Evaluation of Certain Hypergeometric Functions over Finite Fields

arXiv (Cornell University), 2015

8.2. A Pfaff-Saalschütz evaluation formula 8.3. A few analogues of algebraic hypergeometric formulas 9. Quadratic or Higher Transformation Formulas 9.1. Some results related to elliptic curves 9.2. A Kummer quadratic transformation formula 9.3. The quadratic formula in connection with the Kummer relations 9.4. A finite field analogue of a theorem of Andrews and Stanton 9.5. Another application of Bailey cubic transformations 9.6. Another cubic 2 F 1 formula and a corollary 10. An application to Hypergeometric Abelian Varieties 11. Open Questions and Concluding Remarks 11.1. Numeric observations 12. Appendix 12.1. Bailey 3 F 2 cubic transforms 12.2. A proof of a formula by Gessel and Stanton References Index

Proceedings of the American Mathematical Society, 2013

We present explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of 2 F 1 {_{2}}F_1 -hypergeometric functions over F q \mathbb {F}_q for q ≡ 1 ( mod 6 ) q \equiv 1 ( \text {mod}~6) and q ≡ 1 ( mod 4 ) q \equiv 1 ( \text {mod}~4) .

Illinois Journal of Mathematics

We present congruences for Greene's 3 F 2 hypergeometric functions over finite fields, which relate values of these functions to a simple polynomial in the characteristic of the field.

2016

2016

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Journal of Number Theory, 2013

We express the trace of Frobenius of certain families of elliptic curves in terms of Gaussian hypergeometric functions. We also find some special values of 2 F 1 Gaussian hypergeometric functions containing characters of order 4 as parameters.

International Mathematics Research Notices

We establish a simple inductive formula for the trace $${\hbox{ Tr }}_{k}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right),p\right)$$ of the p th Hecke operator on the space $${S}_{k}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right)\right)$$ of newforms of level 8 and weight k in terms of the values of 3 F 2 -hypergeometric functions over the finite field F p . Using this formula when k = 6, we prove a conjecture of Koike relating $${\hbox{ Tr }}_{6}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right),p\right)$$ to the values 6 F 5 (1) p and 4 F 3 (1) p . Furthermore, we find new congruences between $${\hbox{ Tr }}_{k}^{\hbox{ new }}\left({\Gamma }_{0}\left(8\right),p\right)$$ and generalized Apéry numbers.

2017 MATRIX Annals, 2019

We define a finite-field version of Appell–Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier et al. (Hypergeometric functions over finite fields, arXiv:1510.02575v2) in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell–Lauricella functions to establish a finite-field analogue of Koike and Shiga’s cubic transformation (Koike and Shiga, J. Number Theory 124:123–141, 2007) for the Appell hypergeometric function F1, proving a conjecture of Ling Long. We also prove a finite field analogue of Gauss’ quadratic arithmetic geometric mean. We use our multivariable period functions to construct formulas for the number of \\(\\mathbb {F}_p\\)-points on the generalized Picard curves. Lastly, we give some transformation and reduction formulas for the period functions, and consequently for the finite-fi...

Research in Number Theory

We prove two transformations for the p-adic hypergeometric functions which can be described as p-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them to the p-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the p-adic Clausen's transformation. We also deduce special values of certain p-adic hypergeometric functions.

2009

In this work we present an explicit relation between the number of points on a family of algebraic curves over Fq and sums of values of certain hypergeometric functions over Fq. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over Fq in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.