Sato-Tate Distributions of Catalan Curves

TL;DR

This paper investigates the Sato-Tate distributions of Jacobians of Catalan curves defined by specific affine equations, analyzing their statistical properties and Galois endomorphism types.

Contribution

It constructs Sato-Tate groups for these Jacobians, computes their distribution moments, and determines their Galois endomorphism types, advancing understanding of their arithmetic properties.

Findings

Sato-Tate groups are explicitly constructed for Catalan Jacobians.

Statistical and numerical moments of the distributions are computed.

Galois endomorphism types of the Jacobians are determined.

Abstract

For distinct odd primes $p$ and $q$, we define the Catalan curve $C_{p,q}$ by the affine equation $y^q=x^p-1$. In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting distributions of coefficients of their normalized L-polynomials.Catalan Jacobians are nondegenerate and simple with noncyclic Galois groups (of the endomorphism fields over $\mathbb Q$), thus making them interesting varieties to study in the context of Sato-Tate groups. We compute both statistical and numerical moments for the limiting distributions. Lastly, we determine the Galois endomorphism types of the Jacobians using both old and new techniques.