Notes on solutions of KZ equations modulo $p^s$ and $p$-adic limit $s\to\infty$
Notes on solutions of KZ equations modulo $p^s$ and $p$-adic limit $s\to\infty$
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发表时间：
2021-03
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通讯作者：
A. Varchenko
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作者：
A. Varchenko
We consider the differential KZ equations over C in the case, when the hypergeometric solutions are one-dimensional hyperelliptic integrals of genus g. In this case the space of solutions of the differential KZ equations is a 2g-dimensional complex vector space. We also consider the same differential equations modulo p, where p is an odd prime number and s is a positive integer, and over the field Qp of p-adic numbers. We describe a construction of polynomial solutions of the differential KZ equations modulo p. These polynomial solutions have integer coefficients and are p-analogs of the hyperelliptic integrals. We call them the p-hypergeometric solutions. We consider the space Mps of all p-hypergeometric solutions, which is a module over the ring of polynomial quasiconstants modulo p. We study basic properties of Mps , in particular its natural filtration, and the dependence of Mps on s. We show that the p-adic limit of Mps as s → ∞ gives us a g-dimensional vector space of solutions of the differential KZ equations over the field Qp. The solutions over Qp 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 of elliptic integrals. It turns out that in this case the p-adic limit of Mps as s → ∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork’s theory of the classical hypergeometric function overQp and show that our germs of solutions overQp 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 does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. Also in the appendix we follow Dwork and describe the Frobenius transformations of solutions of the KZ equations for 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 = β x(x− 1)(x−α) over the finite field Fp. Here α, β ∈ F×p , α 6= 1. Notice that the same elliptic curves considered over C are used to construct the complex holomorphic solutions of the KZ equations for g = 1.
