Special Values of Hypergeometric Functions and Periods of CM Elliptic Curves

Abstract

Let \(X=X_0^6(1)/W_6\) be the quotient of the Shimura curve \(X_0^6(1)\) by all the Atkin-Lehner involutions. By realizing modular forms on X 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 \(\overline Q\) with complex multiplication.