Endomorphism algebras of Jacobians of certain superelliptic curves

TL;DR

This paper investigates the structure of endomorphism algebras of Jacobians of specific superelliptic curves over characteristic zero fields, revealing they are products of cyclotomic fields using Galois theory.

Contribution

It demonstrates that for certain superelliptic curves, the endomorphism algebras are explicitly characterized as products of cyclotomic fields, expanding understanding of their algebraic structure.

Findings

Endomorphism algebras are products of cyclotomic fields.

Galois theory is used to analyze the algebraic structure.

Results apply to superelliptic curves of the form y^q=f(x).

Abstract

Let $p$ be a prime, and $q$ a power of $p$. Using Galois theory, we show that over a field $K$ of characteristic zero, the endomorphism algebras of the jacobians of certain superelliptic curves $y^q=f(x)$ are products of cyclotomic fields.