2021/03/02 by Alexander Varchenko, Varchenko, Alexander
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2103.01725
openalex publication_date 2021/03/02 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28
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 ps, where p is an odd prime and s is a positive integer, and over the field \mathbb Qp of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space \mathcal Mps of all constructed solutions. We show that the p-adic limit of \mathcal Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over \mathbb Qp. The solutions over \mathbb 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 case g=1 of elliptic integrals. The p-adic limit of \mathcal Mps as s→ ∞ gives us a one-dimensional space of solutions over \mathbb Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over \mathbb Qp 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 Qp. 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 y2= β x(x-1)(x-α) over the finite field \mathbb Fp. Here α,β∈\mathbb Fp^×, α≠ 1.