关于镜像映射的Taylor系数的整性
On the integrality of the Taylor coefficients of mirror maps
摘要：
我们证明了级数${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$的泰勒系数是整数，其中${\bf F}(z)$和${\bf G}(z)+\log(z) {\bf F}(z)$是在$z=0$上具有极大幂单值的某些超几何微分方程的特解.我们还解决的问题，找到最大的整数$u$，使泰勒系数的$(z ^{-1}{\bf q}(z))^{1/u}$仍然是整数。作为结果，我们能够证明许多完整性结果的镜像映射的Calabi-Yau完全交叉加权射影空间，改进和完善以前的结果由廉和丘，和Zudilin的泰勒系数。特别地，我们证明了Zudilin关于这些镜像映射的一般“完整性”猜想。本研究的进一步结果是确定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$q(z)=\exp(F(z)/G(z))$，$F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$和$G(z)=\sum_{m=1} ^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$，$H_n$表示$n$的第三谐波数，条件的猜想，有没有素数$p$和整数$N$，使$p$ -$H_N-1$的adic赋值严格大于3。
英文摘要：
We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, 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 $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general ``integrality'' conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of 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 $q(z)=\exp(F(z)/G(z))$, $F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$ 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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