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 $\alpha ,\beta \in {\mathbb{F}}_p^{\times },\alpha \ne 1$.

Latex 42 pages; v2: an appendix written jointly with Steven Sperber added, references added, abstract and introduction extended; v3: misprints corrected; v4: misprints corrected