Values of Gaussian hypergeometric series

Abstract:

Let $p$ be prime and let $GF(p)$ be the finite field with $p$ elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions \begin{equation*} _{2}F_{1}(x) = _{2}F_{1} \left ( \begin {matrix}\phi , & \phi \\ & \epsilon \end{matrix} | x \right ) \;\;\text {and}\;\; _{3}F_{2}(x)= _{3}F_{2} \left ( \begin {matrix}\phi , & \phi , & \phi \\ & \epsilon , & \epsilon \end{matrix} | x \right ), \end{equation*} where $\phi$ and $\epsilon$ respectively are the quadratic and trivial characters of $GF(p)$. For all but finitely many rational numbers $x=\lambda$, there exist two elliptic curves $_{2}E_{1}(\lambda )$ and $_{3}E_{2}(\lambda )$ for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes $p$ for which $_{2}F_{1}(\lambda )$ is zero; however if $\lambda \neq -1,0, \frac {1}{2}$ or $2$, then the set of such primes has density zero. In contrast, if $\lambda \neq 0$ or $1$, then there are only finitely many primes $p$ for which $_{3}F_{2}(\lambda ) =0$. Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that $_{3}E_{2}(8)$ is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating $_{3}F_{2}(4)$ over every $GF(p)$.