Generalized Legendre curves and Quaternionic Multiplication

TL;DR

This paper investigates abelian varieties from generalized Legendre curves, focusing on their Galois representations, periods, and endomorphism algebras, especially identifying when these contain quaternion algebras.

Contribution

It characterizes conditions under which the endomorphism algebra of certain abelian varieties includes a quaternion algebra, advancing understanding of their algebraic structure.

Findings

Identifies when endomorphism algebras contain quaternion algebras.

Analyzes Galois representations associated with these abelian varieties.

Provides criteria for quaternionic multiplication in specific families.

Abstract

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of $\Bbb Q$ contains a quaternion algebra.