Special values of hypergeometric functions and periods of CM elliptic curves

Abstract

Let X 0 6 ( 1 ) / W 6 X_0^6(1)/W_6 be the Atkin–Lehner quotient of the Shimura curve X 0 6 ( 1 ) X_0^6(1) associated to a maximal order in an indefinite quaternion algebra of discriminant 6 6 over Q \mathbb {Q} . By realizing modular forms on X 0 6 ( 1 ) / W 6 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 Q ¯ \overline {\mathbb {Q}} with complex multiplication.