Quadratic Forms, Galois Cohomology and Function Fields of p-adic Curves

Abstract

Let k be a p-adic field and K a function field of a curve over k. It was proved in ([PS3]) that if p 6 2, then the u-invariant of K is 8. Let l be a prime number not equal to p. Suppose that K contains a primitive l th root of unity. It was also proved that every element in H 3 (K,Z/lZ) is a symbol ([PS3]) and that every element in H 2 (K,Z/lZ) is a sum of two symbols ([Su]). In this article we discuss these results and explain how the Galois cohomology methods used in the proof lead to consequences beyond the u-invariant computation.