Notes on solutions of KZ equations modulo $p^s$ and $p$-adic limit $s\to\infty$

Abstract

We consider the KZ equations over $\mathbb C$ in the case, when the hypergeometric solutions are hyperelliptic integrals of genus $g$. Then the space of solutions is a $2g$-dimensional complex vector space. We also consider the same equations modulo $p^s$, where $p$ is an odd prime and $s$ is a positive integer, and over the field $\mathbb Q_p$ of $p$-adic numbers. We construct polynomial solutions of the KZ equations modulo $p^s$ and study the space $\mathcal M_{p^s}$ of all constructed solutions. We show that the $p$-adic limit of $\mathcal M_{p^s}$ as $s\to\infty$ gives us a $g$-dimensional vector space of solutions of the KZ equations over $\mathbb Q_p$. The solutions over $\mathbb Q_p$ are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case $g=1$ of elliptic integrals. The $p$-adic limit of $\mathcal M_{p^s}$ as $s\to \infty$ gives us a one-dimensional space of solutions over $\mathbb Q_p$ at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over $\mathbb Q_p$ defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over $\mathbb C$ does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over $\mathbb Q_p$. We describe the Frobenius transformations of solutions of the KZ equations for $g =1$ and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations $y^2= \beta \,x(x-1)(x-\alpha)$ over the finite field $\mathbb F_p$. Here $\alpha,\beta\in\mathbb F_p^\times, \alpha \ne 1$.