Evaluation of Certain Hypergeometric Functions over Finite Fields

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\begin{pmatrix} \phi\psi& \psi\\ & \phi \end{pmatrix};x$, $x\in \mathbb{F}_p$, $x\neq 0, 1$, 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 $12$.