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Pré-Publication, Document De Travail Année : 2021

Ranks of elliptic curves over $\mathbb{Q}(T)$ of small degree in $T$

Résumé

We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the factorization and other simple properties of certain polynomials. Finally, we give sharp estimates for the ranks of the considered families and we present several applications, among which there are lists of rational points, generic families with maximal rank and generalizations of former results.

Dates et versions

hal-03334058 , version 1 (03-09-2021)

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Francesco Battistoni, Sandro Bettin, Christophe Delaunay. Ranks of elliptic curves over $\mathbb{Q}(T)$ of small degree in $T$. 2021. ⟨hal-03334058⟩
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