Rational points on quadratic elliptic surfaces

TL;DR

This paper investigates the Mordell-Weil ranks of elliptic surfaces with quadratic polynomial coefficients, establishing an upper bound of 6 and linking the rank to the zeros of a specific polynomial over rationals.

Contribution

It provides a new upper bound for the Mordell-Weil rank of quadratic elliptic surfaces and relates the rank to the zeros of a polynomial over ield.

Findings

Mordell-Weil rank is at most 6 for these surfaces

Rank is controlled by the zeros of a certain polynomial

Infinitely many rational values of T yield rank at least 1

Abstract

We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we prove that the Mordell-Weil rank of each such elliptic surface is at most $6$ over $\mathbb Q$. In fact, we show that the Mordell-Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over $\mathbb Q$.