Modular-type functions attached to Calabi-Yau varieties: integrality properties

Abstract

We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function ${}_n{F}_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. We prove its consequence on the $N$-integrality of the mirror map for the particular cases $1\le n\le 4$. For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle groups with a cusp, and for $n=4$ we prove that $14$ examples of hypergeometric Calabi-Yau equations are the full classification of hypergeometric mirror maps with integral coefficients. As a by-product we get the integrality of the corresponding algebra of modular-type functions. These are natural generalizations of the algebra of classical modular and quasi-modular forms in the case $n=2$.

This paper will appear as an appendix in a monograph of the first author

References in corpus (4)

• Tables of Calabi--Yau equations
• Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds
• Integrality of instanton numbers and p-adic B-model
• Dwork Congruences and reflexive Polytopes