Selmer groups and Tate–Shafarevich groups for the congruent number problem

Abstract

We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve E n W y 2 D x 3 n 2 x. We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate-Shafarevich groups and three large Tate-Shafarevich groups. When combined with a result obtained by Heath-Brown, we show that the mean value of the 2-rank of the large Tate-Shafarevich groups for square-free positive odd integers n Ä X is 1 2 log log X C O.1/, as X ! 1.

On non-congruent numbers with 1 modulo 4 prime factors

“…Moreover, if ψ : E ′ → E is another isogeny, for the composition ψ • ϕ : E → E, then the following diagram of exact sequences commutes (cf. [8] p 5):…”

Joint distribution for the Selmer ranks of the congruent number curves

“…Remark 1.1. Cardinalities of these Selmer groups can thus be expressed as 2 2+s(n) , 2 s Φ (n) and 2 2+s Φ (n) (as seen in [10]).…”

“…which gives a little information as to how likely it is that n is a congruent number. This method has been applied many times (for instance, [2], [3], [4], [5], [6], [10], etc. ).…”

“…Rhoades determined the probability that a square-free integer n with m prime factors has (s Φ (n), s Φ (n)) = (0, 0) in [9]. He utilized a form of representing elements of Selmer groups by even partitions of certain graphs similar to [10]. If n ≡ ±3 (mod 8), then (s Φ (n), s Φ (n)) = (0, 0) if and only if the following three conditions are satisfied:…”

“…In this paper we calculate the joint distribution for (s Φ (n), s Φ (n)) over odd, squarefree integers n as in [7], with the added condition that the integers n have a fixed number of prime factors. This is why we use the description of the Selmer groups from [10]. Then, in Section 3, we will give the distribution in the case when n ≡ ±3 (mod 8) (Theorem 1.2), and in Section 4 when n ≡ ±1 (mod 8) (Theorem 1.3).…”