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

Author:

Laga Jef1,Swaminathan Ashvin A2

Affiliation:

1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge , Wilberforce Road, Cambridge, CB3 0WB, UK

2. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge , MA 02138, USA

Abstract

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 $\mathbb{P}^{1}$.

Publisher

Oxford University Press (OUP)

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