Values of Gaussian Hypergeometric Series

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 2F1(x) =2 F1 „ φ, φ | x « and 3F2(x) =3 F2 „ φ, φ, φ , | x « where φ and respectively are the quadratic and trivial characters of GF (p). For all but finitely many rational numbers x = λ, there exist two elliptic curves 2E1(λ) and 3E2(λ) 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 2F1(λ) is zero; however if λ 6= −1, 0, 1 2 or 2, then the set of such primes has density zero. In contrast, if λ 6= 0 or 1, then there are only finitely many primes p for which 3F2(λ) = 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 3E2(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 3F2(4) over every GF (p).