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

2017/09/18 by Alexander Varchenko, Varchenko, Alexander · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1709.06189

openalex publication_date 2017/09/18 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

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

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