Communications in Number Theory and Physics

Volume 3 (2009)

Number 3

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

Pages: 555 – 591



Christian Krattenthaler (Fakultät für Mathematik, Universität Wien, Austria)

Tanguy Rivoal (Institut Fourier, Université Grenoble 1, Saint-Martin d’Hères, France)


We continue our study begun in“On the integrality of the Taylor coefficients of mirror maps”[Duke Math. J. (to appear)]of the fine integrality properties of the Taylor coefficients of the series${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and${\bf G}(z)+\log(z) {\bf F}(z)$are specific solutions of certain hypergeometric differentialequations 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 $(z ^{-1}{\bf q}(z))^{1/v}$ are stillintegers. In particular, we determine theDwork–Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is thelargest integer such that$q_N(z)^{1/u_N}$ is a series with integercoefficients, where $q_N(z)=\exp(G_N(z)/F_N(z))$,$F_N(z)=\sum _{m=0} ^{\infty} (Nm)!\,z^m/m!^N$ and$G_N(z)=\sum _{m=1} ^{\infty} (H_{Nm}-H_m)(Nm)!\break z^m/m!^N$, with $H_n$ denotingthe $n$th harmonic number, conditional on the conjecture that thereare no prime number $p$ and integer $N$ such that the $p$-adicvaluation of $H_N-1$ is strictly greater than~$3$.

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