Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 086, 16 pages      arXiv:1805.00544      https://doi.org/10.3842/SIGMA.2018.086
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

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

Wadim Zudilin abc
a) Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands
b) School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
c) Laboratory of Mirror Symmetry and Automorphic Forms, National Research University Higher School of Economics, 6 Usacheva Str., 119048 Moscow, Russia

Received May 03, 2018, in final form August 13, 2018; Published online August 17, 2018

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.

Key words: hypergeometric equation; bilateral hypergeometric series; modular form; Calabi-Yau manifold.

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