On the typical rank of elliptic curves over Q(T)

As an application of Turán 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...

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Bibliographic Details
Published inResearch in number theory Vol. 8; no. 4
Main Authors Battistoni, Francesco, Bettin, Sandro, Delaunay, Christophe
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 2022
Springer Nature B.V
Subjects
Curves
Mathematics
Mathematics and Statistics
Number Theory
Upper bounds
14G05
Rank
Elliptic curves
Rational points
11G05
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Summary:As an application of Turán 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 .
ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-022-00377-y