Isotropy of quadratic forms over function fields of p-adic curves.
Parimala, Emory University

Let k be a field of characteristic not 2. The u-invariant of k is defined to be the maximum dimension of anisotropic quadratic forms over k. It is an open question whether finiteness of u(k) implies finiteness of u(k(t)), k(t) denoting the rational function field in one variable over k. This was open until late 90's even for the field Q_p of p-adic numbers. Conjecturally every 9-dimensional quadratic form over Q_p(t) represents zero nontrivially; i.e., u(Q_p(t))=8. This conjecture is settled in the affirmative for nondyadic p-adic fields recently (jointly with Suresh). We shall describe the main steps leading to this result.

Colloquium, Department of Mathematics, Rice University
September 21, 2007

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