Notes on solutions of KZ equations modulo 𝑝^{𝑠} and 𝑝-adic limit 𝑠→∞

Abstract

We consider the differential KZ equations over C \mathbb C in the case, when the hypergeometric solutions are one-dimensional hyperelliptic integrals of genus g g . In this case the space of solutions of the differential KZ equations is a 2 g 2g -dimensional complex vector space. We also consider the same differential equations modulo p s p^s , where p p is an odd prime number and s s is a positive integer, and over the field Q p \mathbb Q_p of p p -adic numbers. We describe a construction of polynomial solutions of the differential KZ equations modulo p s p^s . These polynomial solutions have integer coefficients and are p s p^s -analogs of the hyperelliptic integrals. We call them the p s p^s -hypergeometric solutions. We consider the space M p s \mathcal M_{p^s} of all p s p^s -hypergeometric solutions, which is a module over the ring of polynomial quasi-constants modulo p s p^s . We study basic properties of M p s \mathcal M_{p^s} , in particular its natural filtration, and the dependence of M p s \mathcal M_{p^s} on s s . We show that the p p -adic limit of M p s \mathcal M_{p^s} as s → ∞ s\to \infty gives us a g g -dimensional vector space of solutions of the differential KZ equations over the field Q p \mathbb Q_p . The solutions over Q p \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 special case g = 1 g=1 of elliptic integrals. It turns out that in this case the p p -adic limit of M p s \mathcal M_{p^s} as s → ∞ s\to \infty gives us a one-dimensional space of solutions over Q p \mathbb Q_p at every asymptotic zone. We apply Dwork’s theory of the classical hypergeometric function over Q p \mathbb Q_p and show that our germs of solutions over Q p \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 C \mathbb C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Q p \mathbb Q_p . Also in the appendix we follow Dwork and describe the Frobenius transformations of solutions of the KZ equations for g = 1 g=1 . Using these Frobenius transformations we recover the unit roots of the zeta functions of the elliptic curves defined by the affine equations y 2 = β x ( x − 1 ) ( x − α ) y^2= \beta \,x(x-1)(x-\alpha ) over the finite field F p \mathbb F_p . Here α , β ∈ F p × , α ≠ 1 \alpha ,\beta \in \mathbb F_p^\times , \alpha \ne 1 . Notice that the same elliptic curves considered over C \mathbb {C} are used to construct the complex holomorphic solutions of the KZ equations for g = 1 g=1 .