The second moment of the size of the $2$-Selmer group of elliptic curves

Bhargava, Manjul; Shankar, Arul; Swaminathan, Ashvin

doi:10.48550/arxiv.2110.09063

Cited by 1 publication

(1 citation statement)

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“…The parametrization in [Swa23] is explicitly leading-coefficient-dependent and is, thus, particularly well-suited for applications concerning binary forms with fixed leading coefficient. By generalizing an equidistribution argument that we developed in joint work with Bhargava and Shankar (see [BSS21], where we used the very same parametrization to determine the second moment of the size of the 2-Selmer group of elliptic curves), one can average the bounds in Theorems 1 and 2 over all leading coefficients and, thus, deduce analogues of these theorems for the full family of superelliptic equations of given degree. In this paper, we fix the leading coefficient for two reasons: doing so simplifies the orbit-counting and allows us to prove a stronger result, that within each thin subfamily of superelliptic equations with fixed leading coefficient, most members fail the Hasse principle.…”

Section: Methods Of Proofmentioning

confidence: 99%

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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Section: Methods Of Proofmentioning

confidence: 99%