On the typical rank of elliptic curves over $${\mathbb {Q}}(T)$$

As an application of Turán sieve, we give upper bounds for the number elliptic curves defined over $${\mathbb {Q}}(T)$$ in some families having positive rank, obtaining particular that these form a subset density zero. This confirms Cowan’s conjecture (Cowan Conjecture: 100% surfaces {Q}}$$ have rank Preprint. https://arxiv.org/pdf/2009.08622.pdf , 2020) case $$m,n\le 2$$ .