other hand, the orbits of g = g1 1 g 1 0 correspond to faces. The valence of a face is by definition half the number of edges encountered as one traverses its boundary. Thus λ = 511 is recovered from each of the four dessins in Figure 2.1, with the outer face always having valence five and the two bounded faces having valence one. Let Y be the set of ordered triples (g 0, g 1, g ) in S n such that g 0, g 1, and g respectively have cycle type λ 0, λ 1, and λ, g 0 g 1 g = 1, g 0, g 1 is a transitive subgroup of S n. Then S n acts on Y by simultaneous conjugation, and the quotient is canonically identified with Y. For each of the thirty-one dessins of this section, the monodromy group g 0, g 1 is all of S n. Indeed the only transitive subgroup of S 7 having the three cycle types of Λ is S 7, and the only transitive subgroup of S 12 having the three cycle types of Λ is S Galois action. Let Gal(Q/Q) be absolute Galois group of Q. This group acts naturally on the set X of Belyi maps belonging to any given Λ. In the favorable cleanly marked situation set up in 2.1, one has X Q and the action on X is the restriction of the standard action on Q. A broad problem is to describe various ways in which Gal(Q/Q) may be forced to have more than one orbit. Suppose x, x X respectively give rise to monodromy groups g 0, g 1 and g 0, g 1. If these monodromy groups are not conjugate in S n then certainly x and x are in different Galois orbits. Malle s paper [11] repeatedly illustrates the next most common source of decompositions, namely symmetries with respect to certain base-change operators P 1 t P 1 t. The two splittings in this section do not come from either of these simple sources.

36 32 Hurwitz Belyi maps 2.6. The triple Λ and its associated 24 = splitting. Here we summarize the situation for Λ. Again the computation is completely typical, but the result is atypical. The clean marking on Λ identifies X Λ with the roots of (5x + 4) ( x x x x x x x x x x x x x x x x x x x x x x x ). Figure 2.2. Dessins in Y x,[0,1] P 1 y corresponding to the twenty-four points x X Λ with Λ = (642, , 5322) The twenty-four associated dessins are drawn in Figure 2.2. The cover F 4/5 : P 1 y P 1 t given by (2.6) t = 5 5 y 6 (y 1) 4 (5y + 4) (2y + 1) 3 (5y 2 6y + 2) 2. This splitting of one cover away from the other twenty-three covers is explained in 4.3. is

37 D. Roberts 33 In choosing conventions for using dessins to represent covers, one often has to choose between competing virtues, such as symmetry versus simplicity. Figure 2.2 represents the standard choice when λ 1 has the form 2 a 1 b : one draws the white vertices just as regular points, because they are not necessary for recovering the cover. With this convention there are just three highlighted points in each of the dessins in Figure 2.2: black dots of valence 6, 4, 2 at y = 0, 1, x. The rational cover, with x = 4/5, appears in the upper left Bounds on bad reduction. Let n be a positive integer and let Λ = (λ 0, λ 1, λ ) be a triple of partitions of n as above. Let P loc be the set of primes dividing a part of one of the λ i. Let P glob be the set of primes less than or equal to n. In our two examples P loc = {2, 3, 5} and P glob is larger, by {7} and {7, 11} respectively. Let K Λ be the moduli algebra associated to Λ. Let D Λ be its discriminant, i.e. the product of the discriminants of the factor fields. In our two examples, D Λ = and D Λ = Let P Λ be the set of primes dividing D Λ. Then one always has P Λ P glob. Of course if K Λ = Q, then one has P Λ =. Our experience is that once [K Λ : Q] has moderately large degree, P Λ is quite likely to be all or almost all of P glob, as in the two examples. Suppose now that π : Y P 1 is a Belyi map defined over Q. Then its set P of bad primes satisfies (2.7) P loc P P glob. For our two examples, P coincides with its lower bound {2, 3, 5}. The conceptual explanations of the splitting given in Section 3 also explain why the remaining one or two primes in P glob are primes of good reduction. 3. Hurwitz maps, Belyi pencils, and Hurwitz Belyi maps In 3.1 we very briefly review the formalism of dealing with moduli of maps Y P 1 t with r critical values. A key role is played by Hurwitz covering maps π h : Hur h Conf ν. In 3.2 we introduce the concept of a Belyi pencil u : P 1 {0, 1, } Conf ν and in 3.3 we give three important examples in r = 4. Finally 3.4 combines the notion of Hurwitz map and Belyi pencil in a straightforward way to obtain the general notion of a Hurwitz Belyi map π h,u Hurwitz maps. Consider a general degree n map F : Y P 1 t as in 1.1. Three fundamental invariants are Its global monodromy group G S n. The list C = (C 1,..., C k ) of distinct conjugacy classes of G arising as non-identity local monodromy transformations. The corresponding list (D 1,..., D k ) of disjoint finite subsets D i P 1 over which these classes arise. To obtain a single discrete invariant, we write ν = (ν 1,..., ν k ) with ν i = D i. The triple h = (G, C, ν) is then a Hurwitz parameter in the sense of [22, 2] or [20, 3]. A Hurwitz parameter h determines a Hurwitz moduli space Hur h whose points x index maps F x : Y x P 1 t of type h. The Hurwitz space covers the configuration space Conf ν of all possible

38 34 Hurwitz Belyi maps divisor tuples (D 1,..., D k ) of type ν. The common dimension of Hur h and Conf ν is r = i ν i, the number of critical values of any F x. Sections 2 4 of [22] and Section 3 of [20] provide background on Hurwitz maps, some main points being as follows. There is a group-theoretic formula for a mass m which is an upper bound and often agrees with the degree m of π h : Hur h Conf ν. If all the C i are rational classes, and under weaker hypotheses as well, then the covering of complex varieties descends canonically to a covering of varieties defined over the rationals, π h : Hur h Conf ν. The set P h of primes at which this map has bad reduction is contained in the set P G of primes dividing G. On the computational side, [20] provides many examples of explicit computations of Hurwitz covers. Because the map π h is equivariant with respect to PGL 2 actions, we normalize to take representatives of orbits and thereby replace Hur h Conf ν by a similar cover with three fewer dimensions. Our computations in the previous section for h = (S 7, (322, 421, 511), (1, 1, 1)) and h = (S 12, (642, , 5322), (1, 1, 1)) illustrate the case r = 3. Computations in the cases r 4 proceed quite similarly. The next section gives some simple examples and a collection of more complicated examples is given in the companion paper [21]. Let Out(G, C) be the subgroup of Out(G) which fixes all classes C i in C. Then Out(G, C) acts freely on Hur h. For any subgroup Q Out(G, C), we let Hur Q h be the quotient Hur h /Q and let π Q h : HurQ h Conf ν be the corresponding covering map. One can expect that Q will usually play a very elementary role. For example, it can be somewhat subtle to get the exact degree m of a map π h, but then the degree of π Q h is just m/ Q. We already have the simpler notation Hur h for Hur {e} h. Similarly, following [22], we use as a superscript to represent the entire group Out(G, C). In the literature, Hur h is often called an inner Hurwitz space while Hur h is an outer Hurwitz space. In the entire sequel of this paper, the only Q that we will consider are these two extreme cases. It is important for us to descend to the -level to obtain fullness Belyi pencils. For any ν as above, the variety Conf ν naturally comes from a scheme over Z. Thus for any commutative ring R, we can consider the set Conf ν (R). Section 8 of [22] and then the sequel paper [20] considered R = Q and its subrings Z[1/P] = Z[{1/p} p P ]. From fibers Hur h,u Hur h (Q) above points in Conf ν (Z[1/P]) one gets interesting number fields, the Hurwitz number fields of the title of [20]. Our focus here is similar, but more geometric. A Belyi pencil u is an algebraic map (3.1) u : P 1 v {0, 1, } Conf ν, with image not contained in a single PGL 2 orbit. One can think of v as a time-like variable here. The Belyi pencil u then can be understood as giving r points in P 1 t, typically moving with v. There are ν i points of color i; points are indistinguishable except for color, and they always stay distinct except for collisions at v {0, 1, }. To make the similarity clear, for R a ring let R v = R[v, 1 v(v 1) ]. Then a Belyi pencil can be understood as a point in Conf ν (C v ). We say that the Belyi pencil u is rational if it is in Conf ν (Q( v )). For rational pencils, one has a natural bad reduction set P u. It is the smallest set P with u Conf ν (Z[1/P] v ).

39 D. Roberts 35 Remark 3.1 (A complicated Belyi pencil). In this paper we will actually use only the three very simple Belyi pencils of the next subsection, and also the simple Belyi pencil (10.1). However general Belyi pencils can be much more complicated. For example, consider the eight-tuple (3.2) (( t 6 8vt 3 + 9vt 2 2v 2) (, t 6 3t vt 3 15vt 2 + 9vt 2v 2), ( t 6 6vt vt 4 20vt 3 + 6v 2 t 2 + 9vt 2 6v 2 t + v 2), ( ) ( ) ( ) ) t 4 2t 3 + 2vt v, t 4 4vt + 3v, 2t 3 3t 2 + v, (t), { }. The product of the seven irreducible polynomials presented has leading coefficient 2 and discriminant v 125 (v 1) 125. Thus u : P 1 v {0, 1, } Conf 6,6,6,4,4,3,1,1 is a Belyi pencil with bad reduction set {2, 3} Belyi pencils for r = 4. Three Belyi pencils play a special role in the case r = 4, and we denote them by u 1,1,1,1, u 2,1,1, and u 3,1. For u 1,1,1,1, we keep our standard variable v. To make u 2,1,1 and u 3,1 stand out when they appear in the sequel, we switch the time-like variable v to respectively w and j for them. These special Belyi pencils are then given by (3.3) ({v}, {0}, {1}, { }), (D w, {0}, { }), (D j, { }). Here the divisors D w and D j are the root-sets of t 2 + t + 1 4(1 w) and 4(1 j)t3 + 27jt + 27j respectively. So the three Belyi pencils are all rational, and their bad reduction sets are respectively {}, {2}, and {2, 3}. The images of these Belyi pencils are curves U 1,1,1,1 Conf 1,1,1,1, U 2,1,1 Conf 2,1,1, U 3,1 Conf 3,1. The three curves are familiar as coarse moduli spaces of elliptic curves. Here U 1,1,1,1 = Y(2) parametrizes elliptic curves with a basis of 2-torsion, U 2,1,1 = Y 0 (2) parametrizes elliptic curves with a 2-torsion point, and U 3,1 = Y(1) is the j-line parametrizing elliptic curves. The formulas (3.4) w = (2v 1)2 (3w + 1)3, j = 9 (9w 1) 2 = 22 (v 2 v + 1) v 2 (v 1) 2 give natural maps between these three bases: P 1 v P 1 w P 1 j. Remark 3.2 (The two other 4-point cases). The cases ν = (2, 2) and ν = (4) are complicated by the presence of extra automorphisms. Any configuration (D 1, D 2 ) Conf 2,2 is in the PGL 2 orbit of a configuration of the special form ({0, }, {a, 1/a}). This latter configuration is stabilized by the automorphism t 1/t. Similarly, a configuration (D 1 ) Conf 4 has a Klein four-group of automorphisms. To treat these two cases, the best approach seems to be modify the last two pencils in (3.3) to (D w, {0, }) and (D j { }). Outside of a quick example for ν = (4) in the remark containing (8.10), we do not pursue any explicit examples with such ν in this paper.

40 36 Hurwitz Belyi maps 3.4. Hurwitz Belyi maps. We can now define the objects in our title. Definition 3.3. Let h = (G, C, ν) be a Hurwitz parameter, let Q be a subgroup of Out(G, C), and let π Q h : HurQ h Conf ν be the corresponding Hurwitz map. Let u : P 1 v {0, 1, } Conf ν be a Belyi pencil. Let (3.5) π Q h,u : XQ h,u P1 v be the Belyi map obtained by pulling back the Hurwitz map via the Belyi pencil and canonically completing. A Belyi map obtainable by this construction is a Hurwitz Belyi map. Recall from the end of 3.1 that Q plays a passive role. We usually take Q to be all of Out(G, C), in which case we replace the superscript simply by. In the common case when Out(G, C) is trivial, we can abbreviate further by omitting the superscript. When r = 4 and u is one of the three maps (3.3), then we are essentially not specializing, as we are taking a set of representatives for the PGL 2 orbits on Conf ν. We allow ourselves to drop the u in this situation, writing e.g. π h : X h P 1 j. On the other hand, when r 5 we are truly specializing a cover of (r 3)-dimensional varieties to a Belyi map. Rationality and bad reduction are both essential to this paper. If h and u are both defined over Q, then so is π h,u. If h has bad reduction set P h and u has bad reduction set P u then the bad reduction set of π h,u is within P h P u. All the examples we pursue in this paper satisfy P u P h. 4. The two rational Belyi maps as Hurwitz Belyi maps This section presents some first examples in the setting r = 4, and in particular interprets the two rational Belyi maps of Section 2 as Hurwitz Belyi maps A degree 7 Hurwitz Belyi map: computation and dessins. To realize the Belyi map (2.5) as a Hurwitz Belyi map, we start from the Hurwitz parameter (4.1) h = (S 6, (2 x 1111, 3 0 3, , 4 11), (1, 1, 1, 1)). Here and in the sequel we often present Hurwitz parameters with subscripts which indicate our normalization, without being as formal about markings as we were in Section 2. Thus the subscript 0 in causes us to write y 2 a in the next equation, rather than say y 2 + zy a; this type of normalization on the second-highest order term does not introduce irrationalities. The marked Hurwitz parameter (4.1) tells us to consider rational functions of the form (4.2) F (y) = (y2 a) 3 (b + c + 1) (1 a) 3 (y 2 + by + c) and the equation (4.3) 4y 3 + 5by 2 + 2ay + ab = 4(y 1) 2 (y x). The left side of (4.3) is a factor of the numerator of F (y) and thus its roots are critical points. The right side gives the required locations and multiplicities of these critical points. Equating coefficients of powers of y in (4.3) and solving, we get (4.4) a = 5x x + 2, b = 4 5 (x + 2), c = 4x2 + 5x (x + 2)

41 37 D. Roberts Figure 4.1. Left: The dessin Xh,[0,1] P1x belonging to h = (S6, (21111, 33, 3111, 411), (1, 1, 1, 1)), with real axis pointing up; the integers i are at preimages xi Xh,[0,1] of 1/2 P1v. Right: the dessins Yh,xi,[0,1] P1y for the seven i. Summarizing, we have realized Xh as the complex projective x-line and identified each Yh,x with the complex projective y-line P1y so that the covering maps Fh,x : P1y P1t become (4.5) 3 (x + 2)y 2 5x Fh,x (y) =. 4(2x 1) ( 15(x + 2)y (x + 2)2 y 5(4x2 + 5x + 4)) Since the fourth critical value of Fh,x is just Fh,x (x), we have also coordinatized the Hurwitz Belyi map πh : Xh P1v to (4.6) 2 x3 x2 + 2x 5 πh (x) =. 4(2x 1)(3x 4) Said more explicitly, to pass from the right side of (4.5) to the right side (4.6), one substitutes x for y and cancels x2 + 2x 5 from top and bottom.

42 38 Hurwitz Belyi maps The rational function (4.6) appeared already as (2.5), with its dessin printed in the upper left of Figure 2.1. This connection is our first explanation of why (2.4) splits. It also explains why the bad reduction set of the rational Belyi map is in {2, 3, 5}. Figure 4.1 presents the current situation pictorially, with v = 1/2 chosen as a base point. The elements of X h,1/2 = πh 1 (1/2) are labeled in the box at the left, where the real axis runs from bottom to top for a better overall picture. For each x X h,1/2, a corresponding dessin Y x,[0,1] is drawn to its right. Like the standard dessins of 2.3, these dessins have black vertices, white vertices, and faces. However they each also have five vertices of a fourth type which we are not marking, corresponding to the five parts of the partition The valence of this type of vertex with ramification number e is 2e, so only the extra vertex coming from the critical point with e = 2 is visible on Figure 4.1. The action of the braid group to be discussed in 5.1 can be calculated geometrically from these dessins Cross-parameter agreement. An interesting phenomenon that we will see repeatedly in later sections is cross-parameter agreement. By definition, cross-parameter agreement occurs when two different Hurwitz parameters give rise to isomorphic Hurwitz covers. At present, as mentioned in [20, 3.6], some of the these agreements are explained by the Katz middle convolution operator. However there are unexplained agreements as well that do not seem accidental. Already the phenomenon of cross-parameter agreement occurs for our septic Belyi map, which we realize in a second way as a Hurwitz Belyi map as follows. For the normalized Hurwitz parameter ĥ = (S 5, (2 z 111, 221, , ), (1, 1, 1, 1)), the computation is easier than it was for the Hurwitz parameter h of (4.1). An initial form of ˆF (y), analogous to (4.2), is (4.7) ˆF (y) = (y c) ( y 2 + ay + b ) 2 (1 c)y 2 (a + b + 1) 2. Analogously to (4.5), the covering maps P 1 y P 1 t are (4.8) ˆFz (y) = (4yz + 2y z) ( 2y 2 z y 2 + 6yz yz + 6y + 12z z + 3 ) 2 4y 2 (3z + 2) 5. Analogously to (2.2) the Hurwitz Belyi map P 1 z P 1 v is (4.9) ˆπ(z) = ˆF z (z) = ( 4z 2 + z ) ( 4z z z + 3 ) 2 4z 2 (3z + 2) 5. The map (4.9) agrees with the map (2.2) via the substitution z = (1 2x)/(3x 4). In terms of Figure 4.1, the dessin at the left remains exactly the same, up to change of coordinates. In contrast, the seven sextic dessins at the right would be each replaced by a corresponding quintic dessin Two degree 12 Hurwitz Belyi maps. In our labeling of conjugacy classes mentioned in 1.6, we are accounting for the fact that the five cycles in A 5 fall into two classes,

43 D. Roberts 39 with representatives (1, 2, 3, 4, 5) 5a and (1, 2, 3, 4, 5) 2 = (1, 3, 5, 2, 4) 5b. Consider two Hurwitz parameters, as in the left column: h aa = (A 5, (5a, 311, 221), (2, 1, 1)), (β 0, β 1, β ) = (5331, , 5322), h ab = (A 5, (5a, 5b, 311, 221), (1, 1, 1, 1)), (β 0, β 1, β ) = (642, , 5322). Applying the outer involution of A 5 turns h aa and h ab respectively into similar Hurwitz parameters h bb and h ba, and so it would be redundant to explicitly consider these latter two. It is hard to computationally distinguish 5a from 5b. We will deal with this problem by treating h aa and h bb simultaneously. Thus we working formally with h = (S 5, (5, 311, 221), (2, 1, 1)), ignoring that the classes do not generate S 5. A second problem is that there are only eight parts altogether in the partitions 5, 5, 311, and 221, so the covering curves Y have genus two. To circumvent the genus two problem, we use the braid-triple method, as described later in Section 5. The mass formula [20, 3.5] applies to h, with only the two abelian characters of S 5 contributing. It says that the corresponding cover X h P 1 w has degree C 5 2 C 311 C 221 A 5 2 = = A braid group computation of the type described in 5.2 says X h has two components, each of degree 12. The braid partition triples are given in the right column above. The β τ then enter the formalism of Section 2 as the λ τ there. Conveniently. each cover sought has genus zero, and so the covers are easily computed. The resulting polynomials are f 12aa (w, x) = x 5 ( 9x 2 21x + 16) 3 (x + 3) 2 8 w(x 1) 3 ( 9x 2 12x + 8) 2, f 12ab (w, x) = 5 5 (x 1) 4 x 6 (5x + 4) w(2x + 1) 3 ( 5x 2 6x + 2) 2. Up to the simultaneous letter change y x, t w, the equation f 12ab (w, x) = 0 defines the exact same map as (2.6). The current context explains why this map is defined over Q and has bad reduction at exactly {2, 3, 5} Figure 4.2. Rational degree 12 dessins coming from Hurwitz parameters involving irrational classes. h 12aa yielding partition triple (5331, , 5322) on the left and h 12ab yielding (642, , 5322) on the right Dessins corresponding to f 12aa and f 12ab are drawn in Figure 4.2. The two dessins present an interesting contrast: the dessin on the left of Figure 4.2 is the unique dessin with its

44 40 Hurwitz Belyi maps partition triple, while the dessin on the right is one of the 24 locally equivalent dessins drawn in Figure 2.2. Remark 4.1 (An M 12 specialization). Specializing Hurwitz Belyi maps yields interesting number fields. Except for this remark, we are saying nothing about this application, because we discuss specialization of Hurwitz covers quite thoroughly in [18] and [20]. However the polynomial f 12ab (1/4, x), or equivalently x 12 24x x 8 60x x x x x 2 216, unexpectedly has the Mathieu group M 12 as its Galois group. The splitting field has root discriminant 2 3/2 3 3/2 5 23/ Comparing with the discussion in [19, 6.2], one sees that this field is currently the second least ramified of known M 12 Galois fields, being just slightly greater than the current minimum 2 25/ / / The braid-triple method Our first Hurwitz Belyi map (4.6) was computed with the standard method. After 5.1 gives background on braids, 5.2 and 5.3 describe the alternative braid-triple method, already used twice in 4.3. The two methods are complementary, as we explain in the short 5.4. A key step in the braid-triple method is to pass from a Belyi pencil u to a corresponding braid triple B = (B 0, B 1, B ). In this paper, we use the braid-triple method only for four u, and the corresponding B are given by the simple formulas (5.3), (5.4), (5.5), and (10.2). We do not pursue the general case here; our policy in this paper is to be very brief with respect to braid groups, saying just enough to allow the reader to replicate our computations of individual covers Algebraic background on braid groups. The Artin braid group on r strands is the most widely known of all braid groups, and our summary here follows [22, 3]. The group is defined via r 1 generators and ( r 1) 2 relations: σ (5.1) Br r = σ 1,..., σ r 1 : i σ j = σ j σ i, if i j > 1. σ i σ j σ i = σ j σ i σ j, if i j = 1 The assignment σ i (i, i + 1) extends to a surjection Br r S r. For every subgroup of S r one gets a subgroup of Br r by pullback. Thus, in particular, one has surjections Br ν S ν for S ν = S ν1 S νr. Given a finite group G, let G r G r be the set of tuples (g 1,..., g r ) with the g i generating G and satisfying g 1 g r = 1. The braid group Br r acts on the right of G r by the braiding rule (5.2) (..., g i 1, g i, g i+1, g i+2,... ) σ i = (..., g i 1, g i+1, g g i+1 i, g i+2,... ). The group G acts diagonally on G r by simultaneous conjugation. The actions of Br ν and G commute with one another. Let h = (G, C, ν) be an r-point Hurwitz parameter, with C = (C 1,..., C k ), ν = (ν 1,..., ν k ), and k i=1 ν i = r, all as usual. Consider the subset G h of G r, consisting of tuples with g j C i for i 1 a=1 ν a < j i a=1 ν a. This subset is stable under the action of Br ν G. The group Br ν therefore acts on the right of the quotient set F h = G h /G. The actions of Br ν on F h all factor through a certain quotient HBr ν. Terminology is not important for us here, but for comparison with the literature we remark that HBr ν is the

45 D. Roberts 41 quotient of the standard Hurwitz braid group by its two-element center. Choosing a base point and certain identifications appropriately, the group HBr ν is identified with the fundamental group π 1 (Conf ν, ), in a way which makes the action of HBr ν on F h agree with the action of π 1 (Conf ν, ) on the base fiber πh 1 ( ) Hur h. Let Z be the center of G, this center being trivial for most of our examples. Let Aut(G, C) be the subgroup of Aut(G) which stabilizes each conjugacy class C i in C. Then not only does G/Z act diagonally on G h, but so does the entire overgroup Aut(G, C). The group Out(G, C) introduced in 3.1 is the quotient of Aut(G, C) by G/Z. Quotienting by the natural action of a subgroup Q Out(G, C), gives the base fiber F Q h corresponding to the cover HurQ h Conf ν Step one: computation of braid triples. A Belyi pencil u : P 1 v Conf ν determines, up to conjugacy depending on choices of base points and a path between them, an abstract braid triple (B 0, B 1, B ) of elements of HBr ν. These elements have the property that in any Hurwitz Belyi map π h,u : X h,u P 1, the images of the B τ in their action on F h give the global monodromy of the cover. When F h is identified with {1,..., m}, we denote the image of B τ by b τ S m and its cycle partition by β τ. We call (b 0, b 1, b ) a braid permutation triple. As we have already done before, we call (β 0, β 1, β ) a braid partition triple. For the three 4-point Belyi pencils introduced in (3.3), the abstract braid triples are (5.3) u 1,1,1,1 : (B 0, B 1, B ) = (σ1, 2 σ2, 2 σ2 2 σ 2 1 ), (5.4) u 2,1,1 : (B 0, B 1, B ) = (σ 1, σ1 1 σ 2 2, σ2 2), (5.5) u 3,1 : (B 0, B 1, B ) = (σ 1 σ 2, σ 1 2 σ 2 1, σ 1). The triple for u 1,1,1,1 is given in [8, 5.5.2]. The other two can be deduced by quadratic and then cubic base change, using (3.4). An important point is that, in the quotient group HBr ν, the B 1 for u 2,1,1 and u 3,1 have order 2 and the B 0 for u 3,1 has order 3. It is worth emphasizing the conceptual simplicity of our braid computations. They repeatedly use the generators σ i of (5.1) and their actions on F h from (5.2). However they do not explicitly use the relations in (5.1). Likewise they do not explicitly use the extra relations involved in passing from Br ν to HBr ν. Our actual computations are at the level of the permutations b τ S m rather than the level of the braid words B τ. At the permutation level, all these relations automatically hold. Computationally, we realize F h via a set of representatives in G h for the conjugation action. A difficulty is that the set G h in which computations take place is large. Relatively naive use of (5.1) and (5.2) suffices for the braid computations presented in the next five sections. To work as easily with larger groups and/or larger degrees, a more sophisticated implementation as in [9] would be essential Step 2: passing from a braid triple to an equation. Having computed a braid partition triple (β 0, β 1, β ) belonging to π h,u, one can then try to pass from the triple to an equation for π h,u by algebraic methods. We did this in 4.3 for two partition triples with degree m = 12. For one, as described in 2.6, the desired π h,u is just one of µ = 24 locally equivalent covers, the one defined over Q. The braid partition triples (β 0, β 1, β ) arising in the next five sections have degrees m into the low thousands. The number µ of Belyi maps with a given such braid partition triple is likely to be more than in some cases. The numbers 12 and 24 are therefore being

46 42 Hurwitz Belyi maps replaced by very much larger m and µ respectively. It is completely impractical to follow the purely algebraic approach of getting all the maps belonging to the given (β 0, β 1, β ) and extracting the desired rational one. Instead, there are three more feasible techniques for computing only the desired cover. For almost all the covers in this paper, Step 2 was carried out by a p-adic technique for finding covers defined over Q explained in detail in [10]. Here one picks a good prime p for the cover sought, and first searches for a tame cover with the correct (β 0, β 1, β ) defined over F p. Commonly, one finds several covers, and one cannot yet tell which is the reduction of the cover sought. One then uniquely lifts all these candidates iteratively to Z/p c for some large c. This step requires solving linear equations and is easy. We commonly took c = 50. Then one recognizes the coefficients of the lifted covers as p-adically near rational numbers. In practice this is easy too, and only one of the initial solutions over F p gives small height rational numbers. One concludes by checking that the monodromy of the cover constructed really does agree with the braid permutation triple (b 0, b 1, b ). The efficiency of this p-adic technique decreases rapidly with p. Since the covers we pursue all have a small prime of good reduction, typically 5 or 7, the technique is well adapted to our situation. The second and third technique have been recently introduced, and both are undergoing further development. They take the permutation triple (b 0, b 1, b ) rather than the partition triple (β 0, β 1, β ) as a starting point. Thus they isolate the cover sought immediately, and there is no issue of a large local equivalence class. The technique of [6] centers on power series while the technique of [7] centers on numerically solving partial differential equations. Schiavone used the programs described in [6] to compute (7.5) here. Our tables in Sections 8 10 present braid information going well beyond where one can currently compute equations, in part to provide targets for these developing computational methods Comparison of the two methods. We used the standard method many times in [20] in the context of constructing covers of surfaces. In the present context of curves, the braid-triple method complements the standard method as follows. In the standard method, one can expect the difficulty of the computation to increase rapidly with the genus g Y of Y x and the degree n of the cover Y x P 1 t. In the braid-triple method, these measures of difficulty are replaced by the genus g X of the curve X h,u and the degree m of the cover X h,u P 1 v. The quantities (g Y, n) and (g X, m) are not tightly correlated with each other, and in practice each method has a large range of parameters for which it works well while the other method does not. 6. Hurwitz Belyi maps exhibiting spin separation This section presents three Hurwitz Belyi maps for which we were able to find a defining equation by both the standard and the braid-triple method. Each map has the added interesting feature that the covering curve X h has two components. We explain this splitting by means of lifting invariants. Many of the covers in the next four sections are similarly forced to split via lifting invariants Lifting in general. Decomposition of Hurwitz varieties was studied by Fried and Serre. Here we give a very brief summary of the longer summary given in [20, 4]. The decompositions come from central extensions G of the given group G. The term spin separation

47 D. Roberts 43 is used because many double covers are induced from the double cover Spin n of the special orthogonal group SO n via an orthogonal representation. Let h = (G, C, ν) be a Hurwitz parameter. First, one has the Schur multiplier H 2 (G, Z), always abbreviated in this paper as H 2 (G). Any universal central extension G of G has the form H 2 (G).G. Second, one has a quotient H 2 (G, C) of the Schur multiplier, with H 2 (G, C).G being the largest quotient in which each C i splits completely into H 2 (G, C) different conjugacy classes. Third, one has a torsor H h = H 2 (G, C, ν) over H 2 (G, C). So H h = H 2 (G, C), but the set H h does not necessarily have a distinguished point like the group H 2 (G, C) does. The group Out(G, C) defined in 3.1 acts on the set H h. For any subgroup Q Out(G, C), one has a natural map from the component set π 0 (Hur Q h ) of HurQ h to HQ h. The most common behavior is that these maps π 0 (Hur Q h ) HQ h are bijective. As said already in 3.1, our main interest is in Q = Out(G, C), in which case we replace Out(G, C) by as a superscript. In practice, the key groups H 2 (G, C) and Out(G, C) are extremely small. In the next three subsections H 2 (G, C) has order 2, 3, and 3 respectively, while Out(G, C) has order 1, 2, and 1. We explain lifting in some detail in these subsections and also in 8.2 where H 2 (G, C) and Out(G, C) can be slightly larger A degree 25 = family. Applying the mass formula [20, (3.6)] to a Hurwitz parameter h = (G, C, ν) requires the use of the character table of G. Common choices for G in this paper are A 5 and S 5. Table 6.1 gives the character table for these two groups, as well as their Schur double covers Ã5, and S 5. In this subsection, we use this table to illustrate how mass formula computations for a given Hurwitz parameter h appear in practice, including refinements involving covering groups G. C i C i a 5b χ χ b b χ b b χ χ Orders of classes χ b b χ b b χ χ Table 6.1. Character tables for A 5, S 5, Ã 5, and S 5, with the abbreviations b = ( 1 + 5)/2 and b = ( 1 5)/2. The character table for A 5 is given simply by the upper left 5-by-5 block. The remaining character tables require the use of Atlas conventions. The double cover Ã5 has the listed nine characters. The nine conjugacy classes arise because all but the class 221 splits into two

48 44 Hurwitz Belyi maps classes. We label these classes of Ã5 according to whether the order of a representing element is even (+) or odd ( ). Thus 5a splits into 5a+ and 5a. The printed character values refer to the class with odd order elements. Thus, e.g., χ 8 (311+) = 1 but χ 8 (311 ) = 1. Only the groups A 5 and Ã5 are used in the example of this subsection, but S 5 and S 5 are equally common in the sequel and we explain them here. The group S 5 has seven conjugacy classes, the classes 5a and 5b having merged to a single class 5. The corresponding seven characters are the printed χ 1, χ 4, χ 5, the sum χ 2 + χ 3 extended by zero, and the twists χ 1 ɛ, χ 4 ɛ, and χ 5 ɛ. Here ɛ is the sign character, taking value 1 on A 5 and 1 on S 5 A 5. The cover S 5 has twelve characters, the seven from before and the five new ones χ 6 + χ 7, χ 8, χ 8 ɛ, χ 9, and χ 9 ɛ. For the example of this subsection, let h = (A 5, (311, 5a), (3, 1)). Because of 0 s appearing as character values, only the characters χ 1 and χ 4 appear when evaluating the mass formula: m h = C 1 3 C 2 G 2 5 i=1 χ i (C 1 ) 3 χ i (C 2 ) χ i (1) 2 = ( 1 3 (1) ( 1) 4 2 ) = = 25. Because A 5 does not have a proper subgroup meeting the both the classes 311 and 5a, the desired degree is just the mass, m h = m h = 25. The joint paper [22] was originally planned to include this h as an example. The curves Y x parameterized have genus one but Venkatesh nonetheless computed the Belyi map π h : X P 1 j by the standard method, seeing directly that X breaks into two components, each of genus zero, of degree 15 and 10 over P 1 j. The present author simultaneously used the braid-triple method, using (5.5) to get the braid partition triples (3331, 22222, 541) and (33333, , 5433). Both methods end at the explicit equations (8.4) and (8.5). To explain the splitting, consider the Hurwitz parameters (6.1) h + = (Ã5, (311+, 5a+), (3, 1)), h = (Ã5, (311+, 5a ), (3, 1)). Let (g 1, g 2, g 3, g 4 ) G h. For i = 1, 2, 3, let g i be the unique preimage of g i in Then there is a unique lift g 4 of g 4 which satisfies g 1 g 2 g 3 g 4 = 1. This lift can be in either 5a+ or 5a. In this way one gets a map from G h to H h = Z/2. This invariant does not change under either the braid or conjugation action. The mass formula [20, (3.6)], applied to G now, lets one find the degrees of the factors. In this simple case, where proper subgroups of G are not involved, one has m ± h = 1 C 1 3 C 2 2 G 2 9 i=1 χ i (C 1 ) 3 χ i (C 2 ) χ i (1) 2 = ( (1) ( 1) 4 2 ± ( 1)3 (b + b ) ) 2 2 ± 13 ( 1) 4 2 = 40 ( ± 1 4 ± 1 ) = 40 ( ± 3 ) = 5 (5 ± 1) = 15, Similar mass computations let one properly identify components with lifting invariants in general. We make these identifications, typically with no further comments, in the next two subsections and many times in A degree 70 = family: rational cubic splitting. Let h = (A 7, (22111, 511, 322), (2, 1, 1)).

49 D. Roberts 45 The large singletons 5 and 3 help keep the standard method within computational feasibility. By a direct application of this method, one sees at the end that the degree m is 70 and there is a splitting into two components of degrees 30 and 40. In the braid-triple method the order of events is reversed. Mass formula computations says that the desired X h P1 w has degree 70. A braid group computation using (5.4) says that X h has two components of degrees 30 and 40. The monodromy groups are A 30 and S 40 respectively, with braid partition triples (β 0, β 1, β ) = ( , , ), (β 0, β 1, β ) = ( , 2 20, ). As the total number of parts is 32 and 42 respectively, the genus is zero in each case. The second step in the braid-triple method is challenging, since the smallest prime not in P A7 is 11. This step is only within feasibility because of the splitting 70 = , and the fact that one can compute the two components independently. Explicit equations are ( 7 f 30 (w, x) = x x + 4) ( ) 2 ( ) 2 x 5 (2x + 1) 3 x 2 + 3x + 1 2x 2 + x + 2 ( 5 + w 7x 2 + 6x + 2) ( 3 (5x + 2) 4 14x x x + 2) (x + 2), ( ) 7 ( ) 5 ( ) 4 f 40 (w, x) = x 2 12x + 3 5x 2 15x + 12 x 2 3x + 6 ( 2 4x 2 15x + 15) x(5x 9) ( ) 5 ( ) 4 + w x 2 3 5x 3 45x x 108 ( x x x x x x 27) The polynomial discriminants are D 30 (w) = w 22 (w 1) 14, D 40 (w) = w 29 (w 1) 20. Modulo squares these quantities are 105 and 7w, reflecting the fact that monodromy groups and generic Galois groups are (A 30, S 30 ) in the first case and (S 40, S 40 ) in the second. The group A 7 has the unusually large maximal non-split central extension 6.A 7. For both this subsection and the next, only the subextension 3.A 7 is relevant because all the classes C i split in it, while the class is inert in 2.A 7. In the notation reviewed in 6.1, this reduction is expressed by an identification H 2 (A 7, C) = Z/3. The group Out(A 7, C) is all of Out(A 7 ) = {±1} because the classes 22111, 511, and 322 are all stabilized by the outer involution. The action of {±1} on Z/3 is the nontrivial one where 1 acts by negation. The degree 30 component corresponds to the identity class 0 Z/3 while the degree 40 component corresponds to the orbit of the two nonidentity classes in Z/3. The promised third conceptual explanation of the degree splitting 4 = for h = (A 7, (322, 421, 511), (1, 1, 1)) from (2.4) is in our present context. All three classes split in 3.A 7 while only the last two split in 2.A 7. All three classes are stable under outer involution. So here again Out(A 7, C) = {±1} acts nontrivially on H 2 (A 7, Z) = Z/3. In this case the degree one factor corresponds to 0 Z/3 while the degree three factor corresponds to { 1, 1} Z/3.

50 46 Hurwitz Belyi maps 6.4. A degree 42 = family: irrational cubic splitting. Lando and Zvorkin [8, 5.4] investigated splitting of Hurwitz covers in some generality. The unique splitting in their context that they could not conceptually explain comes from the Hurwitz parameter h = (A 7, (22111, 7a), (3, 1)). Here one has splitting of the form 42 = In this subsection we complement their study of this example by both giving an equation and explaining the splitting. Computing using (5.5), one gets that the two components have the same braid partition triple, namely (6.2) (β 0, β 1, β ) = (3 7, , ). This agreement is in contrast to the previous subsection, where the two components even had different degrees. Lando and Zvonkin speculated (p. 333) that the agreement is explained by the two components being Galois conjugate. Indexing the two maps arbitrarily by ɛ {+, }, each map we seek fits as the right vertical map in a Cartesian square: (6.3) X ɛ P 1 v X ɛ P 1 j. Here the bottom map is the degree six S 3 cover given in (3.4), and so the top map is a degree six S 3 cover as well. There are = 23 parts in all in (6.2), so that the genus of each X ɛ is 0 by the Riemann-Hurwitz formula. Lando and Zvonkin worked first with the base-changed cover. Here the braid partition is ( β 0, β 1, β ), with each β τ = As = 21, the genus is 1. Jones and Zvonkin [5] carried out the S 3 descent as we are doing here. We find via explicit equations that the two components are indeed conjugate with respect to the two choices s = ± 21: ( f 21± (j, x) = x 7 + x 6 (630000s ) + x 5 ( s) + x 4 ( s) + x 3 ( s ) + x 2 ( s) + x(856800s ) s ± j(5239s 21429)x 5 (10x 9) 4 ( 150x 2 + x(40s 15) 8s + 88) 3. Figure 6.1 draws the dessins in P 1 x corresponding to Cover 21+ on the left and its conjugate Cover 21 on the right. The preimages in P 1 x of P 1 j are indicated in the picture by their ramification numbers, with the undrawn P 1 x also being a preimage with ramification number 6. Figure 6.1 gives the correct analytic shape of Figure 3 of [5], and, after base change, the correct shape of Figure 5.15 of [8]. The splitting is induced by the existence of 3.A 7 as in the previous subsection. Again one has an identification H 2 (A 7, C) = Z/3. Here however, because 7a is not stabilized by the outer involution of A 7, the group Out(A 7, C) is trivial. Accordingly one has a natural function from components of X to Z/3. One would generally expect all three preimages to have one ) 3

51 D. Roberts Figure 6.1. Conjugate dessins, with 21+ on left and 21 on right component each. In this case, the preimages of 0, 1, and 1 are respectively empty, X + and X. 7. Hurwitz Belyi maps with G = 2 a 3 b p and ν = (3, 1) In this section, we set up a framework for studying the Hurwitz Belyi maps coming from a systematic collection of 4-point Hurwitz parameters h. Here and in the next two sections, we carry out the first part of the braid-triple method for all these h, obtaining braid permutation triples (b 0, b 1, b ) and thus braid partition triples (β 0, β 1, β ). In many low degree cases, we carry out the second part as well, obtaining a defining equation for the cover Restricting to P = 3 and ν = (3, 1). To respond to the inverse problem of 1.3, we consider only h = (G, C, ν) giving covers defined over Q. To keep our computational study of manageable size we impose two severe restrictions. First, we require that G be almost simple with exactly three primes dividing its order. Second, we restrict attention to the case ν = (3, 1). There are many more cases within computational reach which are excluded because of these two restrictions. The rest of this subsection elaborates on the two restrictions. Almost simple groups divisible by at most three primes. There are exactly eight nonabelian simple groups T for which the set P T of primes dividing T has size at most 3. In all cases, the order has the form 2 a 3 b p and the classical list is as in Table 7.1. References into this classification literature and the complete list in the much more complicated case P T = 4 are in [25]. p T T H 2 A p T T H 2 A 5 A 5 60 = SL 3 (2) 168 = A = SL 2 (8) 504 = W (E 6 ) = SU 3 (3) 6048 = SL 3 (3) 5616 = P SL 2 (17) 2448 = Table 7.1. The eight simple groups of order 2 a 3 b p and related information The column H 2 gives the order of the Schur multiplier H 2 (T ). Non-trivial entries here are the source of spin separation as explained in the previous section. The column A in Table 7.1

52 48 Hurwitz Belyi maps gives the structure of the outer automorphism group of T. So in every case except T = A 6 there are two groups G to consider, T itself and Aut(T ) = T.A. For T = A 6 one has, in Atlas order, the extensions S 6, P GL 2 (9), and M 10, as well as the full extension Aut(A 6 ) = A Attractive features of the case ν = (3, 1). The restriction ν = (3, 1) is chosen for several reasons. First, it makes tables much shorter, and in fact Tables 8.1, 8.3, 9.1, 9.2 are complete. The case ν = (2, 2) would have similar length and the case ν = (4) would be even shorter. However we stay away from both these alternatives as the involutions discussed at the end of 3.3 complicate the situation. Second, covers in a given degree m tend to have considerably smaller genus for ν = (3, 1) than they do for ν = (2, 1, 1) or (1, 1, 1, 1). In fact our tables show that for ν = (3, 1), covers can have genus zero into quite high degree. Third, in the case ν = (3, 1) and (β 0, β 1 ) = (3 m/3, 2 m/2 ), Beukers and Montanus [1] described a method which allows one to solve the given system with m unknowns by first solving an auxiliary system with approximately m/3 unknowns. This method generalizes to the full (3, 1) case of (β 0, β 1 ) = (3 a 1 m 3a, 2 b 1 m 2b ); we used it simultaneously with the p-adic technique sketched in 5.3 to extend the reach of our calculations. Finally, as discussed in 3.3, the base P 1 j is the familiar j-line. Transitive degree-m covers X h P 1 j correspond to index-m subgroups of P SL 2 (Z) and we are in a very classical setting Agreement and indexing. As discussed in 4.2, the interesting phenomenon of cross-parameter agreement says that different Hurwitz parameters can give rise to isomorphic coverings. When the two groups involve different nonabelian simple groups T, as in the initial example of 4.2, we use the term cross-group agreement. We note cross-group agreement in our tables mainly by referencing a common equation. Two covers appearing even for T involving different primes are (7.1) (7.2) f 3,1 (j, x) = (x 4)x 3 + 4j(2x + 1), f 4,3,2 (j, x) = (4x 3 3x + 2) 3 27jx 3 (3x 2) 2. These covers are capable of arising for T of the form 2 3 p for various p because their discriminants are respectively j 2 (j 1) 2 and j 6 (j 1) 4. The covers (7.1), (7.2) illustrate our convention of indexing by the braid partition β. This partition and also β 0 = 3 a 1 m 3a can be read off from the presented polynomial. The remaining partition β 1 = 2 b 1 m 2b governing the factorization of f β (1, x) is then determined by the fact that we give polynomials only in genus zero cases A degree 46 map with bad reduction set {2, 3, 13}. The next two sections focus on Hurwitz Belyi maps coming from groups of order 2 a 3 b p for p {5, 7}. Here and in the next subsection, to give a sense of completeness, we give one map each for p {13, 17}. From h = (P GL 3 (3), (2B, 4B), (3, 1)) we get the braid partition triple (β 0, β 1, β ) = ( , 2 23, ). Our final polynomial is f 132,8,6,3,2,1(j, x) = ( 16x x 3 3x 2 116x 8 ) ( 4096x x x x x x x x x x x x x x ) j(x 2) 3 x 6 (x + 4) 2 (2x 1) ( 3x 2 + 2x 4 ) 13.

53 D. Roberts 49 The discriminant of this polynomial is (j 1) 23 j 28. Modulo squares this discriminant is 39(j 1). The factor of j 1 is known from the outset by the oddness of β 1 and β A degree 54 map with bad reduction set {2, 3, 17}. The Hurwitz parameters (7.3) h 1 = (SL 2 (17), (17a+, 3a ), (3, 1)) and h 2 = (P GL 2 (17), (2B, 6A), (3, 1)) each give conjugate braid permutation triples, with common braid partition triple (7.4) (β 0, β 1, β ) = ( , 2 27, ). An equation was determined by Schiavone using improvements of the techniques described in [6]: (7.5) f 9 3,8 2,4 2,2,1(j, x) ( ) = x x x 8 ( x 17 52x x x x x x x x x x x x x x x x ( ) 4 ( ) 8 ( jx x 2 71x + 32 x 2 + 2x 1 x x 2 48x 8). It seems that this cross-parameter agreement is one of an infinite family indexed by odd primes as follows. Generalize h 1 to (SL 2 (p), (pa+, 3a ), (3, 1)). Generalize h 2 to (P GL 2 (p), (2B, 6A), (3, 1)) when p ±5 (12) and to (P SL 2 (p), (2b, 6a), (3, 1)) when p ±1 (12). Then mass computations confirm that both covers have degree m = p { p if p 1 (3) p if p 2 (3). Braid computations say that indeed the covers are isomorphic at least for p 19. For m = 5 and 7 the degrees are 0 and 18 respectively, these cases arising in 8.1 and 9.1. ) 3 8. Hurwitz Belyi maps with G = 2 a 3 b 5 and ν = (3, 1) In this section we work in the framework set up in 7.1 and present a systematic collection of Hurwitz Belyi maps having bad reduction at exactly {2, 3, 5}. Cross-group agreement. Before getting to the individual groups, we present equations for covers involved in cross-group agreement. The covers (8.1) (8.2) (8.3) ( 3 f 5,3,1 (j, x) = 5 2 5x 3 45x x + 25) jx 3 (3x 25), ( 3 f 5,3,2 (j, x) = 9x 3 + 3x 2 53x + 81) (x + 9) jx 3 (3x 5) 2, ( ) 3 f 5,4,3 (j, x) = 4x 4 24x x 2 48x jx 3 (3x 4) 5

54 50 Hurwitz Belyi maps appear for all three groups. The covers (8.4) ( 3 f 5,4,1 (j, x) = 16x 3 87x x + 16) (16x + 1) jx 4 (x 5), (8.5) ( ) 3 f 5,4,3 2(j, x) = 4 256x x 4 440x x x jx 4 ( 32x x + 80 appear for the simple groups A 5 and A 6. The covers ( 3 (8.6) f 5,1 (j, x) = x 2 5) 3 3 j(2x 5), ( (8.7) f 5 2,4,2(j, x) = 2 7 x 18x 5 144x x 3 224x x 162 ) 3 ) 3 j(2x 9) 2 ( 36x 2 52x 9) 5, appear for A 6 and W (E 6 ) +. Several larger degree covers also appear for both A 6 and W (E 6 ) +. A polynomial for the smallest of these is (8.8) f 10,8 2,6,5,4 2,3(j, x) ( = x x x x x x x x x x x x x x x x ( j 9x 2 10x + 3) ( 4 x 6 (5x 3) 5 3x 2 1) (3x 1) The simple group A 5 = SL2 (4) = P SL 2 (5). Tables 8.1, 8.3, 9.1, and 9.2 have a similar structure, which we explain now drawing on Table 8.1 where T = A 5 for examples. The top left subtable gives degrees of components of Hurwitz Belyi maps Xh P1 j for h = (T, (C 1, C 2 ), (3, 1)). Here C 1 and C 2 are distinct conjugacy classes in T. When the lifting invariant set Hh from 6.1 is trivial, a single number is typically printed. For T = A 5, this triviality occurs exactly if 221 is one of the C i, as from Table 6.1 for A 5, only 221 is inert in the double cover Ã5. When Hh is canonically Z/2, typically two numbers are printed; the top and bottom numbers respectively give the degrees of Xh + and Xh over P 1 j. The remaining subtables on the left sides of Tables 8.1, 8.3, 9.1, and 9.2 similarly give degrees of components of Hurwitz Belyi maps, bit now for h = (T.2, (C 1, C 2 ), (3, 1)). When H h = H h has 2 elements but is not canonically Z/2, typically again two numbers are printed in a column. These numbers are necessarily the same. In general, a number is put in italics when the corresponding component is not defined over Q. For G = S 5, irrationality occurs exactly if {C 1, C 2 } = {41, 32}, a key point being that 41 and 32 each split into two irrational classes in S 5. Other possibilities for H h occur only for T = A 6 and will be discussed in 8.2. In general, if there is splitting beyond that forced by lifting invariants then the corresponding degree is written as a list of the component degrees separated by commas. This extra splitting does not occur on Table 8.1 and we expect it to be rare in general. Indeed for G = A 5 and any (C, ν), it never occurs on the level of the entire r-dimensional Hurwitz cover Hur h Conf ν [4]. ) 3

55 D. Roberts 51 C 1 \C b a b 9 4 5a C 1 \C b # M g β 0 β 1 β Eqn. 2+ A (7.1) 1+ A (8.1) 1+ S 10a (8.2) 2+ S 10b (8.4) 1+ S (8.3) 1+ S (8.5) 1 A (8.9) Two pairs defined over Q( 6) 1 A (8.11) 1 A Table 8.1. Left: Degrees of components of Hurwitz Belyi covers with parameters (G, (C 1, C 2 ), (3, 1)) with G = A 5 or S 5. Right: further information on these covers. We are interested primarily in rational covers and we distinguish non-isomorphic rational covers of the same degree by identifying labels. This convention highlights cross-parameter agreement. Thus on the left half of Table 8.1 the two 4 s and the two 10b s each represent isomorphic covers. The left half of Tables , 9.1, or 9.2, as just described, is well thought of as the Hurwitz half. The right half can then be considered the Belyi half, as it makes no reference to its Hurwitz sources beyond the column #. Here a number printed under # just repeats the number of Hurwitz sources from the left half; a + sign represents cross-group agreement, as it indicates that the cover also arises elsewhere in this paper for a different T. While our focus is on Hurwitz Belyi covers defined over Q, when there is space we include extra lines for Hurwitz Belyi covers not defined over Q. Equations for the first six lines of the top right subtable of Table 8.1 have already been presented in the context of cross-group agreement. An equation for the seventh line is ( (8.9) f 10,6,5,4 2,3(j, x) = x 10 38x x x x x x 4 ) 3 ( ) x x x x 2 14x jx 6 (x 5) 5 ( x 2 4x + 5) 4 (x 9) 3. Note that the four-point covers Y x P 1 t corresponding to the seventh line have genus one, and so (8.9) would be hard to compute by the standard method. The tables of this and the next section give many examples where g Y is large but g X = 0. As we are systematically using the braid-triple method, g Y is irrelevant and the tables present g = g X. Remark 8.1 (The excluded case ν = (4)). Tables 8.1, 8.3, 9.1, and 9.2 exclude the case C 1 = C 2 to stay in the context of ν = (3, 1). The excluded cases (G, C 1, (4)) are interesting too and we mention one of them. For h = (A 5, (311), (4)), the cover Xh + is given

56 52 Hurwitz Belyi maps by f 5,3,1 (j, x) from (8.1) while Xh is empty. This h is our first of three illustrations of a general theorem of Serre [23] as follows. Consider Hurwitz parameters h = (A n, (e 1 1 n e 1,..., e k 1 n e k ), (ν 1,..., ν k )) with all e i odd, so that one has a lifting invariant and thus an equation Xh = X + h X h. Suppose ν i (e i 1) = 2n 2 so that the genus g Y is 0. Then the general theorem says, { { (8.10) If e ν i 8 ±1 X i then Xh + = h ±3 Xh Table 8.1 shows that Xh is empty for (5a, 311) and (5a, 5b) as well, even though g Y > 0 and so Serre s theorem does not apply in these cases. Remark 8.2 (A conjugate pair of irrational covers). Covers not defined over Q arise naturally in our situation, and the left half of Table 8.3 refers to two pairs of irrational covers. Letting s = ± 6, equations for the smaller degree pair are (8.11) f 6,5,4,1 (j, x) ( = (3x + s 3) 225x x 4 s x x 3 s 22662x x 2 s x xs 69741x s jx 5 (5x 9) 4 (53236s )( 15x + 76s + 186) The simple group A 6 = Sp4 (2) = P SL 2 (9). In terms of both its Schur multiplier H 2 = Z/6 and its outer automorphism group A = (Z/2) 2, the group T = A 6 is the most complicated group on Table 7.1. Table 8.2 gives information on conjugacy classes. ) 3 H 2 (A 6 ) = 6 H 2 (S 6 ) = 2 Out(A 6 ) = 2 2 Out(S 6 ) = a 5b H 2 (P GL 2 (9)) = 3 H 2 (M 10 ) = 3 Out(P GL 2 (9)) = 2 Out(M 10 ) = A 811B 10A 10B C 82D Table 8.2. Information on conjugacy classes in A 6 and conjugacy classes not in A 6 of its three overgroups S 6 = A 6.2 1, P GL 2 (9) = A 6.2 2, and M 10 = A The last row gives the number of classes in G mapping to the given class in G. Conventions about the (C 1, C 2 ) entry in the left half of Table 8.3 have been given in 8.1 whenever H 2 (G, C) {1, 2}. The remaining possibilities are as follows. Three entries in a single row separated by semicolons means H h = 3 and Out(G, C) acts trivially on H h, so that Hh = 3 as well. This possibility arises three times, always in the form (a; b; b). By the typeface convention of 8.1, this means that the degree a component is rational and

57 D. Roberts 53 C 1 \C b a; a ; ; ; ; 40 9; 45 51a 60; ; 40 0; 42 C 1 \C a C 1 \C B 10B b 811A 96a 96b A 4, , C 1 \C D ;672 ;672 82C 164;168 ;168 66; 90 ; 90 # M g β 0 β 1 β Eqn. 2+ A (7.1) 1+ A (8.6) 2+ A (8.1) 3+ S 10a (8.2) 1+ S 10b (8.4) 2+ S (8.3) 1+ S (8.5) 2+ A (8.7) 3 S A A (8.8) 2 S S S A A A 96a G 96b A A A A A A pair defined over Q( 10) 1 A Table 8.3. Left: Degrees of components of Hurwitz Belyi covers with parameters (G, (C 1, C 2 ), (3, 1)) with G = A 6, S 6, P GL 2 (9), or M 10. Right: further information on these covers. the degree b components are conjugate. Two entries in a single row separated by semicolons means H h = 3 but Out(G, C) acts nontrivially on H h, so that Hh = 2. This possibility also arises three times, always in the form (c; d). Here both components are rational, as indeed in these three cases c d. Instances of these two situations were described already in 6.4 and 6.3, where degrees were (a; b; b) = (0; 21; 21) and (c, d) = (30; 40) respectively. In these situations, one generally expects a b and c d/2. In the case (5a, 5b), one has H h = Z/6 and Out(G, C) has order two. The non-trivial element of Out(G, C) acts by negation, so that Hh has order four. The natural action of Gal(Q/Q) on H h is trivial, and one would generally expect four rational components. In this case, the natural map π 0 (Xh ) H h is injective but not surjective, and X h has only three components. The cases (42, 51a) and (51b, 42) are similar to (5a, 5b) but now all components are defined

58 54 Hurwitz Belyi maps over Q( 10). The two degree 24 components have their dessin drawn in the website associated to [1]. A blank in the (C 1, C 2 ) slot means that covers belonging to this slot are isomorphic to those of (C1 α, Cα 2 ) for some α in Out(G) Out(G, C). For example the (411, 6) slot is left blank because the cover is the same as that represented by the (411, 321) slot. It is this non-triviality of Out(G) Out(G, C) that makes some of the covers involving 51a and/or 51b rational, even though the classes 51a and 51b are conjugate to each other. Among the further things to note on Table 8.3 are two isomorphic unforced decompositions of the form 46 = Also the cover 96b is unexpectedly nonfull. Finally, a second instance of Serre s theorem (8.10) is (C 1, C 2 ) = (3111, 51a), so that Xh is forced to be empty The simple group W (E 6 ) + = P Sp 4 (3) = P SU 4 (2). The group W (E 6 ) = W (E 6 ) +.2 has twenty-five conjugacy classes. As for all Weyl groups, all the classes are rational. C 1 C 2 M g β 0 β 1 β Eqn. Full low degree covers 3d 4a A (8.6) 3C 9A A (8.1) S (8.12) S (8.2) S (8.3) 6a 4a A (8.7) 6a 2b A (8.13) 3c 9a S (8.14) 4a 2b S (8.15) Agreement with covers coming from A 6 4a 3c1 A (8.8) 6e 6a A a 6a A Large degree genus zero examples of spin separation 3d 5a A d 5a S d 9b S d 9b S An example where all allowed bases appear in β 4D 6G S Table 8.4. Invariants of some covers with G = W (E 6 ) + or W (E 6 )

59 D. Roberts 55 Ten classes are in W (E 6 ) W (E 6 ) + and ten classes stay rational classes in W (E 6 ) +. The remaining five conjugacy classes of W (E 6 ), namely 3ab, 6ab, 6cd, 9ab, and 12ab, split into two classes in W (E 6 ) +. If we were presenting complete tables for ν = (1, 1, 1, 1), there would thousands of lines. Even complete tables for (3, 1) would have hundreds of lines. Accordingly, Table 8.4 presents just some of Hurwitz Belyi covers in a self-explanatory format. One of the new covers has the remarkably small degree nine: (8.12) f 5,4 (j, x) = 5 2 ( 10x x x 100) jx 4. The other three new covers are ( ) 3 (8.13) f 9,6,5,4 (j, x) = 9x 8 72x x 6 104x 5 26x 4 568x x x+729 (8.14) (8.15) j(x 3) 4 x 6 (2x 3) 5, ( f 5 4,4,3(j, x) = 1024x x x x x x 4 f 10,9,5,2 2(j, x) = x x x ( 5 54j 72x 4 508x x x + 768) x 3, ( 3125x x x x x x x x x ) 3 (x 3) jx 9 (5x 6) 5 ( 3x 2 + 2x + 3) 2. In the entire table for T = W (E 6 ) +, there are only twelve integers which can appear as parts for β. The last line of Table 8.4 gives the smallest degree cover where all these integers actually appear. ) 3 9. Hurwitz Belyi maps with G = 2 a 3 b 7 and ν = (3, 1) This section is very parallel in structure to the previous one, and presents a systematic collection of Hurwitz Belyi maps having bad reduction at exactly {2, 3, 7}. Cross-group agreement. Again we present equations for covers involved in cross-group agreement before getting to the individual groups. Now we have only two: (9.1) (9.2) ( 3 f 4,3 (j, x) = 4(x 12) 9x 2 20x 27) jx 3, ( f 7,4,3 2,1(j, x) = 9x 6 126x x 3 63x 2 252x j(3x 2) 4 ( 3x 2 9x + 7) 3 (3x + 14). The cover f 7,4,3 2,1(j, x) was first found by Malle [12] in connection with the group P GL 2 (7). ) 3

60 56 Hurwitz Belyi maps C 1 \C 2 2a 3a 4a 7b 2a a a 18b a 4 18a 16 4 C 1 \C 2 2B 6A 2B 18a 6A 70 # M g β 0 β 1 β Eqn. 1+ A (7.1) 1+ S (9.1) 2+ S 18a (9.2) 1 S 18b (9.3) 1 S A S Two pairs defined over Q( 2) 1 A (9.4) 1 A Table 9.1. Left: Degrees of components of Hurwitz Belyi covers with parameters (G, (C 1, C 2 ), (3, 1)) and G {P SL 2 (7), P GL 2 (7)}. Right: further information on the covers The simple group P SL 2 (7) = SL 3 (2). Equations for the first three Hurwitz Belyi maps have been given already. For the fourth, an equation is ( ) 3 (9.3) f 7,6,3,1 2(j, x) = 9x 6 102x x 4 212x x x + 9 ( ) 2 14 jx 6 (2x 3) 3 9x 2 66x 7. Remark 9.1 (A conjugate pair of irrational covers). The left half of Table 9.1 refers to four pairs of irrational covers. Letting s = ± 2, equations for the smallest degree pair are ( (9.4) f 7,4,3,2 (j, x) = ( 7x + 19s + 27) 49x 5 + x 4 (217s 63) + x 3 (332s 478) + x 2 (154s 658) + x(196s + 147) + 441s jx 4 (35123s )( 2x + s 4) 2 (7s 4x) The simple group SL 2 (8). The group T = SL 2 (8) has outer automorphism group A of order 3. All the corresponding Hurwitz parameters h satisfying the conditions of 7.1 have G = T, as those of the form (T.3, (C 1, C 2 ), (3, 1)) have at least Q( 3) in their field of definition and hence break the rationality restriction in 7.1. Since the Schur multiplier of SL 2 (8) is trivial, there is no spin separation. However Table 9.2 exhibits so many Galois degeneracies that it seems likely that at least some of them are forced by deeper reasons. We describe some of these degeneracies here. Our conventions follow the Atlas: if g 7a, then g 2 7b and g 4 7c; similarly, if g 9a, then g 2 9b and g 4 9c. For the case h = (SL 2 (8), (7b, 7c), (3, 1)), the mass and degree from the mass formula [22, (3.6)] are m = m = 97. A braid calculation gives two degeneracies; first there are two orbits, of size 7 and 90 respectively. Second, the monodromy group for the degree 90 orbit is ) 3

61 D. Roberts 57 C 1 \C 2 2a 3a 7c 9c 2a 18a a a 84 18b, 63 4, b 7, 90 9a 4, 36 4, b 33 # M g β 0 β 1 β Eqn. 3+ A (7.1) 1+ S (9.1) 1 A (9.5) 1 S 18a (9.6) 1+ S 18b (9.2) 3 A (9.7) 2 A A S S S A G (9.8) Table 9.2. Left: Degrees of components of Hurwitz Belyi covers with parameters (SL 2 (8), (C 1, C 2 ), (3, 1)). Right: further information on these covers. m g i M g 1 7b g 2 7b g 3 7b g 4 7c S 7 ( ) ( ) ( ) ( ) G 90 ( ) ( ) ( ) ( ) m g i M g 1 7a g 2 7a g 3 7a g 4 7c A 4 ( ) ( ) ( ) ( ) A 84 ( ) ( ) ( ) ( ) 9 56 SL 2 (8) ( ) ( ) ( ) ( ) 9 56 SL 2 (8) ( ) ( ) ( ) ( ) 1/7 7 S 1 ( ) ( ) ( ) ( ) Table 9.3. Top: Representatives for braid orbits of HBr 3,1 on F h, for h = (SL 2 (8), (7a, 7c), (3, 1)). Bottom: Representatives for braid orbits of HBr 3,1 on F h for h = (SL 2 (8), (7b, 7c), (3, 1)), followed representatives of three degenerate orbits imprimitive, with image inside the wreath produce S 3 S 30. Representatives in G h of the two braid orbits on F h are given in the top part of Table 9.3. The case h = (SL 2 (8), (7a, 7c), (3, 1)), the mass is m = and the degree is m = 88. The degree decomposes, m = 4+84, and the degenerate piece decomposes as well, m m = The two components with g i = 56 have the same monodromy group SL 2 (8), with the rigid braid partition triple (3 3, 2 4 1, ). Representatives in G 4 are given in the bottom part of Table 9.3 for all five orbits. Note that the representative of the orbit with mass 1/7 has

62 58 Hurwitz Belyi maps the very simple form (g, g, g, g 4 ). All the braid computations in this paper involve r-tuples of permutations like the ones exhibited in Table 9.3. Given how differently behaved the last two parameters are, one might expect that the parameters (SL 2 (8), (9a, 9c), (3, 1)) and (SL 2 (8), (9b, 9c), (3, 1)) would be differently behaved as well. However here the Galois degeneracy is in the other direction: not only is m = m = 33 in each case, but the two degree 33 covers are isomorphic. Moreover, these covers are also isomorphic to the cover arising from (SL 2 (8), (3a, 9c), (3, 1)). It is because of the 3-element outer automorphism group that the covers considered above are all rational, despite the fact that 7a, 7b, 7c and 9a, 9b, 9c are defined only over the cyclic cubic fields Q(cos(2π/7)) and Q(cos(2π/9)) respectively. In contrast, the three-element group Out(SL 2 (8)) is not large enough to make the covers indexed by (SL 2 (8), (9a, 7c), (3, 1)) and (SL 2 (8), (7a, 9c), (3, 1)) rational. They are each defined over a cyclic cubic field ramified at both 7 and 9. As reported by Table 9.2, their degrees are 49 and 81 respectively. Like most of the covers in the upper right of Table 9.2, they are full of genus zero. Equations for three covers coming only from SL 2 (8) are (9.5) (9.6) (9.7) ( ) 3 f 9,7 (j, x) = 441x x x 2 140x + 49 ( ) 343x x x 2 468x jx 7, f 9,7,2 (j, x) = (7 4 x 5 441x x x x + 3 7) 3 ( ) 49x 2 + 6x + 9 (x + 3) jx 9 (x 1) 2, ( f 9,7 3,1 3(j, x) = 16x x x x x x x x x x x 1536 ( j(x 1)(x + 2)(x + 8) 8x x 2 9x 8). The cover f 9,7,2 (j, x) was found by Hallouin [3]. For h = (SL 2 (8), (7a, 7b), (3, 1)), an equation for the degree thirty intermediate cover is ( (9.8) f 9,7 2,4,3,1 2(j, x) = 11664x x x x x x x x x x ( 7 ( ) j 8x x + 9) x 4 (x 3) 3x 2 + 6x + 4. ) 3 ) The simple group G 2 (2) = P SU 3 (3). In parallel with the 8.3, the third simple group of order 2 a 3 b 7 is substantially larger than the first and second group. Again we present only some sample Hurwitz Belyi maps, following the format used in 8.3. The first block on Table 9.4 represents cases where the Hurwitz Belyi map has degree 1 and hence is uninteresting in the present context. These three rigid cases are closely related and are studied in detail in [17], starting from Proposition 3.1 there. These three cases serve as a reminder that non-trivial Hurwitz Belyi maps measure a failure of rigidity.

63 D. Roberts 59 C 1 C 2 M g β 0 β 1 β Eqn. Degree one covers corresponding to rigid Hurwitz parameters 3a 4a S x j 4a 4b S x j 4a 2a S x j Genus zero covers of small degree 4a 6a G (7.1) 4c 2a S (9.1) 4a 3b G (7.2) 4c 3a S (9.2) 4D 2B A (9.9) 2B 4D A (9.10) An unforced splitting to two full covers 6a 4b S a 4b S An example where all allowed bases appear in β 8b 2a S Table 9.4. Invariants of some covers with G = G 2 (2) or G 2(2) The last two genus zero covers on Table 9.4 come only from T = P SU 3 (3). Equations are ( ) 3 (9.9) f 8,7,6,3 (j, x) = 4 4x x 6 60x 5 166x x x x ( ) (2x 1) 2x x 49 (9.10) jx 7 (x 2) 6 (x + 4) 3, ( f 12,8 2,7,3,2(j, x) = 64x x x x x x x x x x x x ( ) 4x 4 20x x 2 92x + 49 ( j 2x 2 4x + 3) x 7 (x 2) 3 (x + 1) 2. ) Some 5-point Hurwitz Belyi maps All the explicit Hurwitz Belyi maps presented in the paper so far have had ramification number r = 4. This section presents some examples with r = 5, as a first indication of how things look when r increases.

64 60 Hurwitz Belyi maps A Belyi pencil for ν = (4, 1) yielding 3-2- maps. Sections 7 9 built many Hurwitz Belyi maps from the single Belyi pencil u 3,1 into Conf 3,1. This pencil has the remarkable property that it produces braid permutation triples (b 0, b 1, b ) in S m with b 0 and b 1 of order 3 and 2 respectively. This property kept genera very low in 7-9. Abbreviating k = j 1, let (10.1) s(j, t) = k 2 t 4 6jkt 2 8jkt 3j 2. Define u : P 1 j {0, 1, } Conf 4,1 by j (D 1 (j), { }), with D 1 (j) P 1 t the roots of s(j, t). Let (10.2) B 0 = σ 1 σ 2 σ 2 3, B 1 = (σ 1 σ 2 σ 3 ) 2. A braid calculation says that the abstract braid triple of the Belyi pencil u is (B 0,B 1,B1 1 B 1 0 ), and that B 0 and B 1 likewise have orders 3 and 2 in HBr 4,1 respectively. Two Hurwitz Belyi maps built from u are considered in [20]. First, for h = (S 5, (2111, 5), (4, 1)) the Hurwitz Belyi map π h,u is full and an equation is given in 4.1 there. This Hurwitz Belyi map reappears in Table 10.1 here. For h = (SL 3 (2), (22111, 421), (4, 1)) the degree is 192. After quotienting by the natural action of Out(SL 3 (2)), one gets a full degree 96 map with equation given in [20, 8.2] A table of 3-2- maps from T = A 5. We begin with the smallest nonabelian simple group T = A 5 and build our Hurwitz parameters from G {A 5, S 5 }. Table 10.1 gives all Hurwitz Belyi maps πh,u : X h,u P1 j with h = (G, (C 1, C 2 ), (4, 1)) and u the Belyi pencil (10.1). The complications described at the end of 3.3 arising in the passage from (3, 1) to (4) do not arise when one passes from (4, 1) to (5). Accordingly, Table 10.1 also includes cases of the form h = (G, (C 1 ), (5)), written on the table as h = (G, (C 1, C 1 ), (4, 1)). Otherwise, Table 10.1 has a format very similar to the first two tables in each of Sections 8 and 9. There is one instance of cross-parameter agreement: the Belyi map for (A 5, (5a), (5)) and (A 5, (221), (5)) are isomorphic; this Belyi maps occurs for a third time in the next section, where we get an equation for it. Spin separation is near generic as follows. If (C 1, C 2 ) contains either 221 or 2111, then the Belyi cover Xh,u is always connected. Otherwise both C 1 and C 2 split in the Schur double cover and one has the spin separation X h,u = X + h,u cases X ɛ h,u X h,u. In all has one component except that X + h,u is empty for (C 1, C 2 ) = (5a, 5a) and X h,u has two components for (C 1, C 2 ) = (5a, 5b). Several patterns in Table 10.1 merit comments. First, the first two monodromy groups under the header M are odd, being S 3 and H 9a = 9T 13. However, from the left half of Table 10.1, they arise together as an even intransitive dodecic group. With this packaging, all monodromy groups are even, including H 9b = 9T 11. Second, just like in all the tables in the previous two sections, the exponent on 1 in β 0 is always very small; however, in contrast to these previous tables, the exponent on 1 in β 1 is not always small. Finally, a phenomenon present in the tables of the previous two sections is more visible here because of the different organization: the general nature of β depends on whether G is A 5 or S 5.

65 D. Roberts 61 C 1 \C b a 64 9b b, 9a 5b 0 96 C 1 \C M g X β 0 β 1 β S 3b H 9a H 9b A A A A A A A A A A A A A A A A A A A A A A Table Invariants of π h,u for h = (G, (C 1, C 2 ), (4, 1)) and u the fivepoint pencil (10.1). Top: G = A 5 Bottom: G = S Two unexpectedly similar 3-2- maps built from T = A 6. Consider the two Hurwitz parameters on the left: h 96 = (A 6, (3111), (5)), (β 0, β 1, β ) = (3 36, , ), h 192 = (A 6, (3111, 2211), (4, 1)), (β 0, β 1, β ) = (3 64, , ). These cases are amenable to a standard calculation because the five-point covers Y x P 1 all have genus zero. A mass formula calculation says that the two parameters have their indicated degrees.

66 62 Hurwitz Belyi maps Figure Dessins in X h,u for h = (A 6, (3111), (5)) on the left and h = (A 6, (3111, 2211), (4, 1)) on the right, illustrating the common locations of the three 9 s and the three 15 s. The real axis runs vertically through the center of both pictures. Since the Y x have genus zero, Serre s theorem (8.10) applies and the degree 96 cover X96 := Xh 96,u does not exhibit spin separation, as X 96 = X 96. The braid monodromy computation using (10.2) shows that in fact the monodromy group is A 96. The braid partition triple is as

67 D. Roberts 63 indicated above, and so X96 also has genus zero. The standard computation eventually yields f 15 3,9 3,5,3 6,1(j, x) ( ) 3 = 3x 8 6x 7 60x x 5 110x 4 74x 3 52x 2 10x 1 ( 729x x x x x x x x x x x x x x x x x x x x x x 3 ) x x 3 [ ] 15 [ ] j 3x 3 7x x 1 3x 3 9x 2 + 3x + 1 [x 3] 5 ( 3 x 4 + 8x 3 36x x + 1) [x 1] 3 [x]. The case h = h 192 has monodromy group A 192, braid partition triple as above, and genus zero. There is a remarkable and unexpected similarity between the coefficients of j in the two defining equations, symbolized by 15 3 A 9 3 B A B Here the subscripts 3, 1, and 0 indicate that we are normalizing so that the coordinates induced on X96 and X 192 have some similarity. The unexpected similarity is that the cubic polynomials corresponding to the two A s coincide and likewise the cubic polynomials corresponding to the two B s coincide. All these agreeing factors are bracketed in the two displayed polynomials. The second polynomial is too large to print, but an excerpt containing the part relevant for the current discussion is f 15 3,12 5,9 3,6 8,5,4,3(j, x) ( = x x x x x x x x 64 [ ] 15 ( ) j 3x 3 7x x 1 6x 5 36x x 3 64x x 4 [ ] 9 3x 3 9x 2 + 3x + 1 ( ) 6 9x 8 72x x 6 444x x 4 280x x 2 12x + 1 [x 3] 5 [x] 4 [x 1] 3. Our situation presents many challenges. For example, we have not worked out equations for the four covers of largest degree on Table 10.1 with g X = 0. From the degrees given in the table, 45, 96, 126, and 135, the last three are certainly beyond current implementations of the braid-triple method. However, if some part of these equations could be determined ahead ) 3

68 64 Hurwitz Belyi maps of time, perhaps by understanding better how parts of f 96 (j, x) repeat in f 192 (j, x) as just discussed, these computations might be brought into the range of feasibility. As a second example of a challenge, it would be interesting to build analogs of Table 10.1 both for other simple groups T and other Belyi pencils u. The braid monodromy programs described in [9] would allow one to go quite far. For example, consider the Hurwitz parameter h = (S 6, (6, 51), (4, 1)). Both classes split in the double cover S 6, so one has a decomposition X h,u = X + h,u X h,u. The mass formula applied to the group S 6 says that the degrees are and respectively. Magaard has verified that indeed both X ɛ h,u are full over P1 j, with monodromy groups A and A Expectations in large degree In [22] with Venkatesh and then in the sequel [20], we formulated and supported an unboundedness conjecture for number fields. This final section transposes these considerations from number fields to Belyi maps, with emphasis on phenomena particular to the Belyi map setting Full Belyi maps with at most two bad primes. Consider Belyi maps defined over Q with bad reduction within a given set of primes P. For any prime p and any exponent k, it is elementary to get 3 k different degree p k such Belyi maps P 1 P 1 with monodromy group a p-group and bad reduction set {p} [16]. For any two distinct primes p, l and certain k, mod l-reductions of hypergeometric monodromy representations give degree (l 2k 1)/(l 1) Belyi maps with primitive monodromy group P Sp 2k (l) and bad reduction set {p, l}. Fixing {p, l}, the number of such Belyi maps for a given k can be arbitrarily large. In contrast, it seems very difficult to construct full Belyi maps defined over Q with bad reduction within a two-element set P. Returning to the inverse problem of 1.3, write B P (m) for the number of isomorphism classes of full Belyi maps defined over Q with bad reduction within P. If π contributes to B P (m), then typically the compositions σ π for σ t 1 t, t 1/t = Sym({0, 1, }) all contribute separately, so in a sense the numbers B P (m) are inflated by a factor of six. However the B P (m) enter the unboundedness conjecture below only in a qualitative way, and so this duplication is not important to us. To provide context for the unboundedness conjecture and support the discussion afterwards, we summarize here what we know about the numbers B P (m) for P 2. The trinomial equation y k kty + (k 1)t = 0 gives a Belyi map ramified exactly at the set P k of prime divisors of k(k 1). Thus, as an interesting example, P 9 = {2, 3}. Otherwise one has only the possibilities involving Mersenne primes M r = 2 r 1 and the Fermat primes F r = 2 2r + 1, namely (k 1, k) = (M r, 2 r ) and (k 1, k) = (2 2r, F r ). In [15], we are giving two more sequences of covers T k 1,k and U k 1,k, also ramified exactly at P k. Degrees are now larger, being k(k 1)/2 and (k 1) 2 respectively. Our initial degree 64 example (1.1) is U 8,9. From [14] we know also that B {2,3} (m) is positive for m {28, 33}. Otherwise we do not currently know of any instances with P 2 and m 20 with B P (m) positive beyond the three sequences just described An unboundedness conjecture. The following conjecture is a direct analog of Conjecture 1.1 of [20]:

69 D. Roberts 65 Conjecture Let B P (m) be the number of full degree m Belyi maps defined over Q with bad reduction within P. Suppose that P contains the set of primes dividing the order of a finite nonabelian simple group. Then the numbers B P (m) can be arbitrarily large. Our heuristic argument for Conjecture (11.1) is essentially the same as the argument made in [22] and [20] for its number field analog. Namely we expect that Hurwitz Belyi maps π h,u already give enough maps to make B P (m) arbitrarily large. In more detail, given P as in the conjecture, there is at least one nonabelian finite simple group T with P T P. From Hurwitz parameters h = (G, C, ν), with G of the form T k.a as in [22, 5.1], supplemented if necessary by rational lifting invariants l, there are infinitely many full covers Hur l h Conf ν defined over Q with bad reduction within P. From [18, 8] or [16], there are infinitely many appropriately matching rational Belyi pencils, even with bad reduction set consisting of a single prime. For Conjecture 11.1 to be false, there would be have to be a systematic drop from fullness when one specializes from the full family to the Belyi pencil. We have seen occasional drops from fullness in [20, 6] and on some of the tables in 8-10 here. However these seem to represent a low degree phenomenon, and there is no evidence of systematic drops in asymptotically large degrees. We have already noted an important difference between Hurwitz number fields and Hurwitz Belyi maps in 3.2. Namely for the former, the specialization step is arithmetic, as the ground field becomes Q, but for the latter, the specialization step stays within geometry, as the ground field becomes only C(v). In particular, it seems to us that Conjecture 11.1 is more within reach than its analog, as it may be possible to prove it using braid groups Complements. To conclude very speculatively, say that P is anabelian if it contains the set of primes dividing the order of a finite nonabelian simple group, and abelian otherwise. This terminology seems appropriate to us because we suspect that there are connections between the material in this paper and investigations into anabelian geometry as defined in [2]. Conjecture 11.1 gives a partial qualitative response to the inverse problem set up in 1.3. One could ask for a more complete qualitative response. A guess we find attractive is If P is abelian, then B P (m) is eventually zero. If P is anabelian, then B P (m) is unbounded because of Hurwitz Belyi maps, but still zero for m in a set of density one. We put forward the analogous guess for number fields in [20, 4.6]. The first bulleted statement is supported by the extreme paucity of known Belyi maps contributing to B {p,l} (m), as reported in The second part of the second bulleted statement is motivated by the exponential dependence of the asymptotic mass formula [22, (3-7)] on the multiplicities ν i. Evidence either supporting or opposing this vision would be most welcome. References [1] F. Beukers & H. Montanus, Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps, in Number theory and polynomials, London Mathematical Society Lecture Note Series, vol. 352, Cambridge University Press, 2008, p

70 66 Hurwitz Belyi maps [2] A. Grothendieck, Esquisse d un programme, in Geometric Galois actions, 1, London Mathematical Society Lecture Note Series, vol. 242, Cambridge University Press, 1997, With an English translation on pp , p [3] E. Hallouin, Study and computation of a Hurwitz space and totally real PSL 2 (F 8 )-extensions of Q, J. Algebra 292 (2005), no. 1, p [4] A. James, K. Magaard & S. Shpectorov, The lift invariant distinguishes components of Hurwitz spaces for A 5, Proc. Am. Math. Soc. 143 (2015), no. 4, p [5] G. A. Jones & A. K. Zvonkin, Orbits of braid groups on cacti, Mosc. Math. J. 2 (2002), no. 1, p [6] M. Klug, M. Musty, S. Schiavone & J. Voight, Numerical calculation of three-point branched covers of the projective line, LMS J. Comput. Math. 17 (2014), no. 1, p [7] S. Krämer, Numerical calculation of automorphic functions for finite index subgroups of triangle groups, PhD Thesis, Universität Bonn (Germany), 2015, available from uni-bonn.de/2015/4103/4103.htm. [8] S. K. Lando & A. K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer, 2004, With an appendix by Don B. Zagier, Low- Dimensional Topology, II, xvi+455 pages. [9] K. Magaard, S. Shpectorov & H. Völklein, A GAP package for braid orbit computation and applications, Exp. Math. 12 (2003), no. 4, p [10] G. Malle, Polynomials with Galois groups Aut(M 22 ), M 22, and PSL 3 (F 4 ) 2 2 over Q, Math. Comp. 51 (1988), no. 184, p [11], Fields of definition of some three point ramified field extensions, in The Grothendieck theory of dessins d enfants (Luminy, 1993), London Mathematical Society Lecture Note Series, vol. 200, Cambridge University Press, 1994, p [12], Multi-parameter polynomials with given Galois group, J. Symb. Comput. 30 (2000), no. 6, p [13] G. Malle & B. H. Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer, 1999, xvi+436 pages. [14] G. Malle & D. P. Roberts, Number fields with discriminant ±2 a 3 b and Galois group A n or S n, LMS J. Comput. Math. 8 (2005), p [15] D. P. Roberts, Chebyshev covers and exceptional number fields, in preparation. [16], Fractalized cyclotomic polynomials, Proc. Am. Math. Soc. 135 (2007), no. 7, p [17], Division polynomials with Galois group SU 3 (3).2 = G 2 (2), in Advances in the theory of numbers, Fields Inst. Commun., vol. 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, p [18], Polynomials with prescribed bad primes, Int. J. Number Theory 11 (2015), no. 4, p [19], Lightly ramified number fields with Galois group S.M 12.A, J. Théor. Nombres Bordx 28 (2016), no. 2, p [20], Hurwitz number fields, New York J. Math. 23 (2017), p

71 D. Roberts 67 [21], A three-parameter clan of Hurwitz Belyi maps, Publ. Math. Besançon, Algèbre Théorie Nombres 6 (2018), p [22] D. P. Roberts & A. Venkatesh, Hurwitz monodromy and full number fields, Algebra Number Theory 9 (2015), no. 3, p [23] J.-P. Serre, Relèvements dans Ãn, C. R. Math. Acad. Sci. Paris 311 (1990), no. 8, p [24] J. Sijsling & J. Voight, On computing Belyi maps, Publ. Math. Besançon, Algèbre Théorie Nombres 1 (2014), no. 1, p [25] L. Zhang, G. Chen, S. Chen & X. Liu, Notes on finite simple groups whose orders have three or four prime divisors, J. Algebra Appl. 8 (2009), no. 3, p David P. Roberts, Division of Science and Mathematics, University of Minnesota Morris, Morris, Minnesota, 56267, USA roberts@morris.umn.edu Url :

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73 , A THREE-PARAMETER CLAN OF HURWITZ BELYI MAPS by David P. Roberts Abstract. We study a collection of Hurwitz Belyi maps depending on three integer parameters, finding formulas uniform in the parameters. Résumé. (Une famille d applications d Hurwitz Belyi à trois paramètres) Nous étudions une certaine collection d applications d Hurwitz Belyi dépendant de trois paramètres avec l obtention de formules uniformes. 1. Introduction This paper is a companion to Hurwitz Belyi maps [6]. It makes use of some of the terminology and notation set up in the first three sections there, but is otherwise self-contained. The main fact a reader familiar with Belyi maps has to know from [6] is that Hurwitz Belyi maps are a particularly interesting type of Belyi map that arise in moduli problems. A well-known phenomenon is that certain infinite collections of Belyi maps can be profitably studied simultaneously by means of parameters. We informally refer to such a collection as a clan. For example, the recent papers [4],[5] study Belyi maps which are uniquely determined by partition triples (λ 0, λ 1, λ ), with λ of the form (m, 1,..., 1). These papers find ten clans and ten sporadic examples. Twenty years ago in [1], Couveignes found what in our language we call a four-parameter clan of Hurwitz Belyi maps. To our knowledge, it is the only such clan systematically studied in the literature. However we are convinced from preliminary computations that there are many other natural clans of Hurwitz Belyi maps. Our purpose in this paper is to call attention to the mostly unexplored topic of clans of Hurwitz Belyi maps. We aim to indicate the general nature of all clans by studying the Couveignes clan further. Naturally, we focus on aspects of this clan which are not simple consequences of the results of [1] Mathematics Subject Classification. 14H57, 33E99. Key words and phrases. Belyi map, discriminant, monodromy.

74 70 A three-parameter clan of Hurwitz Belyi maps While the study of Belyi maps always has a very number-theoretic feel, the study of clans of Belyi maps also brings in many notions from the theory of special functions. For example, Jacobi polynomials, Padé approximants, and differential equations are all prominent in [5]. Unlike the cases of [5], a Hurwitz Belyi map in a clan is typically not determined by its partition triple. As a consequence, the special functions from Hurwitz Belyi maps can be quite far removed from classical special functions. Section 2 sets the stage by reviewing some of Couveignes results. Sections 3-7 pursue topics that would be natural to study in any clan. Section 3 first reduces to one fewer parameter to simplify this whole paper, defining Hurwitz Belyi maps (1.1) π a,b,c : X a,b,c P 1. of degree m = 3(a + b + 2c). It then gives a uniform algebraic description of these maps. Section 4 describes the monodromy group of π a,b,c in S m, identifying exactly when it is primitive. Section 5 obtains a discriminant formula from which one obtains the exact set of primes at which π a,b,c has bad reduction. Section 6 discusses wall-crossing phenomena which arise naturally when studying clans. Section 7 compares the Hurwitz Belyi maps (1.1) with other Belyi maps sharing the same ramification partitions. Some of the more complicated formulas in this paper are available in the Mathematica file TPC.m on the author s homepage. Acknowledgements. This work was partially supported by the Simons Foundation through grant # and, in its final stages, by the National Science Foundation through grant DMS I thank Kay Magaard for his contribution to Section Couveignes cubical clan In this section we define Couveignes clan, using notation adapted to our context, and present some of his results Direct description. Let a, b, c, and d be distinct positive integers and set n = a+b+c+d. Consider maps F from the complex projective y-line P 1 y to the complex projective t-line P 1 t given by (2.1) t = F (y) = (1 x 1 y) a (1 x 2 y) b (1 x 3 y) c (1 x 4 y) d. Here the x i are currently unspecificed distinct complex numbers. Clearly, the preimage of 0 P 1 t consists of the points 1/x 1, 1/x 2, 1/x 3, 1/x 4 in P 1 y of respective multiplicities a, b, c, and d. Accordingly, the ramification partition for 0 is λ 0 = (a, b, c, d). Likewise, the preimage of P 1 t is simply P 1 y, giving the ramification partition λ = (n). Note that F (0) = 1. Couveignes starting point is to require also F (0) = F (0) = 0. Explicitly this requirement translates to the following elegant conditions on the x i from [1, 5.1]: (2.2) (2.3) ax 1 + bx 2 + cx 3 + dx 4 = 0, ax bx cx dx 2 4 = 0. The point 0 P 1 y, always in the preimage of 1 P t, now has multiplicity 3. For generic (x 1, x 2, x 3, x 4 ) satisfying (2.2) and (2.3), the ramification partition λ 1 is (3, 1,..., 1). There is then a single remaining critical point y crit on P 1 y. It maps to some point v P 1 t, and the ramification partition for v is λ v = (2, 1,..., 1). The isomorphism type of F depends only on

75 D. Roberts 71 (x 1, x 2, x 3, x 4 ) in the projective curve X a,b,c,d defined by (2.2) and (2.3). There is a natural map (2.4) π a,b,c,d : X a,b,c,d P 1 v. Namely a point (x 1, x 2, x 3, x 4 ) X a,b,c,d is taken into its unique extra critical value v Hurwitz parameter. The previous paragraph fits into the formalism of [6, 3.1] as follows. The Hurwitz parameter describing the desired functions F is h(a, b, c, d) = (S n, (2 1 n 2, a b c d, 3 1 n 3, n), (1, 1, 1, 1)). Attention is thoroughly focused at the moment on the local ramification partitions (λ v, λ 0, λ 1, λ ). The multiplicity vector (1, 1, 1, 1) will remain trivial even in our modifications below. While we only consider F if its global monodromy group is all of S n, this condition turns out to be forced by the local ramification partitions. The maps (2.4), arising as they do from a moduli problem, are Hurwitz Belyi maps Braid monodromy and degree. As a Belyi map, π a,b,c,d has its own ramification partitions. We call them β 0, β 1, and β, to distinguish them from the earlier λ t. This notation is used throughout [6], with the letter β being a reminder that partition triples of Hurwitz Belyi maps can always be computed by braids. The braid ramification partitions are deducible from the figure in [1, 5.2], being (2.5) β 0 = (a + b) 2 (a + c) 2 (a + d) 2 (b + c) 2 (b + d) 2 (c + d) 2, β 1 = n 24, β = (a + b + c) 2 (a + b + d) 2 (a + c + d) 2 (b + c + d) 2. In particular the degree of π a,b,c,d is m = 6n. The total number of parts in (2.5) is 12 + (6 + m 24) + 8 = m + 2, and so X a,b,c,d has genus zero. For other clans of Hurwitz Belyi maps, the degree can grow faster than linearly in the parameters, unlike m = 6(a + b + c + d). The genus of the covering curves can be larger than 0, unlike the genus of X a,b,c,d. In these senses, the Couveignes clan is particularly simple. In the current context, it is better to modify the standard visualization conventions of [6, 2.3], to exploit that the number of parts in β 0 and β is small and independent of the parameters. Accordingly, we now view the interval [, 0] in the projective line P 1 v as the simple bipartite graph. The dessin πa,b,c,d 1 ([, 0]) X a,b,c,d, capturing Couveignes determination [1, 5.2 and 9] of the permutation triple (b 0, b 1, b ) underlying (β 0, β 1, β ), is indicated schematically by Figure 2.1. Note that our visualization is dual to that of Couveignes, as our dessin is formatted on a cube rather than an octahedron Failure of rationality. The Q-curve X a,b,c,d underlying the Riemann surface X a,b,c,d = X a,b,c,d (C) is naturally given in the projective space P 3 by the system (2.2), (2.3). The second equation has no solution in P 3 (R) and so X a,b,c,d (R) is empty. This non-splitting of X over R, which forces non-splitting over Q p for an odd number of primes p, is one of the main focal points of [1].

76 72 A three-parameter clan of Hurwitz Belyi maps a c d b b c a d b d a c d b c a c b d a a c d b Figure 2.1. Schematic indication of Couveignes dessin with parameters (a, b, c, d) based on the combinatorics of a cube. The actual dessin is obtained by replacing each u by u parallel edges More general parameters. Note finally that our requirement that a, b, c, and d are all distinct is just so that the above considerations fit immediately into the formalism of Hurwitz Belyi maps. One actually has natural covers X a,b,c,d P 1 of degree 6n even when this requirement is dropped. These covers have extra symmetries, as illustrated by the rotation ι discussed in the next section. Couveignes allowed the parameters to become zero and negative, and we will do the same in Section The semicubical clan Couveignes does not explicitly give the map X a,b,c,d P 1 v. The map cannot be given simply by a rational function in Q(x), because, as just discussed in 2.4, the curve X a,b,c,d is not isomorphic to P 1 over Q. The fact that all multiplicities are even in the triple (2.5) is necessary for this somewhat rare obstruction. In this section, we simplify by modifying the situation so that the covering curves become isomorphic to P 1 over Q. Then we give corresponding rational functions Restriction to three parameters. For our modified clan, we still require that a, b, and c are distinct. But now we essentially set d = c in Couveignes situation, so that the degree takes the asymmetric form n = a + b + 2c. We thus are now considering the Hurwitz parameters (3.1) h(a, b, c) = (S n, (2 1 n 2, a 0 b 1 c 2, 3 x 1 n 3, n ), (1, 1, 1, 1)). The four subscripts are present for the purposes of normalization and coordinatization. They will enter into our proof of Theorem 3.1 below.

77 D. Roberts 73 The fact that two distinguishable points have now become indistinguishable implies that X a,b,c = X a,b,c,c /ι, where ι is the rotation interchanging c and d in Figure 2.1. The fixed points of this rotation are the upper-left and lower-right white vertices, each with valence c+d = 2c. Thus X a,b,c has degree 3n over P 1 v. The new braid partition triple is (3.2) β 0 = (a + b) (a + c) 2 (b + c) 2 c 2, β 1 = n 12, β = (a + b + c) 2 (a + 2c) 0 (b + 2c) 1. There are three singletons, namely the parts subscripted 0, 1, and. So not only is the Q-curve X a,b,c split, but also our choice of subscripts gives it a canonical coordinate. Because of the equation X a,b,c = X a,b,c,c /ι, we call the clan indexed by h(a, b, c) the semicubical clan Explicit rational functions. To compute π a,b,c for given integers a, b, c as an explicit rational function, we follow the standard procedure illustrated by simple examples in Sections 2 and 4 of [6]. Remarkably, this computation can be done for all a, b, and c at once: Theorem 3.1. For distinct positive integers a, b, c, let n = a + b + 2c and A = 2nx(a + c) + (a + c)(a + 2c) + nx 2 (n c), B = a(a + c) 2anx + nx 2 ( (c n)), C = x 2 (a + b)(n c) 2ax(n c) + a(a + c), D = nx 2 (a + b) + a(a + 2c) 2anx. Then the Hurwitz Belyi map for (3.1) is (3.3) π a,b,c (x) = Proof. The polynomial a a b b A a+c B b+c D c 2 c c 2c n n x a+2c (1 x) b+2c C n c. (3.4) F x (y) = ya (y 1) b (y 2 + ry + s) c x a (x 1) b (x 2 + rx + s) c partially conforms to (3.1), including that F x (x) = 1. From the 3 x we need also that F x(x) = 0 and F x (x) = 0. The derivative condition is satisfied exactly when r = nx3 + (a + 2c)x 2 (a + b)sx + as ((a + b + c)x 2. (a + c)x) The second derivative condition is satisfied exactly when s = n(a + b + c)x4 2n(a + c)x 3 + (a + c)(a + 2c)x 2 (a + b)(a + b + c)x 2 2a(a + b + c)x + a(a + c). The identification of r and s completely determines the maps F x : P 1 y P 1 t. From a linear factor in the numerator of F x(y), one gets that the critical point corresponding to the 2 in the first class 2 1 n 2 in (3.1) is y x = as nx 2. Substantially simplifying F x (y x ) gives the right side of (3.3).

78 74 A three-parameter clan of Hurwitz Belyi maps From the form of the normalized partition tuple (3.2), one knows a priori that (3.5) πa,b,c (1 x) = πb,a,c (x). Indeed, one can check that the simultaneous interchange a b, x 1 x interchanges A and B and fixes C and D. Given this fact, the symmetry (3.5) is visible in the main formula (3.3) Dessins. As with Couveignes cubical clan, the semicubical clan gives covers πa,b,c even when a, b, and c are not required to be distinct. For example, the simplest case is the dodecic cover 2 2 6x2 8x + 3 8x2 8x + 3 6x2 4x + 1. π1,1,1 (x) = 28 x3 (x 1)3 (3x2 3x + 1)3 An example representing the main case of distinct parameters is x2 154x x2 14x x2 42x + 11 π7,6,4 (x) = 214 x15 (x 1)14 (221x2 238x + 77) Let γ(a, b, c) = πa,b,c ([, 0]) P1x. The left part of the Figure 3.1 is a view on γ(1, 1, 1). To obtain the general γ(a, b, c) topologically, one replaces each segment of γ(1, 1, 1) by the appropriate number a, b, or c of parallel segments, so as to create m = 3n = 3a + 3b + 6c edges in total. As an example, the right part of the figure draws the degree sixty-three dessin γ(7, 6, 4). Figure 3.1. Left: γ(1, 1, 1). Right: γ(7, 6, 4). The view on γ(a, b, c) given by (3.1) was obtained via the involution s(x) = x/(2x 1) which fixes 0 and 1 and interchanges and 1/2. The sequence of white, black, and white vertices on the real axis are the points 0,, and 1 in the Riemann sphere. Thus the point not in the plane of the paper is the point x = 1/2. The involution (3.5) corresponds to rotating the figure one half-turn about its central point.

79 D. Roberts Monodromy A degree m Belyi map X P 1 has a monodromy group, which is a subgroup of S m welldefined up to conjugation. A natural question is to determine this monodromy group. This section essentially answers this question for members of the semicubical clan, and also for all other Belyi maps sharing the same ramification partition triple Three imprimitive cases. Suppose in general that X is connected, as all our X a,b,c are. Group-theoretically, we are supposing that the monodromy group is transitive. Then a natural first question is whether X P 1 has strictly intermediate covers. As preparation for the next subsection, we exhibit three settings where there is such an intermediate cover (4.1) X a,b,c δ Y ɛ P 1 v. The monodromy group is imprimitive exactly when there exists a Y as in (4.1), and primitive otherwise. Case 1. Let e = gcd(a, b, c). If e > 1 then the explicit formula (3.3) says that (4.2) π a,b,c (x) = π a/e,b/e,c/e (x) e. Thus one has imprimitivity here, with the cover ɛ naturally coordinatized to y y e. Case 2. Suppose a = b. As a special case of (3.5), the cover π a,a,c has the automorphism x 1 x, corresponding to rotating dessins as in Figure 3.1. To coordinatize Y, we introduce the function y = δ(x) = x(1 x). Then ɛ(y) works out to (4.3) π V a,c(y) = (4ay a + 4cy 2c)c ( 4y 2 (2a + c) 2 4ay(2a + 3c) + a(a + 2c) ) a+c c 2c 2 2a+3c y a+2c (4ay a + 2cy c) 2a+c. The superscript V indicates that π V a,c comes from π a,a,c,c by quotienting by a noncyclic group of order four. The ramification partitions and the induced normalization of π V a,c are (4.4) α 0 = a c (a + c) 2, α 1 = n 6, α = (a + 2c) 0, (2a + c). The covers π V a,c and π V c,a are isomorphic, although our choice of normalization obscures this symmetry. Case 3. Suppose c {a, b}. Via (3.5), it suffices to consider the case b = c. Then while the cover π a,c,c does not have any automorphisms, the original cubical cover π a,c,c,c has automorphism group S 3, This implies that π a,c,c has a subcover ɛ = π S a,c of index three. To coordinatize Y in this case, we use the function Then y = δ(x) = (x 1) 3 (a + c)(a + 2c) x (x 2 (a + c)(a + 2c) 2ax(a + 2c) + a(a + c)). (4.5) π S a,c(y) = ( a)a (y 1) a+c ( a 3 y 2 (a + c) + 2a 2 cy(5a + 9c) + 27c 2 (a + c)(a + 2c) ) c 2 c c c n n y c.

80 76 A three-parameter clan of Hurwitz Belyi maps So the ramification partitions and the induced normalization of π S a,c are (4.6) α 0 = (a + c) 1 c 2 α 1 = 4 1 n 4, α = (a + 2c) c Primitivity. The next theorem says in particular that all π a,b,c not falling into Cases 1-3 of the previous subsection have primitive monodromy. Theorem 4.1. Let a, b, c be distinct positive integers with gcd(a, b, c) = 1. Let π : X P 1 be a Belyi map with the same ramification partition triple (3.2) as π a,b,c. Then π has primitive monodromy. Note that the hypothesis gcd(a, b, c) = 1 excludes Case 1 from the previous subsection. The distinctness hypothesis excludes Cases 2 and 3. Since these cases all have imprimitive monodromy, we will have to use these hypotheses. Proof. The hypothesis gcd(a, b, c) = 1 has implications on the parts of β 0 and β in (3.2). For β 0 it implies gcd(a+b, a+c, b+c, c) = 1 while for β it implies gcd(a+b+c, a+2c, b+2c) {1, 3}. As before, let n = a + b + 2c and m = 3n. The two hypotheses together say that the smallest possible degree m is twenty-one, coming from (a, b, c) = (3, 2, 1). Let Y be a strictly intermediate cover as in (4.1), with X a,b,c replaced by X. Let e be the degree of Y P 1 and write its ramification partition triple as (α 0, α 1, α ). Because gcd(a+b, a+c, b+ c, c) = 1, the cover Y cannot be totally ramified over 0. Because gcd(a + b + c, a + 2c, b + 2c) {1, 3}, it can be totally ramified over infinity only if e = 3. There are then two possibilities, as (α 0, α 1, α ) could be ((1, 1, 1), (3), (3)) or ((2, 1), (2, 1), (3)). The α 1 = 3 in the first possibility immediately contradicts β 1 = (4 3, 1 m 12 ). The α 1 = (2, 1) in the second possibility allows two possible forms for β 1, namely (4 3, 1 6 ) and (4 3 ). But both of these have degree less than twenty-one. So e = 3 is not possible, and thus Y cannot be totally ramified over either. Since α 0 and α both have at least two parts, the minimally ramified partition (2, 1 e 2 ) is eliminated as a possibility for α 1, by the Riemann Hurwitz formula. The candidates (2 2, 1 e 4 ) and (2 3, 1 e 6 ) for α 1 both force X to be a double cover of Y, so that e = m/2; but both are then incompatible with β 1 = (4 3, 1 m 12 ). This leaves (4, 2, 1 m/2 6 ) and (4, 1 m/3 4 ) as the only possibilities for α 1. In the first case, the two critical values of the double cover X Y would have to correspond to the 2 in α 1 and the image of the singleton a + b of β 0 ; the parts of β would have to be those of α with multiplicities doubled; from the form of β in (3.2) this forces a = b, putting us in Case 2 and contradicting the distinctness hypothesis. In the second case, the combined partition α 0 α would have the form (k 1, k 2, k 3, k 4, k 5 ) and the combined partition β 0 β would have the form (3k 1, 2k 2, k 2, 2k 3, k 3, k4 3, k3 5 ) or (3k 1, 3k 2, k3 3, k3 4, k3 5 ). From (3.2), the first possibility occurs exactly when a or b equals c, putting us into Case 3 and contradicting the distinctness hypothesis; the second possibility cannot occur as it is incompatible with the shape of β 0 β in (3.2). We have now eliminated all possibilities for α 1 and so X P 1 has to be primitive Fullness. In [6] we heavily emphasized full Belyi maps, meaning maps with monodromy group the alternating or symmetric group on the degree m. A natural question is whether primitive in Theorem 4.1 can be strengthened to full. In the setting of Theorem 4.1, only S m can appear because β 1 = m 12 in (3.2) is an odd partition.

81 D. Roberts 77 The two smallest degrees of covers as in Theorem 4.1 are m = 21 and m = 24. There are nine primitive groups in degree 21 and five in degree 24, all accessible via Magma s database of primitive groups. None of them, besides S 21 and S 24, contain an element of cycle type m 12. In an to the author on July 5, 2016, Kay Magaard has sketched a proof that, for all m 25, likewise m 12 is not a cycle partition for a primitive proper subgroup of S m. Magaard s proof appeals to Theorem 1 of [2], which has as essential hypothesis that the 1 s in m 12 contribute more than half the degree. Special arguments are needed to eliminate the other possibilities that parts 1, 2, and 3 of Theorem 1 of [2] leave open. Thus, Theorem 4.1 can indeed be strengthened by replacing primitive by full. 5. Primes of bad reduction A natural problem for any Belyi map defined over Q is to identify its set P of primes of bad reduction. In this section, we identify this set for the maps π a,b,c Eleven sources of bad reduction. The dessins γ(a, b, c) have four white vertices: 0, 1, and the roots of D. They have seven black vertices: and the roots of ABC. If any of these eleven points agree modulo a prime p, then the map π a,b,c has bad reduction at p. To study bad reduction, one therefore has to consider some special values, discriminants, and resultants. Table 5.1 gives the relevant information, with value at meaning the coefficient of x 2 in the quadratic polynomial heading the column. A B C D Value at 0 (a + c)(a + 2c) a(a + c) a(a + c) a(a + 2c) Value at 1 b(b + c) (b + c)(b + 2c) b(b + c) b(b + 2c) Value at (a + b + c)n (a + b + c)n (a + b)(a + b + c) (a + b)n Disc. 4bc(a + c)n 4ac(b + c)n 4abc(a + b + c) 8abcn Res. with A 4c 3 n 2 e 4b 2 c 3 e b 2 c 2 (a + 2c) 2 n 2 Res. with B 4a 2 c 3 e a 2 c 2 (b + 2c) 2 n 2 Res. with C a 2 b 2 (a + b) 2 c 2 Table 5.1. Special values, discriminants, and resultants of the four quadratic polynomials A, B, C, D from Theorem 3.1, using the abbreviation e = (a + c)(b + c)(a + b + c) A discriminant formula. Combining the explicit formula (3.3), the general discriminant formula [3, (7.14)], and the elementary facts collected in Table 5.1, one gets the following discriminant formula.

82 78 A three-parameter clan of Hurwitz Belyi maps Corollary 5.1. Let a a b b A a+c B b+c D c v2 c c 2c n n x a+2c (1 x) b+2c C n c be the polynomial whose vanishing defines π a,b,c. Its discriminant is D(a, b, c) = ( 1) (a 1)a/2+(b 1)b/2+c 2 n(c+2n) a 2n2 a 2 +2an n b 2n2 b 2 n+2bn c (10n2 (1+a+b)(a+b+3n))/2 (a + b) (a+b+c 1)n (a + c) an+a+cn+c+n2 n (b + c) bn+b+cn+c+n2 n (a + 2c) (a+2c)2 (b + 2c) (b+2c)2 (a + b + c) (a+b+c)(2(a+b+c)+1) n n(3n+2) v 3n 7 (v 1) 9. In particular, the set P a,b,c of bad primes of π a,b,c is the set of primes dividing (5.1) abc(a + b)(a + c)(b + c)(a + b + c)(a + 2c)(b + 2c)(a + b + 2c). Because of the nature of [3, (7.14)], each factor in the discriminant formula has specific sources on Table 5.1. The discriminants corresponding to π V a,c and π S a,c are given by similar but slightly simpler formulas Responsiveness to the inverse problem of [6]. Let a, b, c be distinct positive integers without a common factor. The fullness conclusion of 4.3 and the identified prime set (5.1) combine to say that the explicit rational Hurwitz Belyi maps π a,b,c of Theorem 3.1 respond in a uniform way to the inverse problem of 1.3 of [6]. All the clans we have looked at seem to share the property that the analog to P a,b,c is relatively sparse, but nevertheless grows with the parameters. In [6], we were most interested in fixing a small P and providing examples of full rational covers in degrees as large as possible. In this direction, clans do not seem to be helpful. The fundamental problem is that in clans the groups G in the Hurwitz parameters are A n or S n and one is increasing n. To obtain a sequence establishing Conjecture 11.1 of [6], we expect that instead one has to fix G, and consider moduli problems in which the number of critical values, always four in this paper, tends to infinity. 6. Allowing negative parameters The formula (3.3) for π a,b,c makes sense for arbitrary integer parameters satisfying abcn 0, although individual factors may switch from numerator to denominator or vice versa. We have seen this extendability in other clans as well, and it is typically associated with complicated wall-crossing phenomena. In this section, we discuss some of the extra Hurwitz Belyi maps obtained by allowing negative parameters. As a special case of (4.2), one has the formula π a, b, c (x) = π a,b,c (x) 1. Using this symmetry, we can and will restrict attention to the half-space c Chambers. Assuming none of the quantities in Table 5.1 vanish, the degree N(a, b, c) of π a,b,c is the total of those quantities on the list 2(c n), 2(a + c), 2(b + c), 2c, a 2c, b 2c, and a + b which are positive. This continuous, piecewise-linear function is homogeneous in the parameters a, b, and c, and so it can be understood by its restriction to c = 1 via N(a, b, c) = cn(a/c, b/c, 1). The left half of Figure 6.1 is a contour plot of N(α, β, 1). Thus, for c 4 fixed, the minimum degree for π a,b,c is 4c, occurring for all (a, b, c) with (a/c, b/c) in the middle black triangle.

83 D. Roberts Figure 6.1. Left: the formal degree of π α,β,1, meaning the quantity degree(π αc,βc,c )/c, drawn in the α-β plane. Contours range from 4 (middle triangle) to 24 (upper right corner). Right: The discriminant locus of the semicubical clan, drawn in the same (α, β) plane. Points are particular (a/c, b/c)- values giving covers appearing in Table 7.1. If (a + c)(a + 2c)(b + c)(b + 2c)(n + c)(n + 2c) = 0, then there is a cancellation among at least a pair of factors, and π a,b,c has degree strictly less than N(a, b, c). If abcn = 0 then, taking a limit, π a,b,c is still naturally defined, and again has degree strictly less than N(a, b, c). Taking c = 1, the lines given by the vanishing of the other nine linear factors in the discriminant formula are drawn in the right half of Figure 6.1. The complement of these lines has 31 connected components, called chambers. The middle chamber is the interior of the triangle given by N(a, b, 1) = 4. As indicated by the caption of Figure 6.1, one can also think of the right half of Figure 6.1 projectively. From this viewpoint, the line at infinity is given by the vanishing of the remaining linear form, i.e., by c = 0. Our main reference [1] already illustrates some of this wall-crossing behavior in the context of distinct a, b, c, and d: the dessins with all parameters positive are described as being a chardon, while the dessins with parameters in certain other chambers are described as being a pomme The central chamber and symmetric coordinates. Each chamber corresponds to a different family of Hurwitz parameters, with corresponding rational functions π a,b,c being uniformly given by (3.4). To study the middle chamber, switch to new parameters (u, v, w) = (c + a, c + b, c n). In the new parameters, the quantity c = u + v + w is still convenient, and we will use it regularly as an abbreviation. The middle chamber is given by the positivity of u, v, and w. We indicate the presence of the new parameters by capital letters, changing h to H, π to Π, and γ to Γ. The normalized Hurwitz parameter (3.1) gets replaced by (6.1) H(u, v, w) = (S 2c, (2 1 2c 2, c 2, 3 x 1 2c 3, (c u) 0 (c v) 1 (c w) ), (1, 1, 1, 1)).

84 80 A three-parameter clan of Hurwitz Belyi maps Simply writing factors with the new parameters in different places corresponding to the new signs, (3.3) becomes (6.2) Π u,v,w (x) = ( 1) c w A u B v C w D c 2 c c 2c (c u) c u (c v) c v (c w) c w x c+u (1 x) c+v. The degree is 4c and the partition triple (3.2) changes to (6.3) β 0 = u 2 v 2 w 2 c 2, β 1 = c 12, β = (c + u) 0 (c + v) 1 (c + w). The symmetry (3.5) in the new parameters takes the similar form Π u,v,w (1 x) = Π v,u,w (x). But now the symmetry (6.4) Π u,v,w (1/x) = Π w,v,u (x) is equally visible. In terms of a sheared version of Figure 6.1 in which the central triangle is equilateral, the symmetries just described generate the S 3 consisting of rotations and flips of this triangle. One has quadratic reduction as in (4.3) whenever two of the parameters are equal. One has cubic reduction as in (4.5) whenever one of the parameters is 2c. Figure 6.2. Left: Γ(1, 1, 1). Right: Γ(4, 3, 2). For dessins, we take Γ(u, v, w) = Π 1 u,v,w([, 0]) with [, 0] = as before. Figure 6.2 is then direct analog of Figure 3.1. The black points on the real axis from left to right are, as before 0,, 1. The unique white point above this axis is connected to 0,, and 1 by respectively u, w, and v edges. To pass from Γ(1, 1, 1) to Γ(u, v, w), one replaces each edge by either u, v, or w parallel edges, illustrated by the example of Γ(4, 3, 2) Degenerations. While symmetric parameters are motivated by the central chamber, they are often better for analysis of the entire clan. As an example, we consider an aspect about degenerations over discriminantal lines, always excluding the intersections of these lines. To begin, consider the numerator of the logarithmic derivative Π u,v,w(x)/π u,v,w (x). In conformity with β 1 = m 12, one gets that this numerator is the cube of a cubic polynomial (u, v, w, x). An easy computation gives (6.5) (u, v, w, x) = wx 3 (w c)(w + c) + 3w(u c)(w c)x 2 + 3u(u c)(w c)x u(u c)(u + c).