Values of Gaussian hypergeometric series

Abstract

Let p p be prime and let G F ( p ) GF(p) be the finite field with p p elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions 2 F 1 ( x ) = 2 F 1 ( ϕ , a m p ; ϕ a m p ; ϵ | x ) and 3 F 2 ( x ) = 3 F 2 ( ϕ , a m p ; ϕ , a m p ; ϕ a m p ; ϵ , a m p ; ϵ | x ) , \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 ϕ \phi and ϵ \epsilon respectively are the quadratic and trivial characters of G F ( p ) GF(p) . For all but finitely many rational numbers x = λ x=\lambda , there exist two elliptic curves 2 E 1 ( λ ) _{2}E_{1}(\lambda ) and 3 E 2 ( λ ) _{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 p for which 2 F 1 ( λ ) _{2}F_{1}(\lambda ) is zero; however if λ ≠ − 1 , 0 , 1 2 \lambda \neq -1,0, \frac {1}{2} or 2 2 , then the set of such primes has density zero. In contrast, if λ ≠ 0 \lambda \neq 0 or 1 1 , then there are only finitely many primes p p for which 3 F 2 ( λ ) = 0 _{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 ) _{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 ) _{3}F_{2}(4) over every G F ( p ) GF(p) .