Sato-Tate Distributions of Catalan Curves

2021/09/15 by Heidi Goodson, Goodson, Heidi · 2 citations
Arts and Humanities · Mathematics · #11G10 #11G20 #11M50 #14G10 #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2109.07417

openalex publication_date 2021/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Degeneracy and Sato-Tate groups of y²=x^p2-1
• Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

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