Work

A Positive Proportion of Monic Odd-Degree Hyperelliptic Curves of Genus g ≥ 4 Have no Unexpected Quadratic Points

Year: 2024

Type: article

Abstract: Let $\mathcal{F}_{g}$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over ${\mathbb{Q}}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_{g}$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_{g}$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^{1}$.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnae184

Authors: Jef Laga, Ashvin A Swaminathan

Institutions: University of Cambridge, Harvard University

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,534

Funder: National Science Foundation