Colliot-Th\'el\`ene's conjecture and finiteness of u-invariants

TL;DR

This paper demonstrates that Colliot-Thélène's conjecture on 0-cycles implies finiteness results for the u-invariant of certain function fields and establishes bounds for Brauer groups over fields of transcendence degree one.

Contribution

It links Colliot-Thélène's conjecture to finiteness of the u-invariant and period-index bounds, providing new implications in algebraic geometry and number theory.

Findings

Finiteness of the u-invariant for function fields over totally imaginary number fields

Period-index bounds for Brauer groups of fields with transcendence degree one

Implication of Colliot-Thélène's conjecture on algebraic invariants

Abstract

We show that Colliot-Th\'el\`ene's conjecture on 0-cycles of degree 1 implies finiteness for the u-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.