Int\'egralit\'e des coefficients de Taylor de racines d'applications miroir

TL;DR

This paper proves the integrality of Taylor coefficients of mirror maps, which are special power series solutions to hypergeometric differential equations, confirming a conjecture by Zhou.

Contribution

It establishes the integrality of Taylor coefficients of mirror maps, advancing understanding of their arithmetic properties and confirming Zhou's conjecture.

Findings

Proved integrality of Taylor coefficients of mirror maps.

Validated a conjecture by Zhou on these coefficients.

Extended techniques from previous related work.

Abstract

We prove the integrality of the Taylor coefficients of roots 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 enables us to prove a conjecture stated by Zhou in "Integrality properties of variations of Mahler measures" [arXiv:1006.2428v1 math.AG]. The proof of these results is an adaptation of the techniques used in our article "Crit\`ere pour l'int\'egralit\'e des coefficients de Taylor des applications miroir", [J. Reine Angew. Math. (to appear)].