Computing points on bielliptic modular curves over fixed quadratic fields

TL;DR

This paper introduces a Mordell-Weil sieve method to compute points on specific bielliptic modular curves over fixed quadratic fields, expanding understanding of these curves over quadratic extensions.

Contribution

The paper develops a Mordell-Weil sieve approach tailored for bielliptic modular curves over quadratic fields, providing explicit computations for selected levels and quadratic discriminants.

Findings

Computed points on X_0(N) over quadratic fields for specified N and d.

Demonstrated the effectiveness of the Mordell-Weil sieve in this context.

Extended knowledge of rational points on modular curves over quadratic fields.

Abstract

We present a Mordell-Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb{Q}(\sqrt{d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and $\lvert d \rvert < 100$.