Concordant pairs in ratios with rank at least two and the distribution of $\theta$-congruent numbers

Abstract

Let $k$ and $\ell$ be distinct nonzero integers. We show that in every congruence class modulo an integer $m>1$, there exist infinitely many integers $n$ such that the Mordell-Weil rank over $\mathbf{Q}$ of the elliptic curve $E(kn,\ell n) : y^{2} = x(x+kn)(x+\ell n)$ is at least two. We also find that for sufficiently large $T$, the number of square-free integers $n$ with $|n| \leq T$ for which the elliptic curve $E(kn, \ell n)$ has rank at least two is at least $\mathcal{O}(T^{2/7})$.