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

TL;DR

This paper investigates the degeneracy of Jacobians of specific algebraic curves and characterizes their Sato-Tate groups using a construction by Shioda, providing new insights into the structure of these abelian varieties.

Contribution

It introduces a novel approach to analyze the Sato-Tate groups of degenerate Jacobians of curves defined by $y^2=x^{p^2}-1$, utilizing Shioda's construction to identify indecomposable Hodge classes.

Findings

Characterization of indecomposable Hodge classes in these Jacobians

Determination of the Sato-Tate groups for the family of curves

Development of techniques inspired by computational examples

Abstract

We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves $C_{p^2}: y^2=x^{p^2}-1$, where $p$ is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper.