A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree

TL;DR

This paper discusses the decomposition of the Jacobian variety of Fermat curves of prime degree, showing that the factors are Jacobians of specific p-gonal curves, building on recent automorphism group analyses.

Contribution

It provides a new perspective by linking the factors in the Jacobian decomposition to Jacobian varieties of p-gonal curves, extending prior automorphism group results.

Findings

Factors in the Jacobian decomposition are Jacobians of p-gonal curves

Builds on recent automorphism group descriptions of Fermat curves

Offers a new geometric interpretation of the decomposition

Abstract

Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree $p$, for $p$ a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain $p$-gonal curves.