Special values of hypergeometric functions and periods of CM elliptic curves

2015/03/27 by Yifan Yang, Yang, Yifan
Mathematics · #11F12 #11G15 #11G18 #33C05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F12 #msc:11G15 #msc:11G18 #msc:33C05

paper · pdf · doi:10.48550/arxiv.1503.07971

The accompanying file is the Magma code used to compute the values of Borcherds forms at CM-points

arxiv created 2015/03/27 · arxiv updated 2015/03/30

Abstract

Let X₀⁶(1)/W₆ be the Atkin-Lehner quotient of the Shimura curve X₀⁶(1) associated to a maximal order in an indefinite quaternion algebra of discriminant 6 over \mathbb Q. By realizing modular forms on X₀⁶(1)/W₆ 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 \mathbb Q with complex multiplication.

Related

• Periods of modular forms and applications to the conjectures of Oda and of Prasanna-Venkatesh
• Ramanujan-type identities for Shimura curves
• Singular Moduli of Shimura Curves
• Hypergeometric functions over F_p and relations to elliptic curves and modular forms
• Exceptional Sets for Certain ₂F₁ Hypergeometric Functions