On the distribution of rank two $\tau$-congruent numbers

Abstract

A positive integer $n$ is the area of a Heron triangle if and only if there is a non-zero rational number $\tau$ such that the elliptic curve \begin{equation*} E_{τ}^{(n)}: Y^{2} = X(X-nτ)(X+nτ^{-1}) \end{equation*} has a rational point of order different than two. Such integers $n$ are called $\tau$-congruent numbers. In this paper, we show that for a given positive integer $p$, and a given non-zero rational number $\tau$, there exist infinitely many $\tau$-congruent numbers in every residue class modulo $p$ whose corresponding elliptic curves have rank at least two.