Capitulation discriminants of genus one curves

TL;DR

This paper explores the invariant theory and arithmetic of genus one curves embedded in projective space, generalizing models for odd degrees, and applies these results to the capitulation problem of the Tate-Shafarevich group.

Contribution

It extends the genus one model framework to arbitrary odd degrees and links the invariant theory to minimal models and the capitulation problem in number theory.

Findings

Genus one models are generalized for all odd degrees.

Every locally soluble genus one curve over Q has a minimal integral model.

Elements of Sha(E/Q)[n] of odd index n split over degree n fields with bounded discriminant.

Abstract

In this paper we study the arithmetic and invariant theory of genus one normal curves embedded in $\mathbb{P}^{n-1}$. We generalize the notion of genus one model of degree $n$, introduced by Cremona, Fisher and Stoll for $n \leq 5$, to arbitrary odd $n$, and describe the invariant theory of a genus one curve of degree $n$ embedded in $ \mathbb{P}^{n-1}$ in terms of the minimal graded free resolution of its homogeneous ideal. We prove that everywhere locally soluble genus one curves over $ \mathbb{Q}$ admit minimal integral models, with the same invariants as those of the minimal model of their Jacobian elliptic curve. We then apply these results to study the capitulation problem for the Tate-Shafarevich group of an elliptic curve $E/\mathbb{Q}$. We prove that every element of $\text{Sha}(E/\mathbb{Q})[n]$ of odd index $n$ splits over a degree $n$ number field $K$, of absolute discriminant at most $c(n) H_E^{2n-2}$, where $H_E$ is the naive height of $E$ and $c(n)$ is a constant only depending on $n$.