M/E23

Let K be a complete discretely valued field with residue field k. (A classical example is when K is a p-adic field and k is a finite field.) Suppose that there is a positive number A such that every system of r quadratic forms defined over k in n variables has a nontrivial zero defined over k whenever n > Ar. (If k is a finite field, one can take A=2.) A long standing open question asks what can be said for systems of quadratic forms defined over K. It could be conjectured that every system of r quadratic forms defined over K in n variables has a nontrivial zero defined over K whenever n > 2Ar. In this talk we will discuss this conjecture and partial results that support this conjecture.

David Leep

University of Kentucky, Lexington, USA