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

2025/07/04 by Finnerty, Kate · 1 citation
#11D45 #11Y50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.03784

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.

Citations

• Ogg's Torsion conjecture: Fifty years later
• Rational points on rank 2 genus 2 bielliptic curves in the LMFDB
• Explicit quadratic Chabauty over number fields
• Quadratic points on modular curves with infinite Mordell--Weil group
• Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13
• Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties
• Explicit Coleman integration for hyperelliptic curves
• The unipotent Albanese map and Selmer varieties for curves

Cited by

• The possible adelic indices for elliptic curves admitting a rational cyclic isogeny

Related

• Quadratic Chabauty for modular curves: Algorithms and examples
• Rational points on rank 2 genus 2 bielliptic curves in the LMFDB
• Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13
• Explicit quadratic Chabauty over number fields
• Quadratic Chabauty for modular curves and modular forms of rank one