Ramanujan Auditorium

Quadratic forms over function fields

R. Parimala

Emory University

Quadratic forms over number fields are well understood via class field theory. Every quadratic form in at least five variables over a totally imaginary number field admits a nontrivial zero. It is an open question whether quadratic forms in large enough
number of variables over function fields of curves over totally imaginary number fields admit a nontrivial zero. The expectation is that every quadratic form in at least nine variables over such a field represents zero nontrivially. Over function fields of
p-adic curves, every quadratic form in nine variables admits a nontrivial zero. We shall explain a method of attack of the problem via Galois cohomology invariants.