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arXiv · 1709.06189

Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

Alexander Varchenko
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Abstract

We consider the Gauss-Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallelly to themselves. We reduce these equations modulo a prime integer $p$ and construct polynomial solutions of the new differential equations as $p$-analogs of the initial hypergeometric integrals. In some cases we interpret the $p$-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field $F_p$. That interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the number of point on an elliptic curve depending on a parameter as a solution of a classical hypergeometric differential equation. We discuss the associated Bethe ansatz.

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Alexander Varchenko. 2017-09-18. Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz. https://arxiv.org/abs/1709.06189

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