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Hypergeometric functions over finite fields
About this Title
Jenny Fuselier, Ling Long, Ravi Kumar Ramakrishna, Holly Swisher and Fang-Ting Tu
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 280, Number 1382
ISBNs: 978-1-4704-5433-3 (print); 978-1-4704-7282-5 (online)
DOI: https://doi.org/10.1090/memo/1382
Published electronically: October 7, 2022
Keywords: Hypergeometric functions,
finite fields,
Galois representations
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. Preliminaries for the Complex and Finite Field Settings
- 3. Classical Hypergeometric Functions
- 4. Finite Field Analogues
- 5. Some Related Topics on Galois Representations
- 6. Galois Representation Interpretation
- 7. A finite field Clausen formula and an application
- 8. Translation of Some Classical Results
- 9. Quadratic or Higher Transformation Formulas
- 10. An application to Hypergeometric Abelian Varieties
- 11. Open Questions and Concluding Remarks
- 12. Appendix
Abstract
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain hypergeometric varieties.- Alan Adolphson and Steven Sperber, On twisted exponential sums, Math. Ann. 290 (1991), no. 4, 713–726. MR 1119948, DOI 10.1007/BF01459269
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