Special values of hypergeometric functions and periods of CM elliptic curves

TL;DR

This paper connects special hypergeometric function values to periods of CM elliptic curves by analyzing Shimura curves, modular forms, and Borcherds forms, providing explicit CM value formulas.

Contribution

It introduces a novel approach to express hypergeometric function values in terms of elliptic curve periods using Shimura curves and Borcherds forms.

Findings

Explicit formulas for hypergeometric values at CM points

Connection between hypergeometric functions and elliptic curve periods

Application of Schofer's formula to Shimura curves

Abstract

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.