Period, index and potential sha

TL;DR

This paper advances the understanding of the period-index obstruction map for genus one curves, demonstrating existence results and implications for arithmetic over global fields, including constructions with prescribed period and index.

Contribution

It introduces new theoretical results on the period-index map, constructs genus one curves with specific period and index over global fields, and explores consequences for Tate-Shafarevich groups.

Findings

Existence of genus one curves with prescribed period and index over number fields.

Construction of infinitely many genus one curves with given period and index over global fields.

Implications for the structure of Tate-Shafarevich groups under field extensions.

Abstract

In this paper we advance the theory of O'Neil's period-index obstruction map and derive consequences for the arithmetic of genus one curves over global fields. Our first result implies that for every pair of positive integers (P,I) with P dividing I and I dividing P^2, there exists a number field K and a genus one curve C over K with period P and index I. Second, let E be any elliptic curve over a global field K, and let P > 1 be any integer indivisible by the characteristic of K. We construct infinitely many genus one curves C over K with period P, index P^2, and Jacobian E. We deduce strong consequences on the structure of Sharevich-Tate groups under field extension.