Endomorphism algebras of factors of certain hypergeometric Jacobians

TL;DR

This paper classifies the endomorphism algebras of factors of hypergeometric Jacobians over characteristic zero fields, revealing they are typically cyclotomic fields, quadratic extensions, or direct sums, extending elliptic curve results.

Contribution

It generalizes the classification of endomorphism algebras from elliptic curves to certain hypergeometric Jacobians, identifying their typical algebraic structures.

Findings

Endomorphism algebras are cyclotomic fields, quadratic extensions, or direct sums.

Most cases follow a predictable algebraic pattern, with few exceptions.

Results extend classical elliptic curve endomorphism classifications.

Abstract

We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field $E=\mathbb{Q}(\zeta_q)$, or a quadratic extension of $E$, or $E\oplus E$. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over $\mathbb{C}$.