There are genus one curves of every index over every infinite, finitely generated field

TL;DR

The paper proves that over any infinite, finitely generated field, there exist genus one curves with any given index, extending the understanding of torsors and their invariants in algebraic geometry.

Contribution

It establishes the existence of genus one curves with any index over all such fields, a significant advancement in the study of torsors and their properties.

Findings

Existence of genus one curves with arbitrary index over specified fields

Construction of infinitely many torsors with prescribed index

Extension of known results from weak Mordell-Weil theorem contexts

Abstract

Every nontrivial abelian variety over a Hilbertian field in which the weak Mordell-Weil theorem holds admits infinitely many torsors with period any $n > 1$ which is not divisible by the characteristic. The corresponding statement with "period" replaced by "index" is plausible but much more challenging. We show that for every infinite, finitely generated field $K$, there is an elliptic curve $E_{/K}$ which admits infinitely many torsors with index any $n > 1$.