\documentclass[reqno,12pt]{amsart} \usepackage{a4wide} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \numberwithin{equation}{section} \newtheorem{theo}{Theorem} \newtheorem{conj}{Conjecture} \newtheorem{coro}{Corollary} \newtheorem{prop}{Proposition} \newtheorem{spec}{Speculation} \newtheorem{lem}{Lemma} \theoremstyle{remark} \newtheorem{Remark}{Remark} \newtheorem{Remarks}[Remark]{Remarks} \def\sd{\sigma} \def\rr{\rho} \def\ro{\textup{o}} \def\io{\infty} \def\gaa{\Gamma} \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\C{\mathcal{C}} \def\de{\delta} \def\ep{\varepsilon} \def\ze{\zeta} \def\et{\eta} \def\th{\theta} \def\vt{\vartheta} \def\io{\iota} \def\ka{\kappa} \def\la{\lambda} \def\rh{\rho} \def\si{\sigma} \def\ta{\tau} \def\te{\theta} \def\ph{\varphi} \def\ch{\chi} \def\ps{\psi} \def\om{\omega} \def\Ga{\Gamma} \def\De{\Delta} \def\Th{\Theta} \def\La{\Lambda} \def\Li{\textup{Li}} \def\Si{\Sigma} \def\Ph{\Phi} \def\Ps{\Psi} \def\Om{\Omega} \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\ii{\infty} \def\fl#1{\left\lfloor#1\right\rfloor} \def\li{\operatorname{Li}} \def\lcm{\operatorname{lcm}} \def\dis{\displaystyle} \def\dd{\textup{d}} \setcounter{tocdepth}{2} \begin{document} \title[]{On the integrality of the Taylor coefficients of mirror maps, II} \author[]{C. Krattenthaler$^\dagger$ and T. Rivoal} \date{\today} \address{C. Krattenthaler, Fakult\"at f\"ur Mathematik, Universit\"at Wien, Nordbergstra{\ss}e~15, A-1090 Vienna, Austria. WWW: \tt http://www.mat.univie.ac.at/\~{}kratt.} \address{T. Rivoal, Institut Fourier, CNRS UMR 5582, Universit{\'e} Grenoble 1, 100 rue des Maths, BP~74, 38402 Saint-Martin d'H{\`e}res cedex, France.\newline WWW: \tt http://www-fourier.ujf-grenoble.fr/\~{}rivoal.} \thanks{$^\dagger$Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and grant S9607-N13, the latter in the framework of the National Research Network ``Analytic Combinatorics and Probabilistic Number Theory''} \subjclass[2000]{Primary 11S80; Secondary 11J99 14J32 33C20} \keywords{Calabi--Yau manifolds, integrality of mirror maps, $p$-adic analysis, Dwork's theory, harmonic numbers, hypergeometric differential equations} \begin{abstract} We continue our study begun in {\em ``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 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 $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q_N(z)^{1/u_N}$ is a series with integer coefficients, where $q_N(z)=\exp(G_N(z)/F_N(z))$, $F_N(z)=\sum _{m=0} ^{\infty} (Nm)!\,z^m/m!^N$ and $G_N(z)=\sum _{m=1} ^{\infty} (H_{Nm}-H_m)(Nm)!\,z^m/m!^N$, 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~$3$. \end{abstract} \maketitle \section{Introduction and statement of results} The present article is a sequel to our article~\cite{kratrivmirror}, where we proved general results concerning the integrality properties of mirror maps. We shall prove here stronger integrality assertions for certain special cases that appear frequently in the literature. For any vector $\mathbf{N}=(N, \ldots, N)$ (with $k$ occurrences of $N$), where $N$ is a positive integer, let us define the power series \begin{equation} F_{\mathbf{N}}(z)= \sum_{m=0}^{\infty} \frac{(Nm)!^k}{m!^{kN}} z^m \label{eq:FN} \end{equation} and \begin{equation} G_{\mathbf{N}}(z)=\sum_{m=1}^{\infty} kN(H_{Nm}-H_m) \frac{(Nm)!^k}{m!^{kN}} z^m, \label{eq:GN} \end{equation} with $H_n:=\sum _{i=1} ^{n}\frac {1} {i}$ denoting the $n$-th harmonic number. The functions $F_{\mathbf{N}}(z)$ and $G_{\mathbf{N}}(z)+\log(z)F_{\mathbf{N}}(z)$ are solutions of the same hypergeometric differential equation with maximal unipotent monodromy at $z=0$. A basis of solutions with at most logarithmic singularities around $z=0$ can then be obtained by Frobenius' method; see~\cite{yoshida}. That differential equation is the Picard--Fuchs equation of a one parameter family of mirror manifolds $W'$ of a complete intersection $W$ of $k$ hypersurfaces $W_1, \ldots, W_k$, all of degree $N$ in $\mathbb{P}^{d+k}(\mathbb{C})$: $W$ is a family of Calabi--Yau manifolds if one chooses $d$ equal to $k(N-1)-1$. The mirrors $W'$ are explicitly constructed in~\cite[Sec.~5.2]{batstrat}. In the underlying context of mirror symmetry, it is natural to define the canonical coordinate $q_{\mathbf{N}}(z):=z\exp\big(G_{\mathbf{N}}(z)/F_{\mathbf{N}}(z)\big)$ and the mirror map $z_{\mathbf{N}}(q)$, which is the compositional inverse of $q_{\mathbf{N}}(z)$. In~\cite{kratrivmirror}, we proved the following result, which settled a conjecture in the folklore of mirror symmetry theory. \begin{equation} \label{eq:folk} \vbox{\hsize14.7cm {\medskip\leftskip1cm\rightskip1cm \em\noindent For any integers $k\ge 1$ and $N\ge 1$, we have $q_{\mathbf N}(z)\in z\mathbb{Z}[[z]]$ and $z_{\mathbf N}(q)\in q\mathbb{Z}[[q]]$, where $N$ is repeated $k$ times in the vector $\mathbf N=(N,\ldots, N)$.~$($\footnote{% In the number-theoretic study undertaken in the present paper, we are interested in the integrality of the coefficients of roots of mirror maps $z(q)$. In that context, $z(q)$ and the corresponding canonical coordinate $q(z)$ play strictly the same role, because $(z^{-1}q(z))^{1/\tau}\in 1+z\mathbb{Z}[[z]]$ for some integer $\tau$ implies that $(q^{-1}z(q))^{1/\tau}\in 1+q\mathbb{Z}[[q]]$, and conversely; see~\cite[Introduction]{lianyau1}. We shall, in the sequel, formulate our integrality results exclusively for canonical coordinates. By abuse of terminology, we shall often use the term ``mirror map'' for any canonical coordinate.}$)$ } } \end{equation} \medskip Lian and Yau \cite[Sec.~5, Theorem~5.5]{lianyau} had proved earlier the particular case of this theorem where $k=1$ and $N$ is a prime number, and Zudilin~\cite[Theorem~3]{zud} had extended their result to any $k\ge 1$ and any $N$ which is a prime power. Zudilin also formulated a more general conjecture, implying \eqref{eq:folk}, which he proved in a particular case. The conjecture was subsequently fully proved as one of the main results in~\cite{kratrivmirror}. For $k=1$, physicists made the observation that, apparently, even the stronger assertion \begin{equation}\label{eq:raflianyau} \big(z^{-1}q_{(N)}(z)\big)^{1/N}\in \mathbb{Z}[[z]] \end{equation} holds. This was proved by Lian and Yau \cite{lianyau2} for any prime number $N$, thus strengthening their result from~\cite{lianyau} mentioned above. The observation \eqref{eq:raflianyau} leads naturally to the more general question of determining the largest integer $V$ such that $\left(z ^{-1}q(z) \right) ^{1/V}\in \mathbb{Z}[[z]]$ for mirror maps $q(z)$ such as $q_{\mathbf{N}}(z)$.~(\renewcommand\thefootnote{1}\footnote{% In the number-theoretic study undertaken in the present paper, we are interested in the integrality of the coefficients of roots of mirror maps $z(q)$. In that context, $z(q)$ and the corresponding canonical coordinate $q(z)$ play strictly the same role, because $(z^{-1}q(z))^{1/\tau}\in 1+z\mathbb{Z}[[z]]$ for some integer $\tau$ implies that $(q^{-1}z(q))^{1/\tau}\in 1+q\mathbb{Z}[[q]]$, and conversely; see~\cite[Introduction]{lianyau1}. We shall, in the sequel, formulate our integrality results exclusively for canonical coordinates. By abuse of terminology, we shall often use the term ``mirror map'' for any canonical coordinate. \newline \indent $^2$Let $q(z)$ be a given power series in $\mathbb{Z}[[z]]$, and let $V$ be the largest integer with the property that $q(z)^{1/V}\in \mathbb Z[[z]]$. Then $V$ carries complete information about {\it all\/} integers with that property: namely, the set of integers $U$ with $q(z)^{1/U}\in \mathbb Z[[z]]$ consists of all divisors of $V$. Indeed, it is clear that all divisors of $V$ belong to this set. Moreover, if $U_1$ and $U_2$ belong to this set, then also $\lcm(U_1,U_2)$ does (cf.\ \cite[Lemma~5]{HeRSAA} for a simple proof based on B\'ezout's lemma).}) A rather general result in this direction has already been obtained in \cite[Theorem~2]{kratrivmirror}. The purpose of the present paper is to sharpen this earlier result for wide classes of special choices of the parameters occurring in \cite{kratrivmirror}. Indeed, our main results, given in Theorems~\ref{thm:3} and \ref{thm:3a} below, provide values of $V$ for infinite families of mirror(-type) maps, which, conditional on widely believed conjectures on the $p$-adic valuations of $H_N$ respectively $H_N-1$, are optimal for these families. \medskip To describe our results, for positive integers $L$ and $N$, we set \begin{equation} G_{L,\mathbf{N}}(z)=\sum_{m=1}^{\infty} H_{Lm} \frac{(Nm)!^k}{m!^{kN}} z^m, \label{eq:GLN} \end{equation} where again $\mathbf N=(N,N,\dots,N)$, with $k$ occurrences of $N$. We then define the mirror-type map $q_{L,\mathbf{N}}(z):=\exp\big(G_{L,\mathbf{N}}(z)/F_{\mathbf{N}}(z)\big)$. Obviously, the mirror map $q_{\mathbf{N}}(z)$ can be expressed as a product of the series $q_{L,\mathbf{N}}(z)$, namely as \begin{equation} \label{eq:truemap} q_{\mathbf{N}}(z)= zq_{N,\mathbf N}^{kN}(z)q_{1,\mathbf N}^{-kN}(z). \end{equation} The special case of the afore-mentioned Theorem~2 from \cite{kratrivmirror} where $\mathbf N=(N,N,\dots,N)$ (with $k$ occurrences of $N$) addressed the above question of ``maximal integral roots'' for the mirror-type map $q_{L,\mathbf N}(z)$. It reads as follows: \begin{equation} \label{eq:NNNN} \vbox{\hsize14cm {\leftskip1cm\rightskip1cm \em\noindent Let $\Th_L:=L!/\gcd(L!, L!\,H_L)$ be the denominator of $H_L$ when written as a reduced fraction. For any positive integers $N$ and $L$ with $L\le N$, we have $q_{L,\mathbf{N}}(z)^{\frac{\Th_L}{N!^k}} \in\mathbb{Z}[[z]].$ } } \end{equation} \medskip As we remarked in \cite{kratrivmirror}, this result is optimal in the case that $L=1$; that is, no integer $V$ larger than $N!^k/\Th_1=N!^k$ can be found such that $q_{L,\mathbf{N}}(z)^{1/V} \in\mathbb{Z}[[z]]$. However, if $L\ge2$, improvements may be possible. Our first main result provides such an improvement. In order to state the result, we need to introduce usual notation for $p$-adic valuation: given a prime number $p$ (and from now on, $p$ will always denote a prime number) and $\alpha\in \mathbf Q_p$, $v_p(\alpha)$ denotes the $p$-adic valuation of $\alpha$. \begin{theo} \label{thm:3} Let $N$ be a positive integer, $\mathbf N=(N,N,\dots,N)$, with $k$ occurrences of $N$, and let $\Xi_1=1$, $\Xi_7=1/140$, and, for $N\notin\{1,7\}$, \begin{equation} \label{eq:Xi} \Xi_N:= {\prod _{p\le N} ^{}}p^{\min\{2+\xi(p,N),v_p(H_N)\}}, \end{equation} where $\xi(p,N)=1$ if $p$ is a Wolstenholme prime {\em(}i.e., a prime $p$ for which $v_p(H_{p-1})\ge3$ {\em(}\renewcommand\thefootnote{3}\footnote{Presently, only two such primes are known, namely $16843$ and $2124679$, and it is unknown whether there are infinitely many Wolstenholme primes or not.}{\em))} or $N$ is divisible by $p$, and $\xi(p,N)=0$ otherwise. Then $q_{N,\mathbf N}(z)^{\frac{1}{\Xi_{N}N!^k}} \in\mathbb{Z}[[z]].$ \end{theo} \begin{Remarks} \label{rem:Xi7} For better comprehension, we discuss the meaning of the statement of Theorem~\ref{thm:3} and its implications; in particular, we address some fine points of the definition of $\Xi_N$. \smallskip (a) The case of $N=1$ is trivial since $q_{1,(1,\dots,1)}(z)=1/(1-z)$. Furthermore, we have $$\Xi_7=\frac {1} {140}=2^{v_2(H_7)}5^{v_5(H_7)}7^{v_7(H_7)},$$ which differs by a factor of $3$ from the right-hand side of \eqref{eq:Xi} with $N=7$ (since $v_3(H_7)=v_3(\frac {363} {140})=1$). \smallskip (b) Let $V_N$ denotes the largest integer such that $\big(z^{-1}q_{N,\mathbf N}(z)\big)^{1/V_N}$ is a series with integer coefficients. Since $q_{N,\mathbf N}(z)=1+H_N N!^kz+\mathcal O(z^2)$, it is clear that $q_{N,\mathbf N}(z)^{1/(p^{v_p(H_N)+1}N!^k)}\notin \mathbb{Z}[[z]]$, so that the exponent of $p$ in the prime factorisation of $V_{N}$ can be at most $v_p(H_N N!^k)$. In Theorem~\ref{thm:3}, this theoretically maximal exponent is cut down to $v_p(\Xi_N N!^k)$. First of all, the number $\Xi_N$ contains no prime factor $p>N$, whereas the harmonic number $H_N$ may very well do so (and, in practice, always does for $N>1$). Moreover, for primes $p$ with $p\le N$ and $v_p(H_N)\ge3$, the definition of $\Xi_N$ cuts the theoretically maximal exponent $v_p(H_NN!^k)$ of $p$ down to $2+v_p(N!^k)$ respectively $3+v_p(N!^k)$, depending on whether $\xi(p,N)=0$ or $\xi(p,N)=1$. In items~(c)--(e) below, we address the question of how serious this cut is expected to be. \smallskip (c) Clearly, the minimum appearing in the exponent of $p$ in the definition \eqref{eq:Xi} of $\Xi_N$ is $v_p(H_N)$ as long as $v_p(H_N)\le 2$. In other words, the exponent of $p$ in the prime factorisation of $\Xi_N$ depends largely on the $p$-adic behaviour of $H_N$. An extensive discussion of this topic, with many interesting results, can be found in \cite{boyd}. We have as well computed a table of harmonic numbers $H_N$ up to $N=1000000$.~(\renewcommand\thefootnote{4}\footnote{The summary of the table is available at {\tt http://www.mat.univie.ac.at/\~{}kratt/artikel/H.html}.}) Indeed, the data suggest that pairs $(p,N)$ with $p$ prime, $p\le N$, and $v_p(H_N)\ge 3$ are not very frequent. More precisely, so far only five examples are known with $v_p(H_N)=3$: four for $p=11$, with $N=848,9338,10583$, and $3546471722268916272$, and one for $p=83$ with \begin{align} \notag &\hbox{\small $N=79781079199360090066989143814676572961528399477699516786377994370\backslash$} \\ &\kern2cm \hbox{\small $78839681692157676915245857235055200779421409821643691818$} \label{eq:boyd} \end{align} (see \cite[p.~289]{boyd}; the value of $N$ in \eqref{eq:boyd}, not printed in \cite{boyd}, was kindly communicated to us by David Boyd). There is no example known with $v_p(H_N)\ge4$. It is, in fact, conjectured that no $p$ and $N$ exist with $v_p(H_N)\ge4$. Some evidence for this conjecture (beyond mere computation) can be found in \cite{boyd}. \smallskip (d) Since in all the five examples for which $v_p(H_N)=3$ we neither have $p\mid N$ (the gigantic number in \eqref{eq:boyd} is congruent to $42$ modulo $83$) nor that the prime $p$ is a Wolstenholme prime, the exponent of $p$ in the prime factorisation of $\Xi_N$ in these cases is $2$ instead of $v_{p}(H_N)=3$. \smallskip (e) On the other hand, should there be a prime $p$ and an integer $N$ with $p\le N$, $v_p(H_N)\ge3$, $p$ a Wolstenholme prime or $p\mid N$, then the exponent of $p$ in the prime factorisation of $\Xi_N$ would be $3$. However, no such examples are known. We conjecture that there are no such pairs $(p,N)$. If this conjecture should turn out to be true, then, given $N\notin\{1,7\}$, the definition of $\Xi_N$ in \eqref{eq:Xi} could be simplified to \begin{equation} \label{eq:Xisimpl} \Xi_N:= {\prod _{p\le N} ^{}}p^{\min\{2,v_p(H_N)\}}. \end{equation} \end{Remarks} Theorem~\ref{thm:3} improves upon \eqref{eq:NNNN} for $L=N$. Namely, if one compares the definition of $\Xi_N$ in \eqref{eq:Xi} with the following alternative way to write the integer $\Th_N$ occurring in \eqref{eq:NNNN}, \begin{equation} \label{eq:ThL} \Th_N= {\prod _{p\le N}^{}}p^{-\min\{0,v_p(H_N)\}}, \end{equation} we see that Theorem~\ref{thm:3} is {\it always} at least as strong as \cite[Theorem~2]{kratrivmirror}, and it is {\it strictly} stronger if $N\ne7$ and $v_p(H_N)\ge 1$ for some prime $p$ less than or equal to $N$. Indeed, the smallest $N\ne 7$ with that property is $N=20$, in which case $v_5(H_{20})=1$. We remark that strengthenings of \cite[Theorem~2]{kratrivmirror} in the spirit of Theorem~\ref{thm:3} for more general choices of the parameters can also be obtained by our techniques but are omitted here. We outline the proof of Theorem~\ref{thm:3} in Section~\ref{sec:2}, with the details being filled in in Sections~\ref{sec:4}--\ref{sec:aux}. As we explain in Section~\ref{sec:DworkKont}, we conjecture that Theorem~\ref{thm:3} cannot be improved if $k=1$, that is, that for $k=1$ the largest integer $t_N$ such that $q_{N,(N)}(z)^{1/t_N}\in\mathbb Z[[z]]$ is exactly $\Xi_N N!$. Propositions~\ref{prop:p>N} and \ref{prop:vp=3} in Section~\ref{sec:DworkKont} show that this conjecture would immediately follow if one could prove the conjecture from Remarks~\ref{rem:Xi7}(c) above that there are no primes $p$ and integers $N$ with $v_p(H_N)\ge 4$. \medskip Even if the series $q_{N,\mathbf N}(z)$ appears in the identity \eqref{eq:truemap}, which relates the mirror map $q_{\mathbf N}(z)$ to the series $q_{L,\mathbf N}(z)$ (with $L=1$ and $L=N$), Theorem~\ref{thm:3} does not imply an improvement over \cite[Corollary~1]{kratrivmirror}, which we recall here for convenience. \begin{equation} \label{eq:cor1} \vbox{\hsize14.7cm {\leftskip1cm\rightskip1cm \em\noindent For all integers $k\ge 1$ and $N\ge 1$, we have $ \big(z^{-1}q_{\mathbf N}(z)\big)^{\frac{\Th_N}{N!^k kN}} \in \mathbb{Z}[[z]], $ where $\Th_N$ is defined in \eqref{eq:NNNN}. } } \end{equation} \medskip\noindent The reason is that the coefficient of $z$ in $(z^{-1}q_{\mathbf N}(z))^{\Th_N/(pN!^kkN)}$ is equal to $\frac {\Th_N} {p}(H_N-1)$, and, thus, it will not be integral for primes $p$ with $v_p(H_N)>0$. Still, there is an improvement of \eqref{eq:cor1} in the spirit of Theorem~\ref{thm:3}. It is our second main result, and it involves primes $p$ with $v_p(H_N-1)>0$ instead. \begin{theo} \label{thm:3a} Let $N$ be a positive integer with $N\ge2$, and let \begin{equation} \label{eq:Om} \Om_N:= {\prod _{p\le N} ^{}}p^{\min\{2+\om(p,N),v_p(H_N-1)\}}, \end{equation} where $\om(p,N)=1$ if $p$ is a Wolstenholme prime or $N\equiv\pm1$~{\em mod}~$p$, and $\om(p,N)=0$ otherwise. Then $\big(z^{-1}q_{\mathbf N}(z)\big)^{\frac{1}{\Om_{N}N!^kkN}} \in\mathbb{Z}[[z]].$ \end{theo} \begin{Remarks} \label{rem:Om} Also here, some remarks are in order to get a better understanding of the above theorem. \smallskip (a) If $v_p(H_N)< 0$, then $v_p(H_N)=v_p(H_N-1)$. Hence, differences in the prime factorisations of $\Xi_N$ and $\Om_N$ can only arise for primes $p$ with $v_p(H_N)\ge0$. \smallskip (b) Since $z^{-1}q_{\mathbf N}(z)=1+(H_N-1)N!^kkNz+\mathcal O(z^2)$, it is clear that $$\big(z^{-1}q_{\mathbf N}(z)\big)^{1/(p^{v_p(H_N-1)+1}N!^kkN)}\notin \mathbb{Z}[[z]].$$ If $\widetilde V_N$ denotes the largest integer such that $\big(z^{-1}q_{\mathbf N}(z)\big)^{1/\widetilde V_NkN}$ is a series with integer coefficients, the exponent of $p$ in $\widetilde V_N$ can be at most $v_p\big((H_N-1)N!^k\big)$. In Theorem~\ref{thm:3a}, this theoretically maximal exponent is cut down to $v_p(\Om_NN!^k)$. Namely, as is the case for $\Xi_N$, the number $\Om_N$ in \eqref{eq:Om} contains no prime factor $p>N$, whereas the difference $H_N-1$ may very well do so (and, in practice, always does for $N>2$). Moreover, for primes $p$ with $p\le N$ and $v_p(H_N-1)\ge3$, the definition of $\Om_N$ cuts the theoretically maximal exponent $v_p\big((H_N-1)N!^k\big)$ of $p$ down to $2+v_p(N!^k)$ respectively $3+v_p(N!^k)$, depending on whether $\om(p,N)=0$ or $\om(p,N)=1$. \smallskip (c) Concerning the question whether there are any primes $p$ and integers $N$ with high values of $v_p(H_N-1)$, we are not aware of any corresponding literature. Our table of harmonic numbers $H_N$ mentioned in Remarks~\ref{rem:Xi7}(c) does not contain any pair $(p,N)$ with $v_p(H_N-1)\ge3$.~(\renewcommand\thefootnote{5}\footnote{The summary of the corresponding table, containing pairs $(p,N)$ with $p\le N$ and $v_p(H_N-1)>0$, is available at {\tt http://www.mat.univie.ac.at/\~{}kratt/artikel/H1.html}.}) In ``analogy'' to the conjecture mentioned in Remarks~\ref{rem:Xi7}(c), we conjecture that no $p$ and $N$ exist with $v_p(H_N-1)\ge4$. It may even be true that there are no $p$ and $N$ with $v_p(H_N-1)\ge3$, in which case the definition of $\Om_N$ in \eqref{eq:Om} could be simplified to \begin{equation} \label{eq:Omsimpl} \Om_N:= {\prod _{p\le N} ^{}}p^{\min\{2,v_p(H_N-1)\}}. \end{equation} \end{Remarks} In view of \eqref{eq:ThL}, Theorem~\ref{thm:3a} improves upon \eqref{eq:cor1}. Namely, Theorem~\ref{thm:3a} is {\it always} at least as strong as \eqref{eq:cor1}, and it is {\it strictly} stronger if $v_p(H_N-1)\ge 1$ for some prime $p$ less than or equal to $N$. The smallest $N$ with that property is $N=21$, in which case $v_5(H_{21}-1)=1$. We sketch the proof of Theorem~\ref{thm:3a} in Section~\ref{sec:Om}. We also explain in that section that we conjecture that Theorem~\ref{thm:3a} with $k=1$ is optimal, that is, that for $k=1$ the largest integer $u_N$ such that $\big(z^{-1}q_{(N)}(z)\big)^{\frac{1}{Nu_N}} \in\mathbb{Z}[[z]]$ is exactly $\Om_N N!$. Propositions~\ref{prop:p>N2} and \ref{prop:vp=32} in Section~\ref{sec:Om} show that this conjecture would immediately follow if one could prove the conjecture from Remarks~\ref{rem:Om}(c) above that there are no primes $p$ and integers $N$ with $v_p(H_N-1)\ge 4$. As a matter of fact, the sequence $(u_{2N})_{N\ge1}$ appears in the On-Line Encyclopedia of Integer Sequences~\cite{oeis}, as sequence {\tt A007757}, contributed around 1995 by R.~E.~Borcherds under the denomination ``Dwork--Kontsevich sequence,'' without any reference or explicit formula for it, however. (\renewcommand\thefootnote{6}\footnote{In private communication, both, Borcherds and Kontsevich could not remember where exactly this sequence and its denomination came from.}) \section{Structure of the paper} We now briefly review the organisation of the rest of the paper. Following the steps of previous authors, our approach for proving Theorems~\ref{thm:3} and \ref{thm:3a} uses $p$-adic analysis. In particular, we make essential use of Dwork's $p$-adic theory (in the spirit of \cite{lianyau2}). Since the details of our proofs are involved, we provide brief outlines of the proofs of Theorems~\ref{thm:3} and \ref{thm:3a} in separate sections. Namely, Section~\ref{sec:2} provides an outline of the proof of Theorem~\ref{thm:3}, while Section~\ref{sec:Om} contains a sketch of the proof of Theorem~\ref{thm:3a}. Both follow closely the chain of arguments used in the proof of \cite[Theorem~2]{kratrivmirror}. In the outline, respectively sketch, the proofs are reduced to a certain number of lemmas. The lemmas which are necessary for the proof of Theorem~\ref{thm:3} are established in Sections~\ref{sec:4}--\ref{sec:aux}, while the lemmas which are necessary for the proof of Theorem~\ref{thm:3a} are contained in Section~\ref{sec:Om}. Section~\ref{sec:DworkKont} reports on the evidence to believe (or not to believe) that the value $t_N$ (defined in the next-to-last paragraph before Theorem~\ref{thm:3a}) is given by $t_N=\Xi_N N!$, $N=1,2,\dots$, while Section~\ref{sec:DK} addresses the question of whether the Dwork--Kontsevich sequence $(u_N)_{N\ge1}$ (defined in the last paragraph of the Introduction) is (or is not) given by $u_N=\Om_N N!$, $N=1,2,\dots$. \section{Outline of the proof of Theorem~\ref{thm:3}}\label{sec:2} In this section, we provide a brief outline of the proof of Theorem~\ref{thm:3}, reducing it to Lemmas~\ref{lem:12}--\ref{lem:congH} and Corollaries~\ref{cor:C1} and \ref{cor:W2}, the proofs of which are postponed to Sections~\ref{sec:4}--\ref{sec:aux}, except that Lemma~\ref{lem:10} has already been established in \cite[Lemma~6]{kratrivmirror}. The whole proof is heavily based on Dwork's $p$-adic theory (as presented by Lian and Yau in \cite{lianyau2}), enhanced in the spirit of~\cite{kratrivmirror}. In particular, there we used the following result (cf.\ \cite[Lemma~10]{kratrivmirror}). \begin{lem}\label{lem:4} Given two formal power series $f(z)\in 1+z\mathbb{Z}[[z]]$ and $g(z)\in z\mathbb{Q}[[z]]$, an integer $\tau\ge 1$ and a prime number $p$, we have $\exp\big(g(z)/(\tau f(z))\big) \in 1+z\mathbb{Z}_p[[z]]$ if and only if \begin{equation}\label{eq:uv=pvu} f(z)g(z^p)-p\,f(z^p)g(z)\in p\tau z\mathbb{Z}_p[[z]]. \end{equation} \end{lem} It follows from Lemma~\ref{lem:4} that Theorem~\ref{thm:3} can be reduced to the following statement: for any prime number $p$, we have $$ F_{\mathbf N}(z)G_{N,\mathbf N}(z^p)-pF_{\mathbf N}(z^p)G_{N,\mathbf N}(z) \in p \Xi_N N!^k z \mathbb{Z}_p[[z]]. $$ We now follow the presentation in~\cite{kratrivmirror} and let $0\le a
0$.
For $p=2$, we observe that
we have always $v_2(H_N)\le 0$ because of Lemma~\ref{lem:H_L}, so that no
improvement over \eqref{eq:firstreduction1} is needed in this case.
Furthermore, Lemma~\ref{lem:3} together with Remarks~\ref{rem:Xi7}(a)
in the Introduction tells us that, if $p=3$, we need an improvement only if
$N=22$. To be precise, for $N=22$ we need to show that
\begin{equation}\label{eq:firstreductionU3}
C(a+3K) \equiv \sum_{j=0}^K B_{\mathbf N}(a+3j)B_{\mathbf N}(K-j)
(H_{N(K-j)}-H_{\lfloor
Na/3\rfloor+Nj}) \mod 3^2 N!^k\mathbb{Z}_3.
\end{equation}
For $p\ge 5$, we should prove
\begin{equation}\label{eq:firstreductionU}
C(a+Kp) \equiv \sum_{j=0}^K B_{\mathbf N}(a+jp)B_{\mathbf N}(K-j)
(H_{N(K-j)}-H_{\lfloor
Na/p\rfloor+Nj}) \mod p^3 N!^k\mathbb{Z}_p,
\end{equation}
and if, in addition, $v_p(H_N)\ge3$ and $p$ is a Wolstenholme prime or
if $v_p(H_N)\ge3$ and $p\mid N$
(the reader should recall the definition \eqref{eq:Xi} of $\Xi_N$), then
we need to show the even
stronger assertion that
\begin{equation}\label{eq:firstreductionUW}
C(a+Kp) \equiv \sum_{j=0}^K B_{\mathbf N}(a+jp)B_{\mathbf N}(K-j)
(H_{N(K-j)}-H_{\lfloor
Na/p\rfloor+Nj}) \mod p^4 N!^k\mathbb{Z}_p.
\end{equation}
%We recall that the congruence \eqref{eq:firstreduction} followed from
%the congruence \eqref{eq:J}. Clearly, one cannot hope for
%improving \eqref{eq:J} to $pH_{J} \equiv H_{\fl{J/p}} \text{ mod }
%p^3\mathbb{Z}_p,$ there are counterexamples. However,
In order to see \eqref{eq:firstreductionU3}--\eqref{eq:firstreductionUW},
we begin by combining our assumption $v_p(H_N)>0$ with
Corollary~\ref{cor:C1}, Lemma~\ref{lem:p 0$,
but not $p=3$ and $N=7$. (The latter case was already discussed.)
We claim that under this condition we have
\begin{equation} \label{eq:logungl}
\fl{N/p}-\fl{\log_p N/p}\ge
\begin{cases}
4,&\text{if }p=5,\\
5,&\text{if }p>5,\text{ or if $p=3$ and $N=22$},\\
6,&\text{if }v_p(H_N)>2.\\
\end{cases}
\end{equation}
Indeed, if $p=3$, then Lemma~\ref{lem:3}
tells us that only the case
$N=22$ needs to be considered, in which case
the above inequality is trivially true.
If $p=5$ then Lemma~\ref{lem:5} says that $N$ must be at
least $20$ because otherwise $v_5(H_N)\le 0$. The inequality
\eqref{eq:logungl} follows immediately. On the other hand, if $p=11$
then it can be checked
(by our table mentioned in Remarks~\ref{rem:Xi7}(c) in the
Introduction, for example) that $N$ must be at least $77$ because otherwise
$v_{11}(H_N)\le 0$. If $p$ is different from $3$, $5$, and $11$,
then, according to
Lemma~\ref{lem:p 0$.
In a similar way as we did for the expression on the left in
\eqref{eq:congrconj2}, we bound the $p$-adic valuation of the
expression on the left in \eqref{eq:congrconj1} from below. For the sake of
convenience, we write $T_1$ for $\max_{1\le\ep\le \lfloor
Na/p\rfloor}v_p(Nj+\ep)$ and $T_2$ for $\fl{\log_p (a+pj)}$.
Since it is somewhat hidden where our assumption $j>0$ enters the
subsequent considerations, we point out to the reader that
$j>0$ implies that $T_2\ge1$; without this property the split of
the sum over $\ell$ into subsums in the
chain of inequalities below would be impossible.
So, using the above notation, we have (the detailed explanations for the
various steps are given immediately after the
following chain of estimations)
{\allowdisplaybreaks
\begin{align} \notag
v_p\Big(&B_{\mathbf N}(a+pj)\left(H_{Nj+\lfloor Na/p\rfloor} -
H_{Nj}\right)\Big)\\
\notag
&=
k\sum _{\ell =1} ^{\infty}\left(\fl{\frac {N(a+pj)}
{p^\ell }}-
N\fl{\frac {a+pj} {p^\ell }}\right)+
v_p\big(H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}\big)\\
&=
\fl{\frac {N(a+pj)} {p }}-
N\fl{\frac {a+pj} {p }}
+
\sum _{\ell =2} ^{\min\{1+T_1,T_2\}}
\left(\fl{\frac {N(a+pj)} {p^\ell }}-
N\fl{\frac {a+pj} {p^\ell }}\right)
\notag\\
\notag
&\kern1cm
+
\sum _{\ell =\min\{1+T_1,T_2\}+1} ^{\infty}
\left(\fl{\frac {N(a+pj)} {p^\ell }}-
N\fl{\frac {a+pj} {p^\ell }}\right)\\
\notag
&\kern1cm
+
(k-1)\sum _{\ell =1} ^{\infty}\left(\fl{\frac {N(a+pj)}
{p^\ell }}-
N\fl{\frac {a+pj} {p^\ell }}\right)
+v_p\big(H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}\big)
\notag
\\
\notag
&\ge
\fl{\frac {Na} {p }}-N\fl{\frac {a} {p }}+\min\{1+T_1,T_2\}-1
+
k\sum _{\ell =T_2+1} ^{\infty}
\left(\fl{\frac {N(a+pj)} {p^\ell }}-
N\fl{\frac {a+pj} {p^\ell }}\right)\\
&\kern1cm
+v_p\big(H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}\big)
\label{eq:ungl1}
\\
&\ge
\fl{\frac {Na} {p }}
+T_1+v_p\big(H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}\big)
+\min\{0,T_2-T_1-1\}
\notag\\
&\kern1cm
+
k\sum _{\ell =\fl{\log_p(a+pj)}+1}
^{\infty}\left(\fl{\frac {N(a+pj)} {p^\ell }}-
N\fl{\frac {a+pj} {p^\ell }}\right)
\label{eq:ungl2}\\
&\ge
\fl{N/p}
+\min\{0,T_2-T_1-1\}
+
k\sum _{\ell =1} ^{\infty}\fl{\frac {N} {p^\ell
}\cdot\frac {a+pj} {p^{\fl{\log_p(a+pj)}}}}
\label{eq:ungl3}\\
&\ge
\fl{N/p}+\fl{\log_p (a+pj)}
-\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}-1
+
k\sum _{\ell =1} ^{\infty}\fl{\frac {N} {p^\ell
}\cdot\frac {a+pj} {p^{\fl{\log_p(a+pj)}}}}
\label{eq:ungl4}
\\
&\ge
\fl{N/p}+\fl{\log_p j}
-\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}
+
k\sum _{\ell =1} ^{\infty}\fl{\frac {N} {p^\ell }}
\label{eq:ungl5}
\\
&\ge
\fl{N/p}+\fl{\log_p j}
-\fl{\log_p N}-\fl{\log_p\left(j+\frac {1} {N}\lfloor Na/p\rfloor\right)}-1
+
k\cdot v_p(N!)
\label{eq:ungl6}\\
&\ge
\fl{N/p}
-\fl{\log_p N}-1+v_p(N!^k).
\label{eq:ungl7}
\end{align}
}%
Here, we used \eqref{eq:sumest} in order to get \eqref{eq:ungl1}.
To get \eqref{eq:ungl3}, we used the inequalities
\begin{equation} \label{eq:ungl6a}
\fl{\frac {Na} {p}}\ge\fl{\frac {N} {p}}
\end{equation}
and
\begin{equation} \label{eq:ungl100}
T_1+v_p\big(H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}\big)\ge0.
\end{equation}
To get \eqref{eq:ungl4}, we used that
$$T_2-T_1-1\ge \fl{\log_p (a+pj)}
-\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}-1$$
and
$$
\fl{\log_p (a+pj)}
-\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}-1=
\fl{\log_p j}-\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}
\le 0,
$$
so that
\begin{equation} \label{eq:ungl6b}
\min\{0,T_2-T_1-1\}\ge
\fl{\log_p (a+pj)}
-\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}-1.
\end{equation}
Next, to get \eqref{eq:ungl5}, we used
\begin{equation} \label{eq:ungl6d}
\fl{\frac {N} {p^\ell }\cdot\frac {a+pj}
{p^{\fl{\log_p(a+pj)}}}}\ge
\fl{\frac {N} {p^\ell }}.
\end{equation}
To get \eqref{eq:ungl6}, we used
\begin{equation} \label{eq:ungl6c}
\fl{\log_p\big(Nj+\lfloor Na/p\rfloor\big)}
\le\fl{\log_p N}+\fl{\log_p\left(j+\frac {1} {N}\lfloor Na/p\rfloor\right)}+1 .
\end{equation}
Finally, we used $\frac {1} {N}\lfloor
Na/p\rfloor<1$ in order to get
\eqref{eq:ungl7}.
If we now repeat the arguments after \eqref{eq:unglA},
then we see that the estimation \eqref{eq:ungl7} implies
\begin{equation} \label{eq:ungl8}
v_p\Big(B_{\mathbf N}(a+pj)\left(H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}\right)\Big)
\ge
v_p\big(\Xi_N N!^k\big).
\end{equation}
This almost proves \eqref{eq:congrconj1}, our lower
bound on the $p$-adic valuation of the number in \eqref{eq:congrconj1}
is just by $1$ too low.
In order to establish that \eqref{eq:congrconj1} is indeed correct, we
assume by contradiction that all the inequalities in the estimations
leading to \eqref{eq:ungl7} and finally to \eqref{eq:ungl8}
are in fact equalities.
In particular, the estimations in \eqref{eq:ungl6a} hold with equality only
if $a=1$, which we shall assume henceforth.
If we examine the arguments after \eqref{eq:unglA} that led us from
\eqref{eq:ungl7} to \eqref{eq:ungl8}, then we see that they prove in
fact
\begin{equation} \label{eq:ungl11}
v_p\Big(B_{\mathbf N}(a+pj)\left(H_{Nj+\lfloor Na/p\rfloor} -
H_{Nj}\right)\Big)
\ge
1+v_p\big(\Xi_N N!^k\big)
\end{equation}
except if $v_p(H_N)\le 0$ and:
\bigskip
{\sc Case 1:} $p=2$ and $\fl{N/2}=1$;
{\sc Case 2:} $p\ge3$ and $p\le N<2p$;
{\sc Case 3:} $p=3$ and $\fl{N/3}=2$;
\bigskip
\noindent
Indeed, if $N\ge 3p$, this is obvious while,
if $2p\le N<3p$, we have $v_p(H_N)=-1$, except if $p=3$ and $N=7$, a case included
in Case~3. Therefore,
if we exclude the case where $p=3$ and $N=7$, then, if $2p\le N<3p$,
we can continue \eqref{eq:ungl7} as
\begin{multline*}
v_p\Big(B_{\mathbf N}(a+pj)\left(H_{Nj+\lfloor Na/p\rfloor} -
H_{Nj}\right)\Big)\\
\ge
2+\fl{\log_p N/p}-\fl{\log_p N}-1+ v_p(N!^k)
\ge
v_p(N!^k)\ge v_p(p\Xi_NN!^k),
\end{multline*}
where we used \eqref{eq:log} in the first step. This is exactly \eqref{eq:ungl11}.
We now show that \eqref{eq:ungl11} holds as well in Cases~1--3, thus
completing the proof of \eqref{eq:congrconj1}.
\medskip
{\sc Case 1}. Let first $p=2$ and $N=2$. We then have
\begin{align*}
\min\{0,T_2-T_1-1\}&=\min\{0,\fl{\log_2(2j+1)}-v_2(2j+1)-1\}\\
&=
\min\{0,\fl{\log_2(2j+1)}-1\}=0>-1,
\end{align*}
in contradiction to having equality in \eqref{eq:ungl6b}.
On the other hand, if $p=2$ and $N=3$, we have
$$H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}
=H_{3j+1} - H_{3j}=\frac {1} {3j+1}.$$
If there holds equality in \eqref{eq:ungl6b}, then $Nj+\fl{Na/p}=3j+1$
must be a power of $2$, say $3j+1=2^e$ or, equivalently,
$j=(2^e-1)/3$. It follows that
$$
\fl{\frac {N} {p }\cdot\frac {a+pj}
{p^{\fl{\log_p(a+pj)}}}}
=\fl{\frac {3} {2 }\cdot\frac {1+2j}
{2^{\fl{\log_2(1+2j)}}}}
=\fl{\frac {3} {2 }\cdot\frac {2^{e+1}+1}
{3\cdot 2^{e-1}}}
=2>1=\fl{\frac {3} {2}}=\fl{\frac {N} {p}},
$$
in contradiction to having equality in \eqref{eq:ungl6d} with $\ell=1$.
\medskip
{\sc Case 2}. Our assumptions $p\ge3$ and $p\le N<2p$ imply
$$
H_{Nj+\lfloor Na/p\rfloor} - H_{Nj}
=H_{Nj+1} - H_{Nj}=\frac {1} {Nj+1}.
$$
Arguing as in the previous case, in order to have equality in
\eqref{eq:ungl6b}, we must have $Nj+1=f\cdot p^e$ for some positive
integers $e$ and $f$ with $0 N$,
then use of Lemmas~\ref{lem:multinomial/N!} and \ref{lem:12}
(cf.\ \eqref{eq:congrconj1})
implies \eqref{eq:110} immediately since $v_p(\Xi_N)\le 0$ in this case.
On the other hand, if $v_p(H_N)>0$ and $p\le N$,
a combination of Corollary~\ref{cor:C1} with Lemma~\ref{lem:p N}
v_p\big(B_{N}(a)H_{Na}\big)=0,
\end{equation}
where $B_N(m)=\frac {(Nm)!} {m!^N}$.
In particular, $p$ does not divide $t_N$.
\end{prop}
\begin{proof}
If $N=1$, we choose $a=1$ to obtain $B_{1}(1)H_{1}=1$. On the other
hand, if $N>1$, we choose $a$ to be the least integer such that $aN\ge p$.
Since then $a N} can be
reformulated as $v_p\big(C(a)\big)=1$.
Since, because of $p>N$,
we have $v_p(N!)=0$ and $v_p(\Xi_N)=0$, it follows that
$$C(a)\notin p^2\Xi_N N!\, \mathbb Z_p.$$
This means that
one cannot increase the exponent of $p$ in \eqref{eq:Ccong}
(with $k=1$) in our special case, and thus $p$ cannot divide $t_N$.
\end{proof}
So, if we hope to improve Theorem~\ref{thm:3} with $k=1$, then it must be by
increasing exponents of prime numbers $p\le N$ in \eqref{eq:Xi}.
It can be checked directly that the exponent of $3$ cannot be
increased if $N=7$. (The reader should recall Remarks~\ref{rem:Xi7}(a)
in the Introduction.)
According to Remarks~\ref{rem:Xi7}(b), an
improvement is therefore only possible if $v_p(H_N)>2$ for some $p\le N$.
Lemmas~\ref{lem:H_L}--\ref{lem:5} in Section~\ref{sec:aux} tell
that this does not happen with $p=2,3,5$, so that the exponents of
$2,3,5$ cannot be improved.
(The same conclusion can also be drawn from \cite{boyd} for many
other prime numbers, but so far not for $83$, for example.)
We already discussed in Remarks~\ref{rem:Xi7}(c)
whether there are any primes $p$ and integers $N$
such that $p\le N$ and $v_p(H_N)\ge3$. We recall that, so far, only five
examples are known, four of them involving $p=11$.
The final result of this section shows that, if $v_p(H_N)=3$, the exponent
of $p$ in the definition of $\Xi_N$ in \eqref{eq:Xi} cannot be increased
so that Theorem~\ref{thm:3} would still hold.
(The reader should recall Remarks~\ref{rem:Xi7}(b).)
\begin{prop} \label{prop:vp=3}
Let $p$ be a prime number and $N$ a positive integer with $p\le N$
and $v_p(H_N)=3$. If $p$ is not a Wolstenholme prime and $p$ does not
divide $N$, then\break
$q_{N,(N)}(z)^{1/p\Xi_N N!}\notin \mathbb Z[[z]]$.
\end{prop}
\begin{proof}
We assume that $p$ is not a Wolstenholme prime and that $p$ does not
divide $N$. In particular, this implies $\xi(p,N)=0$ and thus also
$v_p(\Xi_N)=2$.
By Lemmas~\ref{lem:H_L}--\ref{lem:5}, we can furthermore
assume that $p\ge7$.
Going back to the outline of the proof of Theorem~\ref{thm:3} in
Section~\ref{sec:2}, we claim that
\begin{equation} \label{eq:Cp}
C(p)=\big(B_N(1)H_N-B_N(p)pH_{Np}\big)\notin p^4 N!\,\mathbb{Z}_p,
\end{equation}
where $B_N(m)=\frac {(Nm)!} {m!^N}$.
(The claim here is the non-membership relation; the equality holds by
the definition of $C(\cdot)$ in \eqref{eq:Ccong}.)
This would imply that $C(p)\notin p^2 \Xi_NN!\,\mathbb Z_{p}$,
and thus, by Lemma~\ref{lem:4}
(recall the outline of the proof of Theorem~\ref{thm:3}
in Section~\ref{sec:2}), that
$q_{N,(N)}(z)^{1/p\Xi_N N!}\notin \mathbb Z[[z]]$, as desired.
To establish \eqref{eq:Cp}, we consider
\begin{multline*}
H_N(B_N(1)-B_N(p))\\
=H_NN!\bigg(1-\frac 1{(p-1)!^N}\big(1\cdot 2\cdots (p-1)\big)
\big((p+1)\cdot (p+2)\cdots (2p-1)\big)\cdots\\
\times\cdots
\big((pN-p+1)\cdot (pN-p+2)\cdots (pN-1)\big) \bigg).
\end{multline*}
Using $v_{p}(H_N)=3$ and Wilson's theorem, we deduce
\begin{equation} \label{eq:Zp}
H_N(B_N(1)-B_N(p))\in p^4N!\,\mathbb Z_{p}.
\end{equation}
However, by Lemma~\ref{lem:congH} and the fact that
$v_{p}\big(B_N(p)\big)=v_{p}\big(B_N(1)\big)=v_{p}(N!)$, we obtain
$$B_N (1)H_N-B_N(p)pH_{pN}\hbox{${}\not\equiv{}$}
H_N(B_N(1)-B_N(p))\mod p^4N!\,\mathbb Z_{p}.$$
Together with \eqref{eq:Zp}, this yields \eqref{eq:Cp}.
\end{proof}
To summarise the discussion of this section: if the conjecture in
Remarks~\ref{rem:Xi7}(c) that no prime $p$ and integer $N$ exists with
$v_p(H_N)\ge4$ should be true, then Theorem~\ref{thm:3} with $k=1$ is
sharp, that is, the sequence $(t_N)_{N\ge1}$ is given by
$t_N=\Xi_NN!$.
\section{Sketch of the proof of Theorem~\ref{thm:3a}}
\label{sec:Om}
In this section we discuss the proof of Theorem~\ref{thm:3a}.
Since it is completely analogous to the proof of Theorem~\ref{thm:3}
(see Section~\ref{sec:2}), we content ourselves with pointing out the
differences. At the end of the section, we present analogues of
Propositions~\ref{prop:p>N} and \ref{prop:vp=3}, addressing the
question of optimality of Theorem~\ref{thm:3a} with $k=1$.
First of all, by \eqref{eq:truemap}, we have
$$
\big(z^{-1}q_{\mathbf N}(z)\big)^{1/kN}=\exp(\widetilde G_{\mathbf
N}(z)/F_{\mathbf N}(z)),
$$
where $F_{\mathbf N}(z)$ is the series from the Introduction and
\begin{equation*}
\widetilde G_{\mathbf N}(z):=\sum_{m=1}^{\infty} \frac{(Nm)!^k}{m!^{kN}}
\big(H_{Nm}-H_m\big)\,z^m.
\end{equation*}
We must adapt the proof of Theorem~\ref{thm:3}, as outlined in
Section~\ref{sec:2}. Writing as before
$B_{\mathbf N}(m)=\frac{(Nm)!^k}{m!^{kN}}$, we must consider the sum
\begin{multline} \label{eq:Summe}
\widetilde C(a+Kp):=\sum_{j=0}^K B_{\mathbf N}(a+jp)B_{\mathbf N}(K-j)
\big((H_{N(K-j)}-H_{K-j})-p(H_{Na+Njp}-H_{a+jp})\big) \\
=\sum_{j=0}^K B_{\mathbf N}(a+jp)B_{\mathbf N}(K-j)
(H_{N(K-j)}-pH_{Na+Njp}) \kern4cm\\-
\sum_{j=0}^K B_{\mathbf N}(a+jp)B_{\mathbf N}(K-j)
(H_{K-j}-pH_{a+jp})
\end{multline}
and show that it is in $p\Om_NN!^k\mathbb Z_p$ for all primes $p$, and for all
non-negative integers $K$, $a$, and $j$ with $0\le a 0$, then we use Lemma~\ref{lem:p N2}
v_p\big(B_{N}(a)(H_{Na}-H_a)\big)=0.
\end{equation}
In particular, $p$ does not divide $u_N$.
\end{prop}
So, if we hope to improve Theorem~\ref{thm:3a} with $k=1$, then it must be by
increasing exponents of prime numbers $p\le N$ in \eqref{eq:Om}.
According to Remarks~\ref{rem:Om}(b) in the Introduction, an
improvement is therefore only possible if
$v_p(H_N-1)>2$ for some $p\le N$.
The final result of this section shows that, if $v_p(H_N-1)=3$
(for which, however, so far no examples are known; see
Remarks~\ref{rem:Om}(c)), the exponent
of $p$ in the definition of $\Om_N$ in \eqref{eq:Om} cannot be increased
so that Theorem~\ref{thm:3a} with $k=1$ would still hold.
\begin{prop} \label{prop:vp=32}
Let $p$ be a prime number and $N$ a positive integer with $p\le N$
and $v_p(H_N)=3$. If $p$ is not a Wolstenholme prime and
$N\hbox{${}\not\equiv{}$}\pm1$~{\em mod}~$p$, then
$\widetilde q_{(N)}(z)^{\frac{1}{p\Om_{N}N!}}
\notin\mathbb{Z}[[z]]$.
\end{prop}
Again, the proof is completely analogous to the proof of
Proposition~\ref{prop:vp=3} in Section~\ref{sec:DworkKont},
which we therefore omit.
So, if the conjecture in
Remarks~\ref{rem:Om}(c) that no prime $p$ and integer $N$ exists
with\break
$v_p(H_N-1)\ge4$ should be true, then Theorem~\ref{thm:3a} with $k=1$ is
optimal, that is, the Dwork--Kontsevich sequence $(u_N)_{N\ge1}$ is given by
$u_N=\Om_NN!$.
\section*{Acknowledgements}
The authors are extremely grateful to Alessio Corti
and Catriona Maclean for illuminating discussions
concerning the geometric side of our work,
and to David Boyd for helpful information on
the $p$-adic behaviour of the harmonic numbers $H_N$
and for communicating to us the value \eqref{eq:boyd} from his files
from 1994. They also thank the referees for
an extremely careful reading of the original manuscript.
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