Evaluation of Certain Hypergeometric Functions over Finite Fields

TL;DR

This paper evaluates certain hypergeometric functions over finite fields ${f F}_p$ in terms of Hecke characters linked to CM elliptic curves, for specific character orders, enhancing understanding of their algebraic and arithmetic properties.

Contribution

It provides explicit evaluations of hypergeometric functions over finite fields in terms of Hecke characters associated with CM elliptic curves for specific character orders.

Findings

Explicit formulas for hypergeometric functions over ${f F}_p$

Connections between hypergeometric functions and CM elliptic curves

Evaluation in terms of Hecke characters for orders 3, 4, 6, 8, 12

Abstract

For an odd prime $p$, let $\phi$ 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 $ {}_2F_1\left(\begin{matrix} \phi\psi & \psi\\ & \phi \end{matrix};x\right)$, $x\in {\mathbb F}_p$, $x\neq 0, 1$, 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 $12$.