# Sato–Tate groups of abelian threefolds: a preview of the classification

@article{Fit2019SatoTateGO, title={Sato–Tate groups of abelian threefolds: a preview of the classification}, author={Francesc Fit{\'e} and Kiran S. Kedlaya and Andrew V. Sutherland}, journal={arXiv: Number Theory}, year={2019} }

We announce the classification of Sato-Tate groups of abelian threefolds over number fields; there are 410 possible conjugacy classes of closed subgroups of USp(6) that occur. We summarize the key points of the "upper bound" aspect of the classification, and give a more rigorous treatment of the "lower bound" by realizing 33 groups that appear in the classification as maximal cases with respect to inclusions of finite index. Further details will be provided in a subsequent paper.

#### 12 Citations

Sato-Tate groups of abelian threefolds

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Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It… Expand

Sato-Tate Distributions of Catalan Curves

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For distinct odd primes p and q, we define the Catalan curve Cp,q by the affine equation y = x − 1. In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting… Expand

Determining monodromy groups of abelian varieties

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Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate… Expand

On a local-global principle for quadratic twists of abelian varieties

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Let A and A be abelian varieties defined over a number field k of dimension g ≥ 1. For g ≤ 3, we show that the following local-global principle holds: A and A are quadratic twists of each other if… Expand

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

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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… Expand

Towards the Sato–Tate groups of trinomial hyperelliptic curves

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- 2018

We consider the identity component of the Sato-Tate group of the Jacobians of curves of the form $$C_1:y^2=x^{2g+2}+c, C_2:y^2=x^{2g+1}+cx, C_3:y^2=x^{2g+1} +c,$$ where $g$ is the genus of the curve… Expand

Distribution of values of Gaussian hypergeometric functions

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In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to… Expand

Sato-Tate Distributions of $y^2=x^p-1$ and $y^2=x^{2p}-1$

- Mathematics
- 2020

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form $$ y^2=x^p-1 \text{ and } y^2=x^{2p}-1,$$ where $p$ is an odd prime. Our… Expand

WEIL COHOMOLOGY IN PRACTICE

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These are lecture notes from a course given by Kiran Kedlaya at UC San Diego in fall 2019 on the topic “Weil cohomology in practice”. Thanks to the following students in the course for compiling the… Expand

Computing $L$-polynomials of Picard curves from Cartier-Manin matrices

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- 2021

We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. By determining the density of the set of… Expand

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