Quadratic Chabauty Experiments on Genus 2 Bielliptic Modular Curves in the LMFDB

TL;DR

This paper applies quadratic Chabauty methods to genus 2 bielliptic modular curves in the LMFDB, revealing algebraic irrational points and proposing a conjecture linking modular curve levels to number fields.

Contribution

It demonstrates the effectiveness of quadratic Chabauty techniques on specific genus 2 modular curves and introduces a new conjecture relating curve levels to number field properties.

Findings

Identification of algebraic irrational points on modular curves

Application of quadratic Chabauty over rationals and quadratic fields

Proposal of a conjecture linking modular curve levels to number fields

Abstract

We present results of quadratic Chabauty experiments on genus 2 bielliptic modular curves of Jacobian rank 2 that have recently been added to the LMFDB. We apply quadratic Chabauty methods over both the rationals and quadratic imaginary fields. In a number of cases, the experiments yielded algebraic irrational points among the set of mock rational points. We highlight specific notable examples, including the non-split Cartan modular curve $X^+_{ns}(15)$. Lastly, we offer a conjecture relating the level of the modular curve to the potential number fields over which points can arise.