On the integrality of the Taylor coefficients of mirror maps, II

TL;DR

This paper investigates the integrality properties of Taylor coefficients of mirror maps derived from hypergeometric differential equations, identifying the largest integers for which certain series have integer coefficients, and determining the Dwork-Kontsevich sequence.

Contribution

It extends previous work by explicitly determining the Dwork-Kontsevich sequence and analyzing the integrality of fractional powers of mirror map series under specific conditions.

Findings

Determined the Dwork-Kontsevich sequence $(u_N)$ explicitly.

Established conditions under which fractional powers of mirror map series have integer coefficients.

Provided insights into the p-adic valuation constraints related to harmonic numbers.

Abstract

We continue our study begun in "On the integrality of the Taylor coefficients of mirror maps" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. More precisely, we address the question of finding the largest integer $v$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/v}$ are still integers. In particular, we determine the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(G(z)/F(z))$, $F(z)=\sum_{m=0}^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1}^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.