Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 050, 18 pages      arXiv:1711.05842      https://doi.org/10.3842/SIGMA.2018.050
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Evaluation of Certain Hypergeometric Functions over Finite Fields

Fang-Ting Tu a and Yifan Yang b
a) Department of Mathematics, 303 Lockett Hall, Louisiana State University, Baton Rouge, LA 70803, USA
b) Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, Taipei, Taiwan 10617, ROC

Received November 17, 2017, in final form May 09, 2018; Published online May 19, 2018

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 $12$.

Key words: hypergeometric functions over finite fields; character sums; Hecke characters.

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