A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

Abstract

We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of p and from Weil's general bounds |a(p)| ≤ 2p⁽ᵐ⁻¹⁾/², where m is the weight of the form. Furthermore, the critical L-values of the modular form are predicted to be Q-proportional to the values of a related basis of solutions to the hypergeometric differential equation.