Sato-Tate distributions

Abstract

In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g ≤ 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures presented at the

Sato-Tate Distributions of Catalan Curves

“…It is conjectured that ST(A) is, up to conjugacy in USp(2g), independent of the choice of the prime ℓ and of the embedding of Q ℓ in C and so we will refer to ST(A) as the Sato-Tate group of A (see, for example, [31]). While the Sato-Tate group is a compact Lie group, it may not be connected [15].…”

“…gives a uniform measure of U(1) on θ ∈ [−π, π] (see [31,Section 2]). We can deduce the following pushforward measure…”

“…The generalized Sato-Tate conjecture for an abelian variety predicts the existence of a compact Lie group that determines the limiting distribution of normalized local Euler factors (see, for example, [31]). In our work, the abelian varieties will be the Jacobians of curves, and so we now state the conjecture specifically for Jacobian varieties.…”

“…We begin by defining both the algebraic Sato-Tate group and the Sato-Tate group. We follow the exposition of [9] and [31,Section 3.2]. See also [29,Chapter 8].…”

A database of nonhyperelliptic genus-3 curves over ℚ

“…The real endomorphism algebras were computed by Jeroen Sijsling using an adaptation of the algorithms described in [3]. An abelian threefold J/Q with real endomorphism algebra R × R or R × C over Q is isogenous to the product of an abelian surface A with End(A Q ) = Z and an elliptic curve E (see Table 2 of [32], for example), and it is not hard to show that A, E, and the isogeny J ∼ A × E can all be defined over Q. There is a finite set of possibilities for the isogeny class of E, since its conductor must divide that of J, and by comparing Euler factors one can quickly rule out all but one possibility.…”