##### This material has been published in
Commun. Number Theory Phys.
**3** (2009), 555-591,
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## Christian Krattenthaler and
Tanguy Rivoal

# On the integrality of the Taylor coefficients of mirror maps, II

### (27 pages)

**Abstract.**
We continue our study begun in
"On the integrality of the Taylor
coefficients of mirror maps" of
the fine integrality properties of the Taylor coefficients of the series
**q**(*z*) = *z* exp(**G**(*z*)/**F**(*z*)),
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. More precisely,
we address the question of finding the largest integer *v*
such that the Taylor coefficients of
(*z*^{-1}o**q**(*z*))^{1/v} are still
integers. In particular, we determine the
Dwork-Kontsevich sequence (*u*_{N})_{N>=1},
where *u*_{N} is the
largest integer such that *q*(*z*)^{1/uN}
is a series with integer
coefficients, where *q*(*z*) =
exp(*F*(*z*)/*G*(*z*)),
*F*(*z*) = *\sum _{m=0} ^{\infty}* (*Nm*)!
*z*^{m}/*m*!^{N} and
*G*(*z*) = *\sum _{m=1} ^{\infty}* *(H*_{Nm}-H_{m})(*Nm*)!
*z*^{m}/*m*!^{N}, with *H*_{n} denoting
the *n*-th harmonic number,
conditional on the conjecture that there
are no prime number *p* and integer *N*
such that the *p*-adic
valuation of *H*_{n}-1 is strictly greater than 3.

**Comment.** This is the second part of an originally larger
paper of the same title. The first
part, entitled "On the integrality of
the Taylor coefficients of mirror maps,"
contains integrality assertions for more general classes of mirror
maps, which are however weaker than the results for the more special
families in this second part.

See the supplement to the paper
on the *p*-adic valuation of harmonic numbers
*H*_{L}, and the one
on the *p*-adic valuation of
*H*_{L}-1.

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