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

TL;DR

This paper investigates the ranks of elliptic curves over the rational function field with small degree in T, providing explicit formulas, sharp estimates, and applications such as rational points and maximal rank families.

Contribution

It derives explicit rank formulas for elliptic surfaces over (T) with degree at most 2, and offers sharp estimates and applications for these families.

Findings

Explicit rank formulas depending on polynomial factorization

Sharp bounds for ranks of elliptic families

Applications including rational points and maximal rank families

Abstract

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.