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

TL;DR

This paper explores the modularity of rigid Calabi-Yau manifolds through hypergeometric functions, linking their periods to modular forms and predicting relationships between L-values and hypergeometric solutions.

Contribution

It introduces a hypergeometric perspective on the modularity of Calabi-Yau manifolds, connecting hypergeometric series to modular form coefficients and L-values.

Findings

Coefficients of modular forms can be derived from hypergeometric series modulo p.

Predicted proportionality between L-values and hypergeometric solutions.

Hypergeometric functions encode the periods of 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 be often 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 $|a(p)|\le2p^{(m-1)/2}$, where $m$ is the weight of the form. Furthermore, the critical $L$-values of the modular form are predicted to be $\mathbb Q$-proportional to the values of a related basis of solutions to the hypergeometric differential equation.