## Christian Krattenthaler and
Tanguy Rivoal

# On the integrality of the Taylor coefficients of mirror maps

### (54 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 **q**(*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.
A further outcome of the present study is the determination of 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}(*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} is strictly greater than 3.

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.

**Comment.** This is the original version of a paper which was later divided
into two parts: "On the integrality of
the Taylor coefficients of mirror maps" and
"On the integrality of
the Taylor coefficients of mirror maps, II".
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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