This material has been published in Commun. Number Theory Phys. 3 (2009), 555-591, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by International Press. This material may not be copied or reposted without explicit permission.

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-1oq(z))1/v are still integers. In particular, we determine the Dwork-Kontsevich sequence (uN)N>=1, where uN 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)! zm/m!N and G(z) = \sum _{m=1} ^{\infty} (HNm-Hm)(Nm)! zm/m!N, with Hn 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 Hn-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 HL, and the one on the p-adic valuation of HL-1.

The following versions are available:

Back to Christian Krattenthaler's home page.