Hypergeometric Functions over Finite Fields

2016

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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.

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...

2016

Research in Number Theory

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 [14] 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 [18] for the Appell hypergeometric function F1, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas for the number of Fp-points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell-Lauricella functions.

The Ramanujan Journal, 2012

Let λ ∈ Q \ {0, 1} and l ≥ 2, and denote by C l,λ the nonsingular projective algebraic curve over Q with affine equation given by y l = x(x − 1)(x − λ). In this paper we define Ω(C l,λ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on C l,λ over a finite field and Gaussian hypergeometric series. Finally we give an alternate proof of a result of [13].

Symmetry, Integrability and Geometry: Methods and Applications, 2018

For an odd prime p, let φ denote the quadratic character of the multiplicative group F × p , where F p is the finite field of p elements. In this paper, we will obtain evaluations of the hypergeometric functions 2 F 1 φψ ψ φ ; x , x ∈ F p , x = 0, 1, over F p in terms of Hecke character attached to CM elliptic curves for characters ψ of F × p of order 3, 4, 6, 8, and 12.

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.

Transactions of the American Mathematical Society, 2007

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2 N 7, the N-fold cover of X 0 (N) by X 0 (N 2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6), X 0 (7) are of genus 1. Since their quotients X + 0 (6), X + 0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.