Some families of non-congruent numbers

Abstract

1. Introduction. The elliptic curves E k : y 2 = x(x 2 − k 2 ) with k ∈ Z have been studied extensively, mainly because of their connection with the ancient problem of congruent numbers (see Guy [13] , who found better bounds on the rank of E k by taking the 2-part of the Tate-Shafarevich groups into account. In this article, we will refine the criteria obtained by Lagrange and show that curves E k , where k = pl for primes p ≡ l ≡ 1 mod 8, very rarely have Tate-Shafarevich groups with trivial 2-part.

On Tate-Shafarevich groups of some elliptic curves

“…Also the fact that the fields Q(E k [2]) = Q(i) and Q( E k [2]) = Q( √ 2 ) have class number 1 keeps things simple. Finally we observe that the same technique can be used to construct families of non-congruent numbers (see [11]).…”

“…The first example for such a behaviour I found were the groups ∐ ∐(E/k)[φ] and ∐ ∐( E/k)[ψ] for the curve E : y 2 = x 3 − 68x over the quadratic number field k = Q( √ −43 ), both of order 2 (see [10]). More recently, I have found infinitely many examples over Q using elliptic curves with three rational points of order 2 (see [11]).…”

New series of odd non-congruent numbers

“…Many CN's and non-CN's have been determined (see refs. [1][2][3][4][5][6][7][8][9][10][11][12]), but most of them have at most 4 prime divisors p 1 , p 2 , . .…”

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

“…Clearly one may restrict attention to positive square-free integer n. It is a well-known fact that n is a congruent number if and only if the elliptic curve E n W y 2 D x 3 n 2 x has positive rank over Q ( [25]). Partly because of this relation and among other things, this family of elliptic curves E n has attracted a lot of attention and its arithmetic properties, such as the rank, the associated L-functions, the Selmer groups and Tate-Shafarevich groups related to this curve have been studied extensively (see [1], [4], [5], [8], [11], [13], [14], [15], [16], [20], [21], [23], [27], [29], [30], [32], [33], [34], [35], [37], [38], [42], [43]).…”