A Hypergeometric Version of the Modularity of Rigid CalabiYau Manifolds
Abstract
We examine instances of modularity of (rigid) CalabiYau 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 $a(p)\le2p^{(m1)/2}$, 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.00544
 Bibcode:
 2018arXiv180500544Z
 Keywords:

 Mathematics  Number Theory;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  KTheory and Homology;
 11F33;
 11T24;
 14G10;
 14J32;
 14J33;
 33C20
 EPrint:
 SIGMA 14 (2018), 086, 16 pages