## Geometry Seminar - Abstracts

### Talk

Wednesday 6 June 2018, 16:00-17:00 in HG03.085

**Alexander Varchenko** (University of North Carolina)

*Hyperelliptic integrals modulo p and Cartier-Manin matrices*

### Abstract

The hypergeometric solutions of the Knizhnik-Zamolodchikov (KZ)
equations were constructed almost 30 years ago. The polynomial
solutions of the KZ equations over the finite field F_{p} with
a prime number p of elements were constructed recently. I will
consider the example of the KZ equations whose hypergeometric
solutions are given by hyperelliptic integrals of genus g. It is
known that in this case the total 2g-dimensional space of
holomorphic solutions is given by the hyperelliptic integrals. It
turns out that the recent construction of the polynomial solutions
over the field F_{p} in this case gives only a g-dimensional
space of solutions, that is, a "half" of what the complex analytic
construction gives. It turns out that all the constructed polynomial
solutions over the field F_{p} can be obtained by reduction
modulo p of a single distinguished hypergeometric solution. The
corresponding formulas involve the entries of the Cartier-Manin matrix
of the hyperelliptic curve.

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