Crit\`ere pour l'int\'egralit\'e des coefficients de Taylor des applications miroir

TL;DR

This paper establishes a necessary and sufficient condition for the integrality of Taylor coefficients of mirror maps, generalizing previous results and employing advanced techniques related to hypergeometric differential equations and Landau's function.

Contribution

It provides a new criterion for integrality of mirror map coefficients, extending prior work and utilizing a generalized Dwork theorem for formal congruences.

Findings

Criterion based on Landau's function for integrality

Generalization of Dwork's theorem on formal congruences

Extension of Krattenthaler-Rivoal results

Abstract

We give a necessary and sufficient condition for the integrality of the Taylor coefficients of mirror maps at the origin. By mirror maps, we mean formal power series z.exp(G(z)/F(z)), where F(z) and G(z)+log(z)F(z) are particular solutions of certain generalized hypergeometric differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated to the sequences of factorial ratios) and it generalizes results proved by Krattenthaler-Rivoal in "On the integrality of the Taylor coefficients of mirror maps" (to appear in Duke Math. J.). One of the techniques used to prove this criterion is a generalization of a theorem of Dwork on the formal congruences between formal series, which proved to be insufficient for our purposes.