Rational curves on elliptic surfaces

TL;DR

The paper proves that very general elliptic surfaces with certain properties contain only specific rational curves, implying the Mordell-Weil group of associated elliptic curves over function fields is trivial.

Contribution

It establishes the non-existence of rational curves on very general elliptic surfaces with geometric genus at least 2, extending understanding of their geometric and arithmetic properties.

Findings

No rational curves other than sections and singular fiber components

Mordell-Weil group is trivial for very general elliptic curves of height ≥3

Supports conjectures on rational curves on elliptic surfaces

Abstract

We prove that a very general elliptic surface $\mathcal{E}\to\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\ge2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E/\mathbb{C}(t)$ is a very general elliptic curve of height $d\ge3$ and if $L$ is a finite extension of $\mathbb{C}(t)$ with $L\cong\mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$.