As an application of Turan sieve, we give upper bounds for the number of elliptic curves defined over Q(T) in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (Cowan in Conjecture: 100% of elliptic surfaces over Q have rank zero. Preprint. https://arxiv.org/ pdf/2009.08622.pdf, 2020) in the case m, n <= 2.
On the typical rank of elliptic curves over Q(T)
Battistoni, F;Bettin, S;
2022-01-01
Abstract
As an application of Turan sieve, we give upper bounds for the number of elliptic curves defined over Q(T) in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (Cowan in Conjecture: 100% of elliptic surfaces over Q have rank zero. Preprint. https://arxiv.org/ pdf/2009.08622.pdf, 2020) in the case m, n <= 2.
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