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abstract: 'Let $K$ be a complete discretely valued field with residue field $\kappa$. If char$(K) = 0$, char$(\kappa) = 2$ and $[\kappa : \kappa^2] = d$, we prove that there exists an integer $N$ depending on $d$ such that the $u$-invariant of any function field in one variable over $K$ is bounded by $N$. The method of proof is via introducing the notion of uniform boundedness for the $p$-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to finiteness of the $u$-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over $K$ are uniformly bounded.'
address: |
    Department of Mathematics & Computer Science\
    Emory University\
    400 Dowman Drive NE\
    Atlanta, GA 30322, USA
author:
- 'R. Parimala'
- 'V. Suresh'
title: ' On the $u$-invariant of function fields of curves over complete discretely valued fields'
---

Let $K$ be a complete discretely valued field with residue field $\kappa$ and $F$ a function field in one variable over $K$. Suppose char$(\kappa) \neq 2$. A bound for the $u$-invariant of $F$ in terms of the $u$-invariant of function fields in one variable over 