This material has been published in "Théories galoisiennes et
arithmétiques des équations différentielles,"
L. Di Vizio and T. Rivoal (eds.), Séminaires et Congrès,
Soc. Math. France, vol. 27, Paris, 2011,
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Christian Krattenthaler and
Multivariate p-adic formal congruences and
integrality of Taylor coefficients of mirror maps
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,
we show that the Taylor coefficients of the multi-variable series
q(z)=ziexp(G(z)/F(z)) are integers, where
G(z)+log(zi) F(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
Calabi-Yau equations" of Almkvist, van Enckevort,
van Straten and
In particular, our results prove a conjecture of Batyrev and
[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.
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