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

Abstract

We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture [Cow20] in the case m, n ≤ 2. 2020 Mathematics Subject Classification. 11G05, 14G05. Key words and phrases. Rational elliptic surface, rank, elliptic curves, rational points. 1 Cowan stated the conjecture for the Mahler measure µ, mentioning that one could also choose the height instead. By the inequalities n [n/2] −1 P ≤ µ(P ) ≤ √ n + 1 P for P of degree n (see [BG06, Lemma 1.6.7]) the two choices are equivalent.