On the rank of the fibres of rational elliptic surfaces

TL;DR

This paper investigates the ranks of fibers in rational elliptic surfaces over number fields, proving the existence of infinitely many fibers with ranks significantly higher than the generic fiber.

Contribution

It establishes that for many rational elliptic surfaces, there are infinitely many fibers with ranks at least two greater than the generic fiber's rank.

Findings

Infinitely many fibers have rank ≥ generic rank + 2

Results apply to a large class of rational elliptic surfaces

Advances understanding of rank variation in elliptic surface fibers

Abstract

We consider an elliptic surface $\pi: \mathcal{E}\rightarrow \mathbb{P}^1$ defined over a number field $k$ and study the problem of comparing the rank of the special fibres over $k$ with that of the generic fibre over $k(\mathbb{P}^1)$. We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibres with rank at least equal to the generic rank plus two.