The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

Laga, Jef

doi:10.2140/ant.2022.16.1161

Alg. Number Th.

2022

DOI: 10.2140/ant.2022.16.1161

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The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

Jef Laga¹

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2024

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Cited by 2 publications

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

Laga,

Swaminathan

2024

International Mathematics Research Notices

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

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

Laga,

Swaminathan

2024

International Mathematics Research Notices

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Most odd-degree binary forms fail to primitively represent a square

Swaminathan

2024

Compositio Math.

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Let $F$ be a separable integral binary form of odd degree $N \geq 5$ . A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree- $N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree- $N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$ , ordered by height. For every sufficiently large $N$ , we prove that among equations in the family $\mathscr {F}_N(f_0)$ , more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$ , respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

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The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

Cited by 2 publications

References 34 publications

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

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

Most odd-degree binary forms fail to primitively represent a square

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