关于镜像映射的泰勒系数的完整性，II
On the integrality of the Taylor coefficients of mirror maps, II
摘要：
我们继续我们的研究开始在“关于完整性的泰勒系数的镜像映射”（arXiv：0907.2577）的精细完整性性质的泰勒系数的系列${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$，其中${\bf F}(z)$和${\bf G}(z)+\log(z) {\bf F}(z)$是特定的解决方案，某些超几何微分方程的最大单值在$z=0$。更准确地说，我们解决的问题，找到最大的整数$v$，使泰勒系数的$(z ^{-1}{\bf q}(z))^{1/v}$仍然是整数。特别地，我们确定Dwork-Kontsevich序列$(u_N)_{N\ge1}$，其中$u_N$是最大整数，使得$q(z)^{1/u_N}$是具有整数系数的级数，其中$q(z)=\exp(G(z)/F(z))$、$F(z)=\sum_{m=0}^{\infty} (Nm)! z^m/m!^N$和$G(z)=\sum_{m=1}^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$，其中$H_n$表示$n$次谐波数，条件的猜想，有没有素数$p$和整数$N$，使$p$ -adic估值$H_N-1$严格大于3。
英文摘要：
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.
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