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Solutions of KZ differential equations modulo p

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Abstract

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

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Acknowledgements

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. Institut de Mathématiques de Toulouse – Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse, France

    Vadim Schechtman

  2. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3250, USA

    Alexander Varchenko

Authors
  1. Vadim Schechtman
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  2. Alexander Varchenko
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Correspondence to Alexander Varchenko.

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To Yu.I. Manin with admiration on the occasion of his 80th birthday.

Alexander Varchenko: supported in part by NSF Grants DMS-1362924, DMS-1665239.

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Schechtman, V., Varchenko, A. Solutions of KZ differential equations modulo p. Ramanujan J 48, 655–683 (2019). https://doi.org/10.1007/s11139-018-0068-x

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  • DOI: https://doi.org/10.1007/s11139-018-0068-x

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