关于镜像映射Taylor系数的完整性

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摘要翻译：
证明了级数${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$的Taylor系数是整数，其中${\bf F}(z)$和${\bf G}(z)+\log(z){\bf F}(z)$是某些具有极大单幂单数的超几何微分方程在$z=0$处的具体解。我们还讨论了求最大整数$u$的问题，使得Taylor系数$(z^{-1}{\bf q}(z))^{1/u}$仍然是整数。结果证明了加权射影空间中Calabi-Yau完全交的镜像映射的Taylor系数的许多积分结果，改进和完善了Lian和Yau以及Zudilin的结果。特别地，我们证明了Zudilin关于这些镜像映射的一般的“完整性”猜想。本研究的进一步结果是确定了Dwork-Kontsevich序列$(u_N)_{n\ge1}$,其中$u_N$是最大的整数，使得$q(z)^{1/u_N}$是一个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$g(z)=\sum_{m=1}^{\infty}(H_{Nm}-H_m)(Nm)！Z^m/m！^n$，其中$h_n$表示$n$次调和数，条件是没有素数$p$和整数$n$，使得$h_n-1$的$p$-adic值严格大于3。
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英文标题：
《On the integrality of the Taylor coefficients of mirror maps》
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作者：
Christian Krattenthaler (Universit\"at Wien) and Tanguy Rivoal (CNRS,
  Universit\'e de Grenoble)
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
--
一级分类：Physics        物理学
二级分类：High Energy Physics - Theory        高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
--
一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
--

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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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PDF链接：
https://arxiv.org/pdf/0709.1432

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关键词：Taylor 完整性 coefficients Consequences Differential 证明 完整性 猜想 结果