On the fibres of an elliptic surface where the rank does not jump

TL;DR

This paper discusses the behavior of the Mordell--Weil rank of fibers in a non-constant elliptic surface over b1, showing under certain conjectures that the rank does not jump for infinitely many fibers, exemplified by the Legendre surface.

Contribution

It demonstrates, under the assumption of infinitely many Mersenne primes, that the Legendre elliptic surface has infinitely many fibers where the rank equals the group of sections.

Findings

Conditional on Mersenne primes, the Legendre surface has infinitely many fibers with rank equal to the section group.

Supports the expectation that non-isotrivial elliptic surfaces have infinitely many fibers with constant rank.

Provides a new example under conjectural assumptions where the rank does not jump.

Abstract

For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.