Generalized Legendre curves and Quaternionic Multiplication

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 $\mathbb{Q}$ contains a quaternion algebra.

References in corpus (3)

• Schwarzian differential equations and Hecke eigenforms on Shimura curves
• Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions
• Hypergeometric functions over F_p and relations to elliptic curves and modular forms

Cited by in corpus (5)

• Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions
• A Cubic Transformation Formula for Appell-Lauricella Hypergeometric Functions over Finite Fields
• Modularity and effective Mordell I
• A Whipple ${}_7{F}_6$ formula revisited
• Potentially ${\text{GL}}_2$-type Galois representations associated to noncongruence modular forms