Solutions of KZ differential equations modulo p

  • Vadim Schechtman
  • Alexander VarchenkoEmail author


We construct polynomial solutions of the KZ differential equations over a finite field \({\mathbb F}_p\) as analogs of hypergeometric solutions.


KZ differential equations Multidimensional hypergeometric integrals Polynomial solutions over finite fields 

Mathematics Subject Classification

81R12 (11C08 14H52) 



This article was inspired by lectures on hypergeometric motives by Fernando Rodriguez–Villegas in May 2017 at MPI in Bonn. The authors thank him for stimulating discussions. We were also motivated by the classical paper by Manin [7], from which we learned how to construct solutions of differential equations over \({\mathbb F}_p\) from cohomological relations between algebraic differential forms. The authors thank Buium, Manin, and Zudilin for useful discussions and the referee for comments and suggestions contributed to improving the presentation. The article was conceived during the Summer 2017 Trimester program “K-Theory and Related Fields” of the Hausdorff Institute for Mathematics (HIM), Bonn. The authors are thankful to HIM for stimulating atmosphere and working conditions. The first author is grateful to Max Planck Institute for Mathematics for hospitality during a visit in June 2017.


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Authors and Affiliations

  1. 1.Institut de Mathématiques de Toulouse – Université Paul SabatierToulouseFrance
  2. 2.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA

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