THE l-primary TORSION CONJECTURE FOR ABELIAN SURFACES WITH REAL MULTIPLICATION

THE l-primary TORSION CONJECTURE FOR ABELIAN SURFACES WITH REAL MULTIPLICATION ANNA CADORET Abstrat. We prove that the Bombieri-Lang onjeture implies the l-primary torsion onjeture for abelian surfaes with real multipliation. 2010 Mathematis Subjet Classifiation. Primary: 11G10; Seondary: 14D10, 14D22. 1. Introdution Let O be the ring of integers of a quadrati real field extension of Q with disriminant D and let X(n) denote the oarse moduli sheme for the stak of polarized abelian surfaes with real multipliation by O and µ n -level struture (See Subsetion 2.2 for preise definitions). The irreduible omponents of the X(n) are normal, separated, geometrially onneted surfaes (defined over a number field depending on the involved data) and it is known that the smallest integer n(d) suh that X(n(D)) is of general type is 1 exept for finitely many ases, where it is 2 or 3 (Theorem 2.2). The Bombieri-Lang onjeture (Conjeture 2.1) predits that if X is a surfae of general type over a number field k then the set of k-rational points X(k) is not Zariski-dense in X. Thus, onjeturally, polarized abelian surfaes with real multipliation by O and µ n(d) -level struture defined over k are parametrized by a losed subsheme S X(n(D)) whose irreduible omponents have dimension 1. For suh a sheme S and any abelian sheme A S, works of A. Tamagawa and the author show that, for any integer d 1 and prime l the l-primary rational torsion is uniformly bounded in the fibres at losed points s S whose residue degree [k(s) : k] is d (Theorem 2.4). So, assuming the Bombieri-Lang onjeture, one an expet that for any number field k and prime l the k-rational l-primary torsion of polarized abelian surfaes with real multipliation by O and µ n(d) -level struture defined over k is uniformly bounded. The only diffiulty stems from the fat that, for n(d) 2, the oarse moduli sheme X(n) is not fine hene there is no universal abelian surfae with real multipliation by O and µ n(d) -level struture over it, to whih one ould apply diretly the uniform boundedness result of A. Tamagawa and the author. However, this an be overome by elementary ohomologial and rigidifiation tehnis. The main result we obtain in this short note is the following. Theorem 1.1. Assume that the Bombieri-Lang onjeture holds. Then for any number field k and prime l there exists an integer N := N(O, k, l) suh that, for any polarized abelian surfae A with real multipliation by O and µ n(d) -level struture defined over k: A(k)[l ] l N. In Setion 2 we gather the several ingredients involved in the proof of Theorem 1.1. The proof itself is arried out in Setion 3. Aknowledgements: I would like to thank Pete L. Clark for suggesting the idea of this note as well as Mar-Hubert Niole, Matthieu Romagny and Akio Tamagawa for their interest and onstrutive omments. 2. Preliminaries Given a sheme S over a field k and an integer d 1, we will write S d := {s S [k(s) : k] d}, where k(s) denotes the residue field of S at s. 1

2 ANNA CADORET 2.1. Bombieri-Lang onjeture. Given a field k of harateristi 0, let P(k) (resp. B(k)) denote the ategory of all shemes smooth, projetive and geometrially onneted over k (resp. of all normal shemes separated, of finite type and geometrially onneted over k). Also, let denote the birational equivalene on P(k) (resp. B(k)), it follows from Nagata s ompatifiation theorem and Hironaka s desingularization theorem that the anonial map P(k)/ B(k)/ is bijetive (e.g. [C10, 2.1]). Thus any birational invariant defined on P(k) naturally extends to B(k); this is in partiular the ase for Kodaira dimension and we say that S B(k) is of general type if its Kodaira dimension κ(s) is equal to its dimension dim(s). With this onvention, we an formulate the so-alled Bombieri-Lang onjeture Conjeture 2.1. Let k be a number field and let S B(k) be a surfae of general type. Then the set of k-rational points S(k) is not Zariski-dense in S. Conjeture 2.1 was stated by E. Bombieri for surfaes and generalized by S. Lang for shemes S B(k) of arbitrary dimension under the following more preise form. Let k be a number field and let S B(k) be of general type. Then there existe a losed subsheme Z S, Z S suh that for any finite field extension k k the set S(k ) \ Z(k ) is finite. The Lang onjeture (and even the Bombieri-Lang onjeture) is widely open. The most striking result is that it holds for subvarieties of abelian varieties [F94]. 2.2. The stak of abelian surfaes with real multipliation. Let Q K be a degree g totally real field extension and let O denote the ring of integers of K. Let C and C + denote the lass group and strit lass group of O, respetively. Fix a system J 1,..., J h + of frational ideals of O, whih, endowed with their natural notion of positivity, form a omplete system of representatives of C +. For J one of the J 1,..., J h +, a g-dimensional J -polarized abelian sheme with real multipliation by O is a triple (A, ι, λ), where: - A S is an abelian sheme of relative dimension g; - ι : O End S (A) is a ring homomorphism; - λ : (M A, M + A ) (J, J + ) is a polarization that is, an O-linear isomorphism of étale sheaves between the invertible O-module M A of all symmetri O-linear homomorphisms from A to A and J, identifying the positive one of polarizations M + A with the totally real elements J + in J. Let Q(J ) denote the field of definition of J and S O,g,J spe(q(j )) the étale stak of g-dimensional J -polarized abelian shemes with real multipliation by O. Set S O,g := S O,g,Ji. 1 i h + One an furthermore endow J -polarized abelian shemes with real multipliation by O with a µ n level-struture, that is an injetive O-linear homomorphism of étale sheaves: ɛ : µ n Z D 1 A, where D is the different of Q K. Let S O,g,J (n) and S O,g (n) denote the orresponding staks. The staks S O,g (n), n 0 are Deligne-Mumford staks loally of finite presentation and with finite inertia; let : S O,g,J (n) S O,g,J (n) denote their oarse moduli shemes. For n 3, : S O,g,J (n) S O,g,J (n) is a fine moduli sheme. See [R78], [VdG87, Chap. X], [DP94] for more details. For g = 2, the oarse moduli shemes S O,2,J (n) are 2-dimensional normal shemes, separated and geometrially onneted over Q(J ). The analyti spae assoiated with S O,2,J (n) Q(J ) C an be identified with PSL n 2 (O J ) \ H 2. Here H = {z C im(z) > 0} and PSL n 2 (O J ) = SL n 2 (O J )/{±1}, where SL n 2 (O J ) is the subgroup of SL 2 (k) of matries: ( ) a b with a d mod J, nj and b nj 1. Using this analyti desription, one an show the following [HiVdV74], [HiZ77], [VdG87, Thm. VII.3.3 and VII.3.4]. d

THE l-primary TORSION CONJECTURE FOR ABELIAN SURFACES WITH REAL MULTIPLICATION 3 Theorem 2.2. For any real quadrati field extension Q K with ring of integers O and disriminant D one has - S O,2,J is of general type exept if D =5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 77, 85, 88, 92, 93, 105, 120, 140, 165 (we will then say that D is exeptional). - S O,2,J (2) is of general type exept if D = 5, 8, 12, in whih ase S O,2,J (3) is of general type. We set n(d) = 1 if D is non-exeptional, n(d) = 2 if D is exeptional 5, 8, 12 and n(d) = 3 if D = 5, 8, 12. 2.3. Strong uniform boundedness of l-primary torsion in 1-dimensional families of abelian varieties. The Mordell-Weil theorem asserts that for any number field k and abelian variety A over k the (abelian) group A(k) of k-rational points is finitely generated. In partiular its torsion subgroup A(k) tors is finite. Given a prime l, the l-primary torsion onjeture is a uniform onjetural form of Mordell-Weil theorem for the l-sylow subgroups A(k)[l ] of A(k) tors. Conjeture 2.3. (l-primary torsion onjeture) For any integer δ 1 - Weak form: For any number field k and prime l there exists an integer N := N(δ, l, k) depending only on δ, l and k suh that for any δ-dimensional abelian variety A over k one has A(k)[l ] l N. - Strong form: For any integer d 1 and prime l there exists an integer N := N(δ, l, d) depending only on δ, l and d suh that for any number field k with [k : Q] d and δ- dimensional abelian variety A over k one has A(k)[l ] l N. The δ = 1 ase of the strong l-primary torsion onjeture essentially follows from the fat that for any prime l the gonality of the modular urves Y 1 (l n ) (lassifying ellipti urves with a torsion point of order exatly l n ) goes to + with n. Indeed, ombined with Faltings-Lang-Frey [Fr94], this implies that for any integer d 1 the sets Y 1 (l n ) d are finite for n large enough (depending on d). But if Y 1 (l n ) d is non-empty for all n 0, it follows from the usual ompatness argument that lim Y 1 (l n ) d is non-empty as well, whih ontradits Mordell-Weil theorem. For δ 2, the l-primary torsion onjeture is widely open. However, the following higher dimensional relative variant of the strong l-primary torsion onjeture holds [CT08], [CT09a], [CT09b]. Theorem 2.4. Let k be a number field, S a normal urve, separated, of finite type and geometrially onneted over k and A S an abelian sheme. Then, for any integer d 1 and prime l there exists an integer N := N(A, l, d) depending only on A, l and d suh that for any s S d on has A s (k(s))[l ] l N. When applied to the universal ellipti sheme E y 2 + xy x 3 36 + j 1728 x + 1 j 1728 P1 j \ {0, 1728, + } theorem 2.4 yields the 1-dimensional ase of the l-primary torsion onjeture. 3. Proof of theorem 1.1 We now proeed to the proof of Theorem 1.1 by ombining the ingredients gathered in Setion 2. Fix a finite extension k of Q(J ) and, for simpliity, write n for n(d) and : X (n) X(n) for the pullbak of : S O,2,J (n) S O,2,J (n) via spe(k) spe(q(j )). In any ase, the anonial morphism of staks F : X (6) X (n) indues a ommutative diagram X (6) X(6) F f X (n) X(n),

4 ANNA CADORET where the bottom arrow f : X(6) X(n) is a finite surjetive morphism of surfaes. By Theorem 2.2 (and assuming the Bombieri-Lang onjeture), the irreduible omponents of the Zariski losure X(n) of X(n)(k) in X(n) have dimension 1. Sine X(n) has finitely many irreduible omponents (being of finite type over k), it is enough to bound uniformly the l-primary torsion of the abelian surfaes assoiated with points in X (n)(k) lying over I for eah irreduible omponent I of X(n). So, without loss of generality, we may assume that X(n) is irreduible. 3.1. dim(x(n)) = 0 (i.e. X(n) is a singleton). Sine : X (n) X(n) is a oarse moduli sheme, it indues a bijetion X (n)(k)/ X(n)(k). In partiular, any two A i (= (A i, ι i, λ i, ɛ i )) X (n)(k), i = 1, 2 beome isomorphi in X (n)(k). The onlusion then follows from Lemma 3.1 below applied to the forgetful funtor X (n) A 2 and the fat that for any abelian variety A over a number field k and integer d 1 the set is finite [CT10, Lemma 3.19]. A d A(k) tors Lemma 3.1. Let S be any sheme, let A δ S denote the étale stak of δ-dimensional abelian varieties over S and let X S be an algebrai stak with quasi-finite inertia 1. Then for any morphism A : X A δ of fibered ategories, there exists an integer := (δ) 1 suh that for any S-field k and x, y X (k) suh that x k and y k are isomorphi in X (k), there exists a finite extension k k x,y with degree [k x,y : k] suh that A(x) kx,y and A(y) kx,y beome isomorphi in A δ (k x,y ). Here, given a morphisme of S-shemes V U and an objet x X (U), we write x V for the image of x via the anonial pull-bak funtor X (U) X (V ). For instane, with this notation, A(x) kx,y is nothing but A(x) k k x,y. Proof. Reall first that there exists an integer B(δ) 1 suh that for any field k and δ-dimensional abelian variety A over k, any finite subgroup G of Aut(A) has order B(δ). Indeed, let l be a prime different from the harateristi of k. Then, sine End(A) ats faithfully on T l (A) [M86, Lemma 12.2], G is a finite subgroup of GL 2δ (Z l ). But suh subgroups have order bounded by a onstant depending only on δ and l [S65, LG 4.27, Thm. 5]. The onlusion thus follows from the fat that we an take l = 2 or 3. Given x X (k), write Tw(x/k) for the set of isomorphism lasses of twists of x over k that is pairs (x, φ), where x X (k) and φ : x k x k is an isomorphism in X (k). One has a ommutative diagram of pointed sets: Tw(A(x)/k) β H 1 (Γ k, Aut(A(x) k )) α Tw(x/k) a H 1 (Γ k, A(Aut(x k ))) b H 1 (Γ k, Aut(x k )), where β is an isomorphism of pointed sets. By the quasi-finite inertia assumption, the group Aut(x k ) is finite so A(Aut(x k )) is a finite subgroup of Aut(A(x k )) and, by the observation above, has order B(δ). Set k x := k ker(ϕ), where ϕ : Γ k S(A(Aut(x k ))) is the permutation representation assoiated with the Γ k -set A(Aut(x k )). Then [k x : k] B(δ)! and A(Aut(x k )) is a trivial Γ kx -module so H 1 (k x, A(Aut(x k ))) an be identified with the set of orbits of Hom Group (Γ kx, A(Aut(x k ))) 1 Reall that the inertia stak of X is defined to be IS(X ) := X X S X X X. Hene, in partiular, it follows from the definition of the fibre produt in the 2-ategory of S-groupoids that for any geometri point x X (Ω) the fibre I S(X ) x an be identified with Aut(x). The assumption that X has finite inertia is only there to ensure that the automorphism group of geometri points is finite.

THE l-primary TORSION CONJECTURE FOR ABELIAN SURFACES WITH REAL MULTIPLICATION 5 under the right ation by inner automorphisms of A(Aut(x k )). Now, given y Tw(x/k), set k x,y := k ker(ba(y) kx ) x then [k x,y : k] B(δ)!B(δ) and ba(y) kx,y is trivial. Sine β is an isomorphism of pointed sets and the above diagram ommutes, this implies that βα(y) kx,y is trivial as well. So (δ) = B(δ)!B(δ) works. 3.2. dim(x(n)) = 1. Let X(6) denote the pullbak of X(n) X(n) via f : X(6) X(n). Up to onsidering separately the (finitely many) irreduible omponents of X(6), we may assume that X(6) is irreduible. From Subsetion 3.1, we may freely remove finitely many points from X(n) hene also assume that X(6) is a urve smooth (separated and of finite type) over k. Eventually, we may freely replae k by any finite extension k k hene assume that X(6) is geometrially onneted over k. As : X (6) X(6) is a fine moduli sheme, there is a setion X (6) X(6) orresponding to a J -polarized abelian sheme with real multipliation by O and µ 6 -level struture (A, ι, λ, ɛ) X(6). For simpliity, write again A := A X(6) X(6) X(6) for the pullbak of A X(6) via X(6) X(6). Then, from Theorem 2.4, for any prime l and integer d 1 there exists an integer N := N(A, l, d) suh that for any x X(6) d one has A x (k(x))[l ] l N. This is true, in partiular, for d = e (2), where e denotes the degree of f : X(6) X(n). But for any A = (A, ι, λ, ɛ) X (n)(k) above some x X(n)(k) and any x f 1 (x), A k k and F (A x k(x ) k) beome isomorphi in X (n)(k). So the onlusion, again, follows from Lemma 3.1 applied to the forgetful funtor X (n) A 2. Remark 3.2. (1) The proof given here applies as it is to other similar situations. For instane if A δ,γ denotes the stak of δ-dimensional abelian varieties endowed with a degree γ 2 polarization and : A δ,γ A δ,γ its oarse moduli sheme then, for any number field k and 2-dimensional subsheme S A δ,γ suh that S B(k) is of general type, the l-primary torsion of the abelian varieties assoiated with points in A δ,γ (k) lying over S is uniformly bounded. (2) Even in the fine moduli situation, that is A S is an abelian sheme with S B(k) of general type, our proof does not show that the Bombieri-Lang onjeture implies that for integer d 2 and prime l the l-primary k(s)-rational torsion in the fibres A s, s S d is uniformly bounded. The obstrution omes from the fat that the set S d might be Zariski dense in S even if for eah finite extension k k with degree [k : k] d the set S(k ) is not. Referenes [C10] A. Cadoret, Kodaira dimension of abstrat modular surfaes, preprint 2010. [CT08] A. Cadoret and A. Tamagawa, A uniform boundedness of p-primary torsion on abelian ahemes, preprint 2008. [CT09a] A. Cadoret and A. Tamagawa, A uniform open image theorem for l-adi representations I, preprint 2009. [CT09b] A. Cadoret and A. Tamagawa, A uniform open image theorem for l-adi representations II, preprint 2009. [CT10] A. Cadoret and A. Tamagawa, On a weak variant of the geometri torsion onjeture, preprint 2010. [DP94] P. Deligne and G. Pappas, Singularités des espaes de modules de Hilbert, en les aratéristiques divisant le disriminant, Compositio Math. 90, 1994, p. 59 79. [F94] G. Faltings, The general ase of Lang s onjeture, in Barsotti s symposium in algebrai geometry, Aademi Press, 1994, p. 175 182. [Fr94] G. Frey, Curves with infinitely many points of finite degree, Israel Journal of Math. 85, 1994, p. 79 83. [HiVdV74] F. Hirzebruh and A. Van de Ven, Hilbert modular surfaes and the lassifiation of algebrai surfaes, Inventiones Math. 23 No 1, 1974, p. 1 29. [HiZ77] F. Hirzebruh and D. Zagier, Classifiation of Hilbert modular surfaes, in: Complex analysis and algebrai geometry, Iwanami Shoten and Cambridge Univ. Press, 1977, p. 43 77. [M86] J.S. Milne, Abelian varieties, in Arithmeti geometry (Cornell/Silverman, ed.), Springer-Verlag, 1986, p. 103 150. [R78] M. Rapoport, Compatifiations de l espae de modules de Hilbert-Blumenthal, Compositio Math. 36, 1978, p. 255 335. [S65] J.-P. Serre, Lie algebras and Lie groups, W.A. Benjamin, 1965.

6 ANNA CADORET [VdG87] G. Van der Geer, Hilbert Modular Surfaes, E.M.G. 16, Springer, 1987. anna.adoret@math.u-bordeaux1.fr Institut de Mathématiques de Bordeaux - Université Bordeaux 1, 351 Cours de la Libération, F33405 TALENCE edex, FRANCE.