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 k. If char(K) = 0, char(k) = 2 and the 2-rank of k is 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.