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Evaluation of Certain Hypergeometric Functions over Finite Fields (1711.05842v2)
Published 15 Nov 2017 in math.NT
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 $12$.