This material has been published in
Duke Math. J. 151 (2010), 175-218,
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Christian Krattenthaler and
Tanguy Rivoal
On the integrality of the Taylor coefficients of mirror maps
(34 pages)
Abstract.
We show that the Taylor coefficients of the series
q(z) = z exp(G(z)/F(z))
are integers, where F(z) and
G(z) + log(z) F(z)
are specific solutions of certain hypergeometric differential
equations with maximal unipotent monodromy at z=0.
We also address the question of finding the largest integer u
such that the Taylor coefficients of
(z-1q(z))1/u
are still integers.
As consequences, we are able to prove numerous integrality
results for the Taylor coefficients of mirror maps
of Calabi-Yau complete intersections in weighted projective spaces,
which improve and refine previous results by Lian and Yau, and by Zudilin.
In particular, we prove the general ``integrality'' conjecture of
Zudilin about these mirror maps.
Comment. This is the first part of an originally larger
paper of the same title. The second
part, entitled "On the integrality of
the Taylor coefficients of mirror maps, II,"
contains further refinements of the integrality
assertions in the first part for more restricted choices of the
parameters. This work has been extended to multivariable mirror
maps in On the integrality of the Taylor
coefficients of mirror maps in several variables".
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