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 [GRAPHICS] where phi and epsilon respectively are the quadratic and trivial characters of GF(p). For all hut finitely many rational numbers x = lambda, there exist two elliptic curves E-2(1)(lambda) and E-3(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 F-2(1)(lambda) is zero; however if lambda not equal -1,0, 1/2 or 2, then the set of such primes has density zero. In contrast, if lambda not equal 0 or 1, then there are only finitely many-primes p for which F-3(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 E-3(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 Apery numbers, as well as a few new ones, and we answer a question of Koike by evaluating F-3(2)(4) over every GF(p).