An Étale Aspect of the Theory of Étale Theta Functions. Yuichiro Hoshi. December 2, 2015

Transcription

1 Introduction to Inter-universal Teichmüller Theory II An Étale Aspect of the Theory of Étale Theta Functions Yuichiro Hoshi RIMS, Kyoto University December 2, 2015 Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

2 Notation and Terminology For an odd prime number l, F ± l an F ± l = (F l ) + {±1}, F l = F l /{±1}, l = F l = l 1 2 -group a set S equipped with a {±1}-orbit of S F l For a topological group G, H i (G, A) = lim H G: open subgps of finite index H i (H, A) For a p-adic local field k, the µ-kummer structure of G k O µ {Im ( (O k )H = O k H O k O µ k a poly-(iso)morphism A B a set consisting of (iso)morphisms A B ) }H Gk : open subgps the full poly-isomorphism A B the poly-isom Isom(A, B) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26 k

3 Fundamental Strategy (cf p23 of I) is, for instance, a log-shell, a theta function, or a κ-coric function Start with a usual/existing (ie, a Frobenius-like ) Construct links by means of such Frobenius-like objects Take an étale-like object closely related to (eg, π temp 1 (X v ) for a theta function cf II and III) Give a multiradial mono-anabelian algorithm of reconstructing from the étale-like object, ie, construct a suitable étale-like Establish multiradial Kummer-detachment of, ie, a suitable Kummer isomorphism Frob-like étale-like Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

4 Fundamental Strategy (cf p23 of I) is, for instance, a log-shell, a theta function, or a κ-coric function Start with a usual/existing (ie, a Frobenius-like ) Construct links by means of such Frobenius-like objects Take an étale-like object closely related to (eg, π temp 1 (X v ) for a theta function cf II and III) Give a multiradial mono-anabelian algorithm of reconstructing from the étale-like object, ie, construct a suitable étale-like Establish multiradial Kummer-detachment of, ie, a suitable Kummer isomorphism Frob-like étale-like Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

5 p, l: prime numbers k: a p-adic local field, ie, [k : Q p ] <, st 1 k E: an elliptic curve/k which has split multiplicative reduction/o k q Ok : the q-parameter of E log X = (E, {o} E): the smooth log curve/k determined by E {±1} E, hence also X log X log log C = [X log /{±1}] Assumptions 2, p, and l are distinct prime numbers E[2l](k) = E[2l](k) ( µ l (k) k and q = q 1/2l k) C log is a k-core ( one may apply elliptic cuspidalization) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

6 One obtains a comm diagram of conn d log étale tempered coverings Ÿ log (1) µl Ÿ log (2) µ 2 (3) µ 2 Y log (4) µl Y log (5) l Z (6) l Z X log (7) µl X log (8) Fl X log (9) {±1} (10) {±1} C log Csp( ) = the set of cusps of ( ) (11), deg=l C log Irr( ) = the set of irreducible comp of the special fiber of ( ) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

7 The three squares are cartesian The composite Y log (6) log (8) X X log : the covering determined by the dual semi-graph of the special fiber of X log Z = Gal(Y log /X log ) = Z; Csp(Y log ), Irr(Y log ): Z-torsors log (8) X X log : the intermediate covering corresp g to l Z Z F l = Gal(X log /X log ) = F l ; Csp(X log ), Irr(X log ): F l -torsors Fix an Csp(X log ), ie, the zero cusp of X log a structure of elliptic curve on the underlying scheme of X log log (9) X C log = [X log /{±1}] zero cusp of X log a cusp of C log, ie, the zero cusp of C log Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

8 log (7) X X log totally ramifies at Csp(X log ), Gal = µ l Csp(X log ) Csp(X log ), Irr(X log ) Irr(X log ) zero cusp of X log a cusp of X log, ie, the zero cusp of X log Ÿ log (3) Y log : the double covering determined by ü = u 1/2 Csp(Ÿ log ) 2:1 Csp(Y log ), Irr(Ÿ log ) Irr(Y log ) Aut k (C log ) = Aut k (C log ) = {id} Aut k (X log ) = Gal(X log /C log ) = {±1} Aut k (X log ) = Gal(X log /X log ) Gal(X log /C log ) = F l {±1} Aut k (X log ) = Gal(X log /X log ) Gal(X log /C log ) = µ l {±1} ( Csp(X log ) Aut k(x log) = {the zero cusp}) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

9 Labels of Cusps and Components geometry of X log /k det a natural str of F ± l -gp on Csp(Xlog ), hence also on Irr(X log ), Csp(X log ), Irr(X log ), ie, each element of these sets is labeled by an F l up to {±1} {±1} LabCusp ± = Csp(X log ) F l ( (F ± l =) Autk (X log )) Fix a lifting Irr(Y log ) of 0 Irr(X log ) (Such liftings form an (l Z)-torsor) Such a lifting det Z Irr(Y log ) Irr of Ÿ log, Ÿ log, Y log, ie, each Irr of Y log, Ÿ log, Ÿ log, Y log is labeled by an Z Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

10 Evaluation Points!µ X(k): a 2-torsion whose closure intersects 0 Irr(X log ) µ Y Y (k): a unique lifting of µ whose closure inter 0 Irr(Y log ) ξa Y Y (k): the image of µ Y by a Z = Gal(Y log /X log ) Definition an evaluation point of Ÿ log (resp Ÿ log ) labeled by a Z a (necessarily k-rat l) lifting Ÿ (resp Ÿ ) of ξy a Y (k) an evaluation point of X log labeled by a LabCusp ± the image X of an evaluation point of Ÿ log labeled by a lifting Z of a LabCusp ± Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

11 Theta Functions The function Θ(ü) = q 1 8 ( 1) n q 1 2 (n+ 1 2 )2 ü 2n+1 n Z on 0 Irr(Ÿ log ) uniquely extends to a meromorphic function Θ on the stable model of Ÿ the zero divisor of Θ = c Csp(Ÿ log ) [c] the pole divisor of Θ = a Z =Irr(Ÿ log ) a 2 ord k (q) 2 [a] An l-th root Θ on Ÿ of Θ(an evaluation pt labeled by 0) Θ 1 (Note: Θ(an ev pt labeled by 0) = Θ(another ev pt labeled by 0)) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

12 Special Values Θ on Ÿ : Θ(an evaluation point labeled by j) µ 2l q j j ev pt q 4 q 1 q q 4 q j2 mod µ 2l l } l {{ 0 1 l 1 l } LabCusp ± {±1} = Fl Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

13 Π ( ) : the log étale π 1 of ( ) log ( ) = Ker(Π ( ) G k ), ie, the geom log étale π 1 of ( ) log Π tp ( ) : the log tempered π 1 of ( ) log tp ( ) = Ker(Π tp ( ) G k), ie, the geom log temp d π 1 of ( ) log For N 1, if J G k, then J[µ N ] = µ N (k) J a tautological splitting s J : J J[µ N ] of J[µ N ] J a natural homomorphism H 1 (J, µ N (k)) Out(J[µ N ]) Thus, by the Kummer theory, we have: D Y k k /(k ) N H 1 (Π tp Y, µ N(k)) Out(Π tp Y [µ N]) = Im(k ), Gal(Y log /X log ) = l Z Out(Π tp Y [µ N]) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

14 Θ = [ X, X ]/[ X, [ X, X ]] Θ = ab X ab X, ie, = Ẑ(1) ( = lim µ n n (k)) l Θ = Ẑ(1) η Θ H 1 (Π tp Ÿ, Θ): the Kummer class of a suitable O k Θ η Θ H 1 (Π tp Ÿ, l Θ) st η Θ Ÿ = Im( η Θ ) in H 1 (Π tp Ÿ, Θ), ie, the Kummer class of an O k Θ 1 η Θ,l Z µ 2 H 1 (Π tp Ÿ, l Θ): the orbit of η Θ by Gal(Ÿ log /X log ) = Π tp X /Πtp Ÿ = l Z µ 2 ( indep of the choice of a lifting Irr(Y log ) of 0 Irr(X log )) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

15 Thus, relative to (l Θ ) Z Z/NZ each η Θ,l Z µ 2 by scheme theory µ N (k), mod N H 1 (Π tp Ÿ, (l Θ) Z Z/NZ) can be obtained as s Θ s Ÿ Π tp for some s Θ: Ÿ Ÿ s Θ Ÿ, 1 µ N (k) Π tp Ÿ [µ N] s Π tp Ÿ Π tp Ÿ 1 s Θ Ÿ : Πtp Ÿ s Θ Ÿ Π tp Ÿ [µ N] Π tp Y [µ N] : a (mod N) theta section Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

16 Mono-theta Environments and Associated Cyclotomes Definition A (mod N) model mono-theta environment a triple (Π tp Y [µ N], D Y Out(Π tp Y [µ N]), {γ Im(s Θ Ÿ ) γ 1 } γ µn (k) ) A (mod N) mono-theta environment an isomorph M Θ N = (Π, D Π Out(Π), s Θ Π ) of a mod N model mono-theta env The subgroup of Π (of M Θ N ) corresp g to µ N(k) Π tp Y [µ N] is group-theoretic Π µ (M Θ N ): the exterior cyclotome The subquotient of Π (of M Θ N ) corresponding to l Θ is group-theoretic (l Θ )(M Θ N ): the interior cyclotome Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

17 Algorithmic Reconstruction Π : an isomorph of Π tp X M Θ N : a mod N mono-theta environment func l Π topological gp corresponding to the topological gp algorithm Π tp Ÿ, Πtp Ÿ, Πtp Y, Πtp Y, Πtp X, Πtp X, Πtp C, Πtp C, G k, l Θ func l Π a subset corresponding to the subset algorithm (l Z µ 2 )-orbit of O k Θ H1 (Π tp Ÿ, l Θ) func l Π a mod N mono-theta environment algorithm M Θ N func l algorithm a topological group corresponding to Πtp X Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

18 Rigidity Properties of Mono-theta Environments Cyclotomic Rigidity Discrete Rigidity Constant Multiple Rigidity Isomorphism Class Compatibility Frobenioid Structure Compatibility Cyclotomic Rigidity M Θ N func l algorithm a canonical (l Θ)(M Θ N ) Z Z/NZ Π µ (M Θ N ), ie, (l Θ ) Z Z/NZ µ N (k) by scheme theory (cf a suitable Kmm isom Frob-like étale-like of p3) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

19 Discrete Rigidity By means of the var surj µ N (k) µ M (k) (M N), one may ine the notion of a projective system of mono-theta env {M Θ N } N 1 proj system = the natural proj system of model mono-theta env Constant Multiple Rigidity M Θ N func l the subset corresponding to the subset algorithm θ = (l Z µ 2 )-orbit of µ l Θ H 1 (Π tp Ÿ, l Θ) of (l Z µ 2 )-orbit of O k Θ H1 (Π tp Ÿ, l Θ) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

20 Pointed Inversion Automorphisms Consider a pair (ι X, µ X ) of a unique ι X Aut k (X) of order two, ie, 1 LabCusp ± an evaluation point µ X of X log labeled by 0 LabCusp ± Definition A pointed inversion automorphism of Ÿ log a lifting (ιÿ, µÿ ) on Ÿ log of (ι X, µ X ) st ι 2 = id, ι(µ) = µ A group-theoretic pointed inversion automorphism a group-theoretic pair (ι, D) associated to a (ιÿ, µÿ ) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

21 Thus, such a (ι, D) det a lifting Irr(Y log ) of 0 Irr(X log ) In particular: θ = (l Z µ 2 )-orbit of µ l Θ θ ι = µ 2l -multiples of a Θ (where ( ) ι is the set of ι-invariants ), which thus implies that H 1 (Π tp Ÿ, l Θ) restriction to D H 1 (D, l Θ ) θ ι (Kummer class of) µ 2l, as well as, for a decomp subgp D j Π tp Ÿ labeled by j (for (ι, D)), H 1 (Π tp Ÿ, l Θ) restriction to D j H 1 (D j, l Θ ) θ ι (Kummer class of) µ 2l q j2 (The operation of Galois evaluation wrt D, as well as D j ) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

22 Reconstructions of Étale Theta Functions via Mono-theta Env M Θ = {M Θ N } N: a projective system of mono-theta environments (l Θ )(M Θ ) = lim( (l Θ )(M Θ N ) (l Θ )(M Θ M ) ) Π µ (M Θ ) = lim N Π µ (M Θ N ) By the cycl rig: (l Θ )(M Θ ) Π µ (M Θ ) By the cons mult rig: θ(m Θ ) H 1 (Π tp Ÿ (MΘ ), (l Θ )(M Θ )) (θ(m Θ ) ) θ(m Θ ) H 1 (Π tp Ÿ (MΘ ), (l Θ )(M Θ )) ined by { η H 1 n η θ for some n 1 } By (l Θ )(M Θ ) Π µ (M Θ ): θ env (M Θ ) θ env (M Θ ) H 1 (Π tp Ÿ (MΘ ), Π µ (M Θ )) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

23 Reconstructions of Constant Portions via Mono-theta Environments Π : an isomorph of Π tp X func l algorithm M = M Θ (Π ): a proj system Since X log is of strictly Belyi type, by Belyi cuspidalization, a functorial algorithm for reconstructing, from Π, an isomorph Π G = G k (Π ) k(π ) k(π ) H 1 (G, (l Θ )(Π )) of the Π tp X G k k k Kummer H 1 (G k, l Θ ) O µ k(π ) O k(π ) O k(π ) By (l Θ )(Π ) (l Θ )(M) Π µ (M): k(π ) H 1 (G, (l Θ )(Π )) k(m) O µ k(m) O k(m) O k(m) k(m) H 1 (G, Π µ (M)) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

24 Reconstructions of Splittings via Mono-theta Environments (O θ env )(M) = O k(m) + θ env (M) H 1 (Π tp Ÿ (M), Π µ(m)) In particular, for a gp-th c pt d inv aut (ι, D) for Π tp Ÿ (M), H 1 (Π tp Ÿ (M), Π µ(m)) Gal ev wrt D H 1 (D, Π µ (M)) (O θ env )(M) ι O k(m) (ie, Gal ev labeled by 0 LabCusp ± ) determines a splitting (O θ env )(M) ι /O µ k(m) = O µ ( ) k(m) θ env (M) ι /O µ k(m) Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

25 Thus, in summary, we obtain: A Local Multiradial Algorithm Related to A multiradial algorithm as follows: Étale Theta Functions coric data: an isomorph (G O µ, µ-kmm) of G k O µ k radial data: (Π Π µ (M Θ (Π )), a coric data, α µ, µ ) for an isomorph Π of Π tp X, where α µ, µ is the pair of the full poly-isomorphism G G and Π µ (M Θ (Π )) Z (Q/Z) zero O µ output: the radial data and: Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26

26 A Local Multiradial Algorithm Related to Étale Theta Functions The proj system of mono-theta environments M = M Θ (Π ) The subsets O k(m) (O θ env )(M) H 1 (Π tp Ÿ (M), Π µ(m)) The set of group-th c pointed inversion automorphisms {(ι, D)} The splittings for the various (ι, D) (O θ env )(M) ι /O µ = O µ ( ) k(m) k(m) θ env (M) ι /O µ k(m) via the operation of Galois evaluation wrt D The diagram Π µ (M) Z (Q/Z) nat l O µ nat l O µ zero O µ full k(m) k(π ) k(π ) poly O µ Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, / 26