The parity of the rank of the mordell-weil group

The size of Selmer groups for the congruent number problem

“…As described there, the parity condition yields stronger average bounds for s(D) than can otherwise be obtained. As described in the introduction, it is possible to extract this result from the work of Cassels [4] and Birch and Stephens [1]. However the proof given here seems to be of independent interest, firstly because it is entirely elementary, and secondly because the method can be applied to twists of other curves with rational 2-torsion, even when there is no complex multiplication.…”

“…As will be clear from the proof of Theorem 2 we have This parity condition, which was included as an unproved hypothesis in the first paper [5] of this series, can in fact be derived from results of Cassels [4] and Birch and Stephens [1], as noted by Birch and Swinnerton-Dyer [2; page 95]. Specifically, one can check that, in the notation of Birch and Swinnerton-Dyer, one has s(D) = λ * + λ 1 − 2.…”

“…Using the work of Cassels [4] one may show that λ * has the same parity as λ, again in the notation of Birch and Swinnerton-Dyer. Finally, according to a result of Birch and Stephens [1], one sees that λ + λ 1 is even for D ≡ 1 or 3 (mod 8), and odd when D ≡ 5 or 7 (mod 8). Birch and Swinnerton-Dyer refer to λ + λ 1 − 2 as the 'number of first descents', however it should be noted that this is not in general the same as our s (D).…”

On the Birch-Swinnerton-Dyer quotients modulo squares

“…To our best knowledge, over number fields Theorem 1.3 is the first general result of this kind, except for the work [7], [11] on curves with a p-isogeny. In contrast, the p-parity conjecture over ‫ޑ‬ was known in almost all cases, thanks to Birch, Stephens, Greenberg and Guo [3], [15], [16] (E CM), Kramer, Monsky [22], [26] (p D 2), Nekovář [28] (p potentially ordinary or potentially multiplicative) and Kim [18] (p supersingular). The results for Selmer groups in dihedral and false Tate curve extensions are similar to those recently obtained by Mazur-Rubin [23] and Coates-Fukaya-Kato-Sujatha [7], [8] Finally, we will need a slight modification of c.E=K/.…”

On quadratic families of CM elliptic curves

“…From the theory of quadratic forms and the root number formula [1] it follows that there is a coprime integer pair (u 0 , v 0 ) such that the corresponding curve E u 0 ,v 0 has root number −1. Consequently, from our theorem it follows that the quadratic form Q(u, v) represents infinitely many congruent numbers.…”