Special Values of Hypergeometric Functions and Periods of CM Elliptic Curves

Let X = X 0 6 ( 1 ) ∕ W 6 $$X=X_0^6(1)/W_6$$ be the quotient of the Shimura curve X 0 6 ( 1 ) $$X_0^6(1)$$ 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 Q ¯ $$\overline Q$$ with complex multiplication.