Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

TL;DR

This paper computes the Sato-Tate groups of Jacobian varieties of certain hyperelliptic curves over Q, extending existing results and connecting gamma function values with abelian variety invariants.

Contribution

It provides the first computation of Sato-Tate groups for these Jacobians and generalizes previous results to degenerate cases, also extending Gross-Koblitz formulas.

Findings

Computed Sato-Tate groups for Jacobians of y^2=x^m+1 over Q

Generalized Sato-Tate group descriptions to degenerate abelian varieties

Extended Gross-Koblitz formula relating gamma function values

Abstract

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. In this paper, we compute the Sato-Tate group of $J_m$. Currently, there is no general algorithm that computes this invariant. We also describe the Sato-Tate group of an abelian variety, generalizing existing results that apply only to non-degenerate varieties, and prove an extension of a well-known formula of Gross-Koblitz that relates values of the classical and $p$-adic gamma functions at rational arguments.