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I was hoping somebody might be able to point me to a good reference on Gaussian hypergeometric functions defined over a finite field. the reason I'm interested is that I've encountered sums of the form $$\sum_{t\in (\mathbb{F}_p)^{\times}} \phi(t)\phi(t-4)\phi(At^2+Bt+C),$$ where $\phi$ is the Dirichlet character. These sums are naturally arising from the representation theory of a particular subgroup of $\mbox{GL}_4(\mathbb{F}_p).$ Anyhow, it seems that $$_2F_1(x)=p^{-1}\phi(-1)\sum_{t\in (\mathbb{F}_p)^{\times}} \phi(y)\phi(1-y)\phi(1-xy),$$ which is close to the sums I'm interested in evaluating. Any suggestions are welcome.

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    $\begingroup$ I thnk you are essentially counting points on $y^2=t(t-4)(At^2+Bt+C)$, which is generally an elliptic curve? $\endgroup$ Feb 28, 2016 at 5:02

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