The Diophantine equationy2= x(x2+ 21Dx+ 112D2)and the conjectures of Birch and Swinnerton-Dyer

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 functionswhere 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 2 E 1 (λ) and 3 E 2 (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 2 F 1 (λ) is zero; however if λ = −1, 0, 1 2 or 2, then the set of such primes has density zero. In contrast, if λ = 0 or 1, then there are only finitely many primes p for which 3 F 2 (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 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).

show abstract

“…By (10) this is the 42-quadratic twist of an elliptic curve with conductor 49, and by the work of Rajwade [21] we find that 3 a 2 p;…”

Section: Corollarymentioning

confidence: 87%

Values of Gaussian hypergeometric series

Ono

1998

Trans. Amer. Math. Soc.

111

View full text Add to dashboard Cite

show abstract

“…In a series of papers beginning in the late 1960's and continuing into the 1980's, Rajwade and co-authors (see for example [25,26,27,28,29]) dealt with elliptic curves over Q with complex multiplication by the ring of integers in Q( √ −d) for some small values of d, including d = 1, 2, 3, 7, 11, 19, using cyclotomy and the theory of complex multiplication.…”

Section: Theorem 25 (Gauss and Othersmentioning

confidence: 99%