Strong u-invariant and Period-Index bound for complete ultrametric fields

TL;DR

This paper investigates the u-invariant and strong u-invariant of complete ultrametric fields and their function fields, establishing bounds and relations based on residue fields, with implications for Brauer dimension bounds.

Contribution

It provides a new description of u-invariants for ultrametric fields and their function fields in terms of residue fields, along with bounds for Brauer-$l$-dimensions.

Findings

u-invariant of $k$ described via residue field invariants

Bounds established for Brauer-$l$-dimensions of $k$ and $F$

Results extend to function fields over $k$

Abstract

Let $k$ be a complete ultrametric valued field. Let u($k$) (resp. u_s($k$)) denote the u-invariant (resp. the strong u-invariant) of $k$. We give a description of this invariant for $k$ in terms of the u-invariant (resp. the strong u-invariant) of its residue field. Let $C$ be a curve over $k$ and $F$ = $k(C)$. We prove similar results for the u-invariant of $F$. For $l$ a prime away from the characteristic of the residue field of $k$, we obtain bounds for the Brauer-$l$-dimensions of $k$ and $F$.