Evaluation of Certain Hypergeometric Functions over Finite Fields

Abstract

For an odd prime $p$, let $\varphi$ denote the quadratic character of the multiplicative group ${\mathbb{F}}_p^{\times }$, where ${\mathbb{F}}_p$ is the finite field of $p$ elements. In this paper, we will obtain evaluations of the hypergeometric functions ${}_2{F}_1\left(\begin{array}{cc}\varphi \psi & \psi \\ & \varphi \end{array};x\right)$, $x\in {\mathbb{F}}_p$, $x\ne 0,1$, over ${\mathbb{F}}_p$ in terms of Hecke character attached to CM elliptic curves for characters $\psi$ of ${\mathbb{F}}_p^{\times }$ of order $3$, $4$, $6$, $8$, and $12$.

References in corpus (3)

• A Trace Formula for Certain Hecke Operators and Gaussian Hypergeometric Functions
• A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform
• Jacobsthal identity for Q(sqrt(-2))

Cited by in corpus (1)

• Distribution of values of Gaussian hypergeometric functions