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

Abstract

The Sato-Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this paper, we explicitly compute the auto-correlation functions of Sato-Tate distributions for genus 2 curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups Sp(2m) for all m ∈ Z ≥1 , which have interest in their own rights.

Sato-Tate groups of abelian threefolds

“…In this case we focus on the computation of the moments, which will suffice for our purposes. Since none of the elementary approaches used so far seems to work to compute M e 1 ,e 2 ,e 3 (JU(3)), we will instead take an approach recently introduced by Lee and Oh [LO20]. We have not attempted to apply this method to the other groups in the extended classification, although this seems an interesting problem.…”

“…From a theoretical point of view, formula (6.3) makes it conceivable to obtain closed formulas for M e 1 ,e 2 ,e 3 (G). In fact in dimension 2, Lee and Oh [LO20] derive closed formulas for m H (χ λ * ) for each possible Sato-Tate group H of an abelian surface. From a computational perspective, however, computing the right-hand side of (6.3) is extremely expensive as the number of terms of χλ grows very rapidly in m, and in fact (6.3) is the result of an almost miraculous cancellation.…”

Auto-correlation functions for unitary groups

“…Inspired by the approach of Bump and Gamburd, in a previous paper [LO20], the authors computed the auto-correlation functions of characteristic polynomials for Sato-Tate groups H ≤ USp(4), which appear in the generalized Sato-Tate conjecture for genus 2 curves. The result of [LO20] can be described as follows. Let H ≤ USp(4) be a Sato-Tate group.…”

“…where the coefficients m (b+2z,b) are the multiplicities of the trivial representation in the restrictions χ Sp(4) (b+2z,b) H and are explicitly given in the paper [LO20] for all the Sato-Tate groups of abelian surfaces. Exploiting the representation-theoretic meaning of m (b+2z,b) , the authors obtained this result by establishing branching rules for χ Sp(4) (b+2z,b) H .…”