多元镜像映射Taylor系数整性的判定
Criterion for the integrality of the Taylor coefficients of mirror maps in several variables
摘要：
本文给出了形式幂级数$q_i({\mathbf z})=z_i\exp(G_i({\mathbf z})/F({\mathbf z}))$的Taylor系数在原点处可积的一个充要条件，其中${\mathbf z}=(z_1,...,z_d)$和$G_i({\mathbf z})+\log(z_i)F({\mathbf z})$，$i=1,...,d$是某些微分方程的A-系的特解.这个标准是基于朗道函数的分析性质（这是经典的阶乘比的序列相关联），它概括了标准的情况下，一个变量中提出的“关键是为了l 'intégralité des coefficients de Taylor des applications miroir”[J. Reine Angew. Math.]。用来证明这一准则的技术之一是德沃克关于形式级数之间的形式同余的定理的一个版本的推广，由Krattenthaler和Rivoal在“Multivariate $p$ -adic formal congruences and integrality of Taylor coefficients of mirror maps”[arXiv：0804.3049v3，math.NT]中证明。该准则涉及Almkvist，van Enckevort，van Straten和Zudilin的“Tables of Calabi-Yau equations”[arXiv：math/0507430 v2，math.AG]中所列的新的一元镜像映射的Taylor系数的完整性。
英文摘要：
We give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series $q_i({\mathbf z})=z_i\exp(G_i({\mathbf z})/F({\mathbf z}))$, with ${\mathbf z}=(z_1,...,z_d)$ and where $F({\mathbf z})$ and $G_i({\mathbf z})+\log(z_i)F({\mathbf z})$, $i=1,...,d$ are particular solutions of certain A-systems of differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated with the sequences of factorial ratios) and it generalizes the criterion in the case of one variable presented in "Critère pour l'intégralité des coefficients de Taylor des applications miroir" [J. Reine Angew. Math.]. One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on the formal congruences between formal series, proved by Krattenthaler and Rivoal in "Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps" [arXiv:0804.3049v3, math.NT]. This criterion involves the integrality of the Taylor coefficients of new univariate mirror maps listed in "Tables of Calabi--Yau equations" [arXiv:math/0507430v2, math.AG] by Almkvist, van Enckevort, van Straten and Zudilin.
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