Criterion for the integrality of the Taylor coefficients of mirror maps in several variables

TL;DR

This paper establishes a necessary and sufficient criterion for the integrality of Taylor coefficients of multivariable mirror maps, based on Landau's function and generalizing previous one-variable results, with implications for Calabi-Yau equations.

Contribution

It generalizes the integrality criterion for mirror map coefficients to several variables using Landau's function and advanced p-adic congruence techniques.

Findings

Provides a criterion based on Landau's function for multivariable mirror maps

Extends previous one-variable integrality results to multiple variables

Links integrality to properties of specific differential equations and Calabi-Yau models

Abstract

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 "Crit\`ere pour l'int\'egralit\'e 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.