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

We continue our study begun in “On the integrality of the Taylor coefficients of mirror maps” (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series $\mathbf{q}(z)=z\exp (\mathbf{G}(z)\mathrm{/}\mathbf{F}(z))$, where $\mathbf{F}(z)$ and $\mathbf{G}(z)+\log (z)\mathbf{F}(z)$ 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 $(z ^{-1}{\bf q}(z))^{1/v}$ are still integers. In particular, we determine the Dwork-Kontsevich sequence $(u_N{)}_{N\ge 1}$, where $u_N$ is the largest integer such that $q(z{)}^{1\mathrm{/}{u}_N}$ is a series with integer coefficients, where $q(z)=\exp (G(z)\mathrm{/}F(z))$, $F(z)={\sum }_{m=0}^{\mathrm{\infty }}(Nm)!z^m\mathrm{/}m{!}^N$ and $G(z)={\sum }_{m=1}^{\mathrm{\infty }}(H_{Nm}-H_m)(Nm)!z^m\mathrm{/}m{!}^N$, 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.