Sato-Tate groups of some weight 3 motives

Abstract

Abstract. We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h 3,0 = h 2,1 = h 1,2 = h 0,3 = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).

Sato-Tate groups of 𝑦²=𝑥⁸+𝑐 and 𝑦²=𝑥⁷-𝑐𝑥.

“…Using the explicit representation of ST(C 2 ) given in Corollary 5.8, one may compute moments sequences for the characteristic polynomial coefficients a 1 , a 2 , a 3 using the techniques described in §3.2 of [FKS13]; the first eight moments are listed below: The a 1 moments closely match the corresponding moment statistics listed in Table 2 in the cases where c is generic, as expected. We also computed moment statistics for a 1 , a 2 and a 3 by applying the algorithm of [HS14b] …”

“…Using the explicit representation of ST(C 1 ) given in Corollary 5.3 one may compute moment sequences using the techniques described in §3.2 of [FKS13]. The table below lists moments not only for a 1 , but also for a 2 and a 3 , where a i denotes the coefficient of T i in the characteristic polynomial of a random element of ST(C 1 ) distributed according to the Haar measure (these correspond to normalized L-polynomial coefficients of Jac(C 1 )): The a 1 moments closely match the corresponding moment statistics listed in Table 1 in the cases where c is generic, as expected.…”

Auto-correlation functions of Sato-Tate distributions and identities of symplectic characters

“…Kedlaya and Sutherland [KS09] and later with Fité and Rotger [FKRS12] made a conjectural, exhaustive list of 55 compact subgroups of USp(4) that would classify all the distributions of Euler factors for abelian surfaces, and called the groups Sato-Tate groups. Later, when they considered certain motives of weight 3 [FKS16], two other groups were added to the list of Sato-Tate groups that are subgroups of USp(4). They determined the moment sequences c k (m; 2), k = 1, 2, for each Sato-Tate group by expressing them as combinations of some sequences.…”

“…Since then, the Sato-Tate conjecture for abelian surfaces defined over Q, which covers 34 Sato-Tate groups, has been established by C. Johansson and N. Taylor [Joh17,Tay20] except for the generic case USp(4). Actually, those three groups, which do not arise for abelian surfaces, and two other subgroups of USp(4) appear when certain motives of weight 3 are considered in [FKS16]. Thus all the 57 groups are interesting in number theory and arithmetic geometry, and we will call all of them Sato-Tate groups in what follows.…”

Sato-Tate distributions

“…There are abelian varieties over K that are not the Jacobian of any curve defined over K, and L-functions that can be written as Euler products over primes of K that are not the L-function of any abelian variety. One can more generally consider the distribution of normalized Euler factors of motivic L-functions, which we also expect to be governed by the Haar measure of a Sato-Tate group associated to the underlying motive, as defined in [76,77]; see [26] for some concrete examples in weight 3.…”