Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps

Abstract

We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\mathbf{z}=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\mathbf{z})=z_i\exp (G(\mathbf{z})/F(\mathbf{z}))$ are integers, where $F(\mathbf{z})$ and $G(\mathbf{z})+\log (z_i)F(\mathbf{z})$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multi-variable mirror maps of Calabi-Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the "Tables of Calabi-Yau equations" [arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.

28 pages, AmS-LaTeX; title changed; the original manuscript was restructured

References in corpus (3)

• Integrality of instanton numbers and p-adic B-model
• Integrality of instanton numbers
• Critère pour l'intégralité des coefficients de Taylor des applications miroir

Cited by in corpus (8)

• Integrality theorems in the theory of topological strings
• Criterion for the integrality of the Taylor coefficients of mirror maps in several variables
• Some integrality properties in local mirror symmetry
• On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps
• Integrality Properties of Variations of Mahler Measures
• Algebraic independence of $G$-functions and congruences "à la Lucas"
• Criterion for the integrality of hypergeometric series with parameters from quadratic fields
• Integrality Properties of Open-Closed Mirror Maps