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

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.