Towards a precise Sato-Tate conjecture in genus 2

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Towards a precise Sato-Tate conjecture in genus 2 Kiran S. Kedlaya Department of Mathematics, Massachusetts Institute of Technology Department of Mathematics, University of California, San Diego kedlaya@mit.edu, kedlaya@ucsd.edu http://math.mit.edu/~kedlaya/papers/talks.shtml Explicit Methods in Number Theory Oberwolfach, July 19, 2011 Joint work in progress with Francesc Fité, Victor Rotger, Andrew V. Sutherland and with Grzegorz Banaszak. Supported by NSF (CAREER grant DMS-0545904), DARPA (grant HR0011-09-1-0048), MIT (NEC Fund), UCSD (Warschawski chair). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 1 / 21

Advertisement Come to the Algorithmic Number Theory Symposium (ANTS-X), July 9-13, 2012, at the University of California, San Diego! We expect to have some funding for graduate students and postdocs. Watch for news at the web site: http://math.ucsd.edu/~kedlaya/ants10 Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 2 / 21

Contents 1 Introduction 2 Definition of the Sato-Tate group 3 Classification of Sato-Tate groups 4 Numerical evidence for equidistribution 5 Complements Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 3 / 21

Contents Introduction 1 Introduction 2 Definition of the Sato-Tate group 3 Classification of Sato-Tate groups 4 Numerical evidence for equidistribution 5 Complements Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 4 / 21

Notations Introduction Throughout, let A be an abelian variety of dimension g over a number field k. Write the L-function of A (or rather the part coming from primes of good reduction) as an Euler product L(A/k, s) = p L p (A/k, q s ) 1 (q = Norm k/q (p)) Here L p is an integer polynomial of degree 2g. If we renormalize by setting L p (A/k, T ) = L p (A/k, q 1/2 T ), then L p (A/k, T ) has its roots in conjugate pairs on the unit circle. In particular, it occurs as the characteristic polynomial of a unique conjugacy class in the unitary symplectic group USp(2g). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 5 / 21

Introduction The Sato-Tate conjecture: imprecise to precise Conjecture (Imprecise Sato-Tate) For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). One way to make this precise is to give a recipe for H; more on that soon. For g = 2, we also do the following. We classify subgroups consistent with this construction. (There are 55 of them up to conjugacy.) We correlate these groups with the Galois action on endomorphism algebras. (This leaves 52 possibilities, of which 34 can occur over Q.) We obtain strong numerical evidence for the resulting predictions. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 6 / 21

Introduction The Sato-Tate conjecture: imprecise to precise Conjecture (Imprecise Sato-Tate) For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). One way to make this precise is to give a recipe for H; more on that soon. For g = 2, we also do the following. We classify subgroups consistent with this construction. (There are 55 of them up to conjugacy.) We correlate these groups with the Galois action on endomorphism algebras. (This leaves 52 possibilities, of which 34 can occur over Q.) We obtain strong numerical evidence for the resulting predictions. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 6 / 21

Introduction The Sato-Tate conjecture: imprecise to precise Conjecture (Imprecise Sato-Tate) For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). One way to make this precise is to give a recipe for H; more on that soon. For g = 2, we also do the following. We classify subgroups consistent with this construction. (There are 55 of them up to conjugacy.) We correlate these groups with the Galois action on endomorphism algebras. (This leaves 52 possibilities, of which 34 can occur over Q.) We obtain strong numerical evidence for the resulting predictions. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 6 / 21

Introduction The Sato-Tate conjecture: imprecise to precise Conjecture (Imprecise Sato-Tate) For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). One way to make this precise is to give a recipe for H; more on that soon. For g = 2, we also do the following. We classify subgroups consistent with this construction. (There are 55 of them up to conjugacy.) We correlate these groups with the Galois action on endomorphism algebras. (This leaves 52 possibilities, of which 34 can occur over Q.) We obtain strong numerical evidence for the resulting predictions. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 6 / 21

Introduction Example: the case of dimension 1 Conjecture For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). Suppose g = 1, so A = E is an elliptic curve. If E has complex multiplication defined over k, then the conjecture holds with H = SO(2) (classical). If E has complex multiplication not defined over k, then the conjecture holds with H = N(SO(2)) (not connected!). If E does not have complex multiplication, then one expects H = USp(2) = SU(2). Known for k totally real (Barnet-Lamb, Clozel, Gee, Geraghty, Harris, Taylor). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 7 / 21

Introduction Example: the case of dimension 1 Conjecture For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). Suppose g = 1, so A = E is an elliptic curve. If E has complex multiplication defined over k, then the conjecture holds with H = SO(2) (classical). If E has complex multiplication not defined over k, then the conjecture holds with H = N(SO(2)) (not connected!). If E does not have complex multiplication, then one expects H = USp(2) = SU(2). Known for k totally real (Barnet-Lamb, Clozel, Gee, Geraghty, Harris, Taylor). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 7 / 21

Introduction Example: the case of dimension 1 Conjecture For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). Suppose g = 1, so A = E is an elliptic curve. If E has complex multiplication defined over k, then the conjecture holds with H = SO(2) (classical). If E has complex multiplication not defined over k, then the conjecture holds with H = N(SO(2)) (not connected!). If E does not have complex multiplication, then one expects H = USp(2) = SU(2). Known for k totally real (Barnet-Lamb, Clozel, Gee, Geraghty, Harris, Taylor). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 7 / 21

Introduction Example: the case of dimension 1 Conjecture For fixed A/k, as p varies over prime ideals of good reduction, the L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on some closed subgroup H of USp(2g). Suppose g = 1, so A = E is an elliptic curve. If E has complex multiplication defined over k, then the conjecture holds with H = SO(2) (classical). If E has complex multiplication not defined over k, then the conjecture holds with H = N(SO(2)) (not connected!). If E does not have complex multiplication, then one expects H = USp(2) = SU(2). Known for k totally real (Barnet-Lamb, Clozel, Gee, Geraghty, Harris, Taylor). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 7 / 21

Contents Definition of the Sato-Tate group 1 Introduction 2 Definition of the Sato-Tate group 3 Classification of Sato-Tate groups 4 Numerical evidence for equidistribution 5 Complements Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 8 / 21

Definition of the Sato-Tate group The Mumford-Tate group Let G k be the absolute Galois group of k. For l a prime, let ρ A/K,l : G k GSp 2g (Q l ) be the l-adic Galois representation of A. Let G l be the image of ρ A/k,l. Let Gl Zar denote the Zariski closure of G l in GSp 2g (Q l ); note that G l is open in Gl Zar (Q l ) (Bogomolov). Conjecture (Mumford-Tate) The identity component G Zar,0 l of Gl Zar equals the Mumford-Tate group, the minimal algebraic group over Q whose R-points contain the action of C on H 1 (A C, Q) Q R coming from the complex structure on the latter. In particular, G Zar,0 l should descend to an algebraic group over Q which does not depend on l. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 9 / 21

Definition of the Sato-Tate group An l-adic Sato-Tate group (after Serre) Let G l,1 be the kernel of G l GSp 2g (Q l ) Q l, where the last map comes from the Weil pairing. Let Gl,1 Zar denote the Zariski closure of G l,1 in GSp 2g (Q l ). It is generally not connected! Conjecture (Algebraic Sato-Tate) The group Gl,1 Zar descends to an algebraic group G Zar 1 over Q which does not depend on l. We call G Zar 1 the algebraic Sate-Tate group of A/k, denoted AST(A/k). The Sato-Tate group ST(A/k) is a maximal compact subgroup of AST(A/K) Q C. Conjecture (Refined Sato-Tate) The L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on ST(A/k). (A stronger statement gives an equidistribution on Cl(ST(A/k)).) Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 10 / 21

Definition of the Sato-Tate group An l-adic Sato-Tate group (after Serre) Let G l,1 be the kernel of G l GSp 2g (Q l ) Q l, where the last map comes from the Weil pairing. Let Gl,1 Zar denote the Zariski closure of G l,1 in GSp 2g (Q l ). It is generally not connected! Conjecture (Algebraic Sato-Tate) The group Gl,1 Zar descends to an algebraic group G Zar 1 over Q which does not depend on l. We call G Zar 1 the algebraic Sate-Tate group of A/k, denoted AST(A/k). The Sato-Tate group ST(A/k) is a maximal compact subgroup of AST(A/K) Q C. Conjecture (Refined Sato-Tate) The L p (A/k, T ) are equidistributed for the image on Cl(USp(2g)) of the Haar measure on ST(A/k). (A stronger statement gives an equidistribution on Cl(ST(A/k)).) Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 10 / 21

Definition of the Sato-Tate group Sato-Tate and endomorphism algebras Theorem-in-progress (with Banaszak) For g 3, the algebraic Sato-Tate conjecture holds for A/k. This is obtained by proving an open image theorem relating AST(A/k) to an upper bound defined using End Q (A k ). The argument applies to a large class of cases in which each simple factor of A k has enough endomorphisms, by following the treatment of the Mumford-Tate conjecture in these cases by Banaszak-Gajda-Krasoń. As a corollary of the proof, one identifies AST(A/k)/ AST(A/k) 0 with Gal(L/k) for L the minimal extension of k for which End(A L ) = End(A k ). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 11 / 21

Contents Classification of Sato-Tate groups 1 Introduction 2 Definition of the Sato-Tate group 3 Classification of Sato-Tate groups 4 Numerical evidence for equidistribution 5 Complements Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 12 / 21

Classification of Sato-Tate groups Properties of Sato-Tate groups Under standard conjectures, the group G = ST(A/k) has these properties. ST1) The real Lie group G is the intersection of USp(2g) with a reductive Q-algebraic subgroup of Sp 2g. ST2) There exists a homomorphism θ : U(1) G 0 such that θ(u) has eigenvalues u, u 1 each with multiplicity g, and θ( 1) = 1 USp(2g). The image of such a homomorphism θ is called a Hodge circle. ST3) The group G 0 is generated by the Hodge circles. ST4) Each component of G contains an element whose characteristic polynomial P(T ) is such that P(T / m) has rational coefficients for some positive integer m. For given g, this limits ST(A/k) to finitely many groups up to conjugacy. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 13 / 21

Classification of Sato-Tate groups Properties of Sato-Tate groups Under standard conjectures, the group G = ST(A/k) has these properties. ST1) The real Lie group G is the intersection of USp(2g) with a reductive Q-algebraic subgroup of Sp 2g. ST2) There exists a homomorphism θ : U(1) G 0 such that θ(u) has eigenvalues u, u 1 each with multiplicity g, and θ( 1) = 1 USp(2g). The image of such a homomorphism θ is called a Hodge circle. ST3) The group G 0 is generated by the Hodge circles. ST4) Each component of G contains an element whose characteristic polynomial P(T ) is such that P(T / m) has rational coefficients for some positive integer m. For given g, this limits ST(A/k) to finitely many groups up to conjugacy. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 13 / 21

Classification of Sato-Tate groups Properties of Sato-Tate groups Under standard conjectures, the group G = ST(A/k) has these properties. ST1) The real Lie group G is the intersection of USp(2g) with a reductive Q-algebraic subgroup of Sp 2g. ST2) There exists a homomorphism θ : U(1) G 0 such that θ(u) has eigenvalues u, u 1 each with multiplicity g, and θ( 1) = 1 USp(2g). The image of such a homomorphism θ is called a Hodge circle. ST3) The group G 0 is generated by the Hodge circles. ST4) Each component of G contains an element whose characteristic polynomial P(T ) is such that P(T / m) has rational coefficients for some positive integer m. For given g, this limits ST(A/k) to finitely many groups up to conjugacy. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 13 / 21

Classification of Sato-Tate groups Properties of Sato-Tate groups Under standard conjectures, the group G = ST(A/k) has these properties. ST1) The real Lie group G is the intersection of USp(2g) with a reductive Q-algebraic subgroup of Sp 2g. ST2) There exists a homomorphism θ : U(1) G 0 such that θ(u) has eigenvalues u, u 1 each with multiplicity g, and θ( 1) = 1 USp(2g). The image of such a homomorphism θ is called a Hodge circle. ST3) The group G 0 is generated by the Hodge circles. ST4) Each component of G contains an element whose characteristic polynomial P(T ) is such that P(T / m) has rational coefficients for some positive integer m. For given g, this limits ST(A/k) to finitely many groups up to conjugacy. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 13 / 21

Classification of Sato-Tate groups Properties of Sato-Tate groups Under standard conjectures, the group G = ST(A/k) has these properties. ST1) The real Lie group G is the intersection of USp(2g) with a reductive Q-algebraic subgroup of Sp 2g. ST2) There exists a homomorphism θ : U(1) G 0 such that θ(u) has eigenvalues u, u 1 each with multiplicity g, and θ( 1) = 1 USp(2g). The image of such a homomorphism θ is called a Hodge circle. ST3) The group G 0 is generated by the Hodge circles. ST4) Each component of G contains an element whose characteristic polynomial P(T ) is such that P(T / m) has rational coefficients for some positive integer m. For given g, this limits ST(A/k) to finitely many groups up to conjugacy. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 13 / 21

Classification of Sato-Tate groups Classification in genus 2 Theorem (with Fité, Rotger, Sutherland) For g = 2, there are exactly 55 conjugacy classes of subgroups G of USp(4) satisfying (ST1)-(ST4). In this list, G 0 {SO(2) (32 cases), SU(2) (10 cases), SO(2) SO(2) (8 cases), SO(2) SU(2) (2 cases), SU(2) SU(2) (2 cases), USp(4) (1 case)}. (Hint: USp(4)/{±1} = SO(5).) The cases G 0 = SO(2) come from finite subgroups of SU(2)/{±1} = SO(3). The icosahedral group fails (ST4), but the tetrahedral and octahedral groups survive! Problem Carry out the same exercise for g = 3. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 14 / 21

Classification of Sato-Tate groups Matching to Galois type For g = 2, one can determine ST(A/k) in terms of End Q (A k ), using both the algebra structure and the G k -action (though the former plays a surprisingly small role). The right package of data (definition omitted) is what we call the Galois type. Theorem-in-progress (with Fité, Rotger, Sutherland) There are exactly 52 Galois types. Of these, exactly 34 can be realized with k = Q. Detailed information about the distributions over Q can be found here: http://math.mit.edu/~drew/exceptionaldistributions.html Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 15 / 21

Contents Numerical evidence for equidistribution 1 Introduction 2 Definition of the Sato-Tate group 3 Classification of Sato-Tate groups 4 Numerical evidence for equidistribution 5 Complements Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 16 / 21

Numerical evidence for equidistribution Computing L p (A/k, T ) To test the refined Sato-Tate conjecture numerically, one chooses A/k and then computes L p (A/k, T ) for many prime ideals p. For example, for k = Q, for A the Jacobian of a hyperelliptic curve, we can compute L p (A/k, T ) for all p 2 30. (This is far more data than is really needed!) The key technique is Sutherland s highly optimized version of the baby steps-giant steps algorithm; see our ANTS 8 paper. Methods of p-adic cohomology, as optimized by David Harvey, do not beat this until p 2 40. Schoof-type methods, as optimized by Gaudry-Schost, kick in even later except maybe in some special cases, e.g., real multiplication (Gaudry-Kohel-Smith). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 17 / 21

Numerical evidence for equidistribution Give me a moment... To deal with this mass of data, we find it easiest to use moment statistics. Theorem For any group G USp(2g) satisfying (ST1)-(ST4), the expected value of any symmetric function of the eigenvalues is an integer. For G given explicitly, it is easy to compute these expected values for powers of the first or second coefficient of the characteristic polynomial. We then use these to test equidistribution. See again http://math.mit.edu/~drew/exceptionaldistributions.html for animations illustrating many cases. Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 18 / 21

Contents Complements 1 Introduction 2 Definition of the Sato-Tate group 3 Classification of Sato-Tate groups 4 Numerical evidence for equidistribution 5 Complements Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 19 / 21

Complements What can be proved in dimension 2? Current technology on modularity of Galois representations seems incapable of proving Sato-Tate in even a single case with g = 2 and ST(A/k) = USp(4). (It seems crucial to have distinct Hodge-Tate weights, i.e., to have a regular motive.) However, it should be possible to handle every case with g = 2 and ST(A/k) USp(4). The most delicate cases are when ST(A/k) contains SU(2) SU(2); these should be tractable using Rankin-Selberg convolution, as observed by Michael Harris. (This does not generalize to SU(2) SU(2) SU(2).) Problem Prove the refined Sato-Tate conjecture for all A/k with k totally real, g = 2, and ST(A/k) USp(4). Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 20 / 21

Complements Finding Mumford s fourfolds Mumford described a family of abelian fourfolds for which the endomorphism algebra is trivial but the Mumford-Tate group is not. This discrepancy should be reflected in the Sato-Tate distribution. Problem Identify the Sato-Tate distribution corresponding to Mumford s fourfolds, and compute moments. Then find A with g = 4 for which the Sato-Tate distribution appears to have matching moments. It is not clear over which number fields one should find such examples, or what form they may take. (E.g., it seems unlikely that they occur as Jacobians.) Kiran S. Kedlaya (MIT/UCSD) Towards a precise Sato-Tate conjecture Oberwolfach, July 19, 2011 21 / 21

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