On the distribution of rank two \(\tau\)-congruent numbers (Q2408425)

Let \(\tau\) be a nonzero rational number: a positive integer \(n\) is called \textit{\(\tau\)-congruent number} if the elliptic curve \(E_\tau^{(n)}: y^2=x(x-n\tau)(x-n\tau^{-1})\) has a rational point of order \(\neq 2\) (it is equivalent to the existence of an Heron triangle with area exactly \(n\), see [\textit{E. H. Goins} and \textit{D. Maddox}, Rocky Mt. J. Math. 36, No. 5, 1511--1524 (2006; Zbl 1137.11039)]). The paper deals with rank 2 \(\tau\)-congruent numbers, i.e., those \(n\) such that the curve \(E_\tau^{(n)}\) has rank at least 2. Picking arbitrary integers \(a\), \(b\), \(p\) and \(k\) with \((a,b)=1\) and \(p>0\), the author defines an explicit function \(n(\ell)\) (with \(\ell\in \mathbb{Z}\)) such that 1. \(n(\ell)\equiv k\pmod{p}\); 2. for almost all \(\ell\in\mathbb{Z}\), the curve \(E_\tau^{(n(\ell))}\) has rank at least 2. Part 2 is proved by checking that the image of the usual 2-descent map has order at least 16 (it is known that the order of the image divides \(2^{\mathrm{rank}(E_\tau^{(n(\ell))})+2}\)), while part 1 and the arbitrarity of \(k\) and \(p\) show that for any congruence class \([k]\) modulo \(p\) there are infinitely many rank 2 \(\tau\)-congruent numbers \(n\) in \([k]\).