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,
pp. 279-307,
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Christian Krattenthaler and
Tanguy Rivoal
Multivariate p-adic formal congruences and
integrality of Taylor coefficients of mirror maps
(27 pages)
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 z=(z1,z2,...,zd),
we show that the Taylor coefficients of the multi-variable series
q(z)=ziexp(G(z)/F(z)) are integers, where
F(z) and
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
"Tables of
Calabi-Yau equations" 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.
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