Sato–Tate distributions and Galois endomorphism modules in genus 2

Abstract

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp (4); this Sato-Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato-Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the R-algebra generated by endomorphisms of A Q (the Galois type), and establish a matching with the classification of Sato-Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato-Tate groups for suitable A and k, of which 34 can occur for k = Q. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over Q whenever possible), and observe numerical agreement with the expected Sato-Tate distribution by comparing moment statistics.

Sato–Tate distributions of twists ofy2=x5−xandy2=x6+ 1

“…We devote §4 to the proofs of assertions (i), (ii) and (iii), which follow from Corollary 4.17, Proposition 4.15, and Proposition 4.16, respectively. The final assertion (v) follows from (iii) and (iv): it is enough to check that for each of the 41 possibilities of T (C), the formulas obtained for µ[a i (C)] coincide with the ones obtained for Φ i, * (µ(ST(Jac(C)))) in [FKRS12].…”

“…It follows from [FKRS12,Prop. 9] that the eigenvalues of θ Q (A)(τ ) are 1, 1, 1, 1, ζ r , ζ r , ζ r , ζ r .…”

“…For A of arbitrary dimension, the invariance of ST(A) under quadratic twisting follows easily from the definition of ST(A) given in [Se12] (see also [FKRS12]), in terms of the image of the ℓ-adic representation attached to A.…”

“…It follows that if C is a twist of either of these curves, then ST(Jac(C)) also has identity component U(1). In fact, the representative curves for all 32 of the U(1) cases listed in [FKRS12] are actually twists of one of the two curves in (1.2) (possibly using an extended field of definition).…”

“…The classification in [FKRS12] gives an explicit description of each of the 52 Sato-Tate groups that can and do arise in genus 2, as subgroups of USp(4), of which 32 have identity component (isomorphic to) U(1). The two curves listed in (1.2) both appear in [FKRS12], where they are shown to have SatoTate groups with identity component U(1). It follows that if C is a twist of either of these curves, then ST(Jac(C)) also has identity component U(1).…”