Conic bundles and Mordell--Weil ranks of elliptic surfaces

TL;DR

This paper investigates the structure of elliptic surfaces over number fields, providing bounds on their Mordell--Weil ranks by analyzing conic bundle structures over the projective line.

Contribution

It introduces bounds for the Mordell--Weil rank of elliptic curves over function fields using conic bundle techniques, a novel approach in this context.

Findings

Established lower bounds for Mordell--Weil ranks

Derived upper bounds based on conic bundle analysis

Connected geometric structures to arithmetic rank properties

Abstract

Let $k$ be a number field and $\mathcal{E}$ an elliptic curve defined over the function field $k(T)$ given by an equation of the form $y^2 = a_3x^3 + a_2x^2 + a_1x + a_0$, where $a_i \in k[T]$ and $deg(a_i) \leq 2$. We explore the conic bundle structure over the $x$-line to obtain lower and upper bounds for the Mordell--Weil rank of $\mathcal{E}(k(T))$.