# Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations

## TL;DR

This paper proves a special case of Mazur's conjecture related to the density of rational points on certain elliptic curve fibrations, and explores rational curves on Kummer surfaces with broader implications for rational points on superelliptic curves.

## Contribution

It introduces a simplified construction of a rational curve on Kummer surfaces and extends the analysis to more general surfaces, advancing understanding of rational points and Mazur's conjecture.

## Key findings

- Rational points are dense on specific elliptic fibrations.
- A new algebraic approach simplifies the construction of rational curves on Kummer surfaces.
- Results imply existence of rational points on twists of superelliptic curves.

## Abstract

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. For the proof, we investigate a highly singular rational curve on the Kummer surface $K$ associated to a product of two elliptic curves over $\mathbb{Q}$, which previously appeared in publications by Mestre, Kuwata-Wang and Satg\'e. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves. Finally, we give a proof of Mazur's conjecture for the Kummer surface $K$ without any restrictions on the $j$-invariants of the two elliptic curves.