Special values of hypergeometric functions and periods of CM elliptic curves

Yang, Yifan

doi:10.1090/tran/7134

Special values of hypergeometric functions and periods of CM elliptic curves
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Trans. Amer. Math. Soc. 370 (2018), 6433-6467 Request permission

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.

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