A criterion for the integrality of the Taylor coefficients of mirror maps in several variables

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) (z) = _(z i) exp (_(G i) (z) / F (z)), with z = (_(z1,..., z d_) and where F (z) and _(G i) (z) + log (_(z i)) F (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 sequences of factorial ratios) and it generalizes the criterion in the case of one variable presented in [E. Delaygue, Critère pour l'intégralité des coefficients de Taylor des applications miroir, J. Reine Angew. Math. 662 (2012) 205-252]. One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on formal congruences between formal series, proved by Krattenthaler and Rivoal in [C. Krattenthaler, T. Rivoal, Multivariate p-adic formal congruences and integrality of Taylor coefficients of mirror maps, in: L. Di Vizio, T. Rivoal (Eds.), Théories Galoisiennes et Arithmétiques des équations Différentielles, in: Séminaire et Congrés, vol.27, Soc. Math. France, Paris, 2011, pp.279-307].