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

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 "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.

Cited by in corpus (7)

• Calabi--Yau Operators
• Arithmetic properties of Apéry-like numbers
• On logarithmic solutions of A-hypergeometric systems
• On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps
• On the integrality of factorial ratios and mirror maps
• Criterion for the integrality of hypergeometric series with parameters from quadratic fields
• On integrality properties of hypergeometric series