Isotropy of quadratic forms over function fields of p-adic curves.
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.
Parimala, Emory University
Colloquium, Department of Mathematics, Rice University
September 21, 2007
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