Elliptic curves and special values of Gaussian hypergeometric series

Certain transformations for hypergeometric series in the p-adic setting

“…By definition, results involving hypergeometric functions over finite fields are often restricted to primes in certain congruence classes. For example, the expressions for the trace of Frobenius map on certain families of elliptic curves given in [1,2,5,10,11] are restricted to such congruence classes. In [12], McCarthy defined a function n G n [· · · ] which can best be described as an analogue of hypergeometric series in the p-adic setting.…”

Summation identities and special values of hypergeometric series in the p-adic setting

“…Consequently, results involving hypergeometric functions over finite fields are often restricted to primes in certain congruence classes. For example, the expressions for the trace of Frobenius map on families of elliptic curves given in [1,2,9,16,17] are restricted to certain congruence classes to facilitate the existence of characters of specific orders. To overcome these restrictions, in [18,19], the third author defined a function n G n [· · · ] in terms of quotients of the p-adic gamma function which can best be described as an analogue of hypergeometric series in the p-adic setting.…”

Evaluation of Certain Hypergeometric Functions over Finite Fields

“…for some integer t. Applying Lemma 3.5 to the curves y 2 = x 7 + x and y 2 = x 3 + x, we obtain ψ p (−1) j=1, 3,5,7,9,11 J φ, ψ j p = −χ 1 (P) − χ 1 P − χ(P) − χ P − t, ψ p (−1) J φ, ψ 3 p + J φ, ψ 9 p = −χ 1 (P) − χ 1 P and consequently ψ p (−1) j=1,5,7,11 J φ, ψ j p = −χ(P) − χ P − t.…”