| dc.description | Let F be a eld of characteristic 2. For a quadratic form over F, let F( ) denote the
function eld of the projective quadric given by . If B is a bilinear form over F, we denote by
F(B) the function eld of the projective quadric given by the diagonal quadratic form associated
to B. For any eld extension K=F, there exists two homomorphisms iK : Wq(F) �����! Wq(K)
and jK : W(F) �����! W(K) induced by the inclusion F K. A natural problem in the
algebraic theory of quadratic (bilinear) forms consists in computing the kernels of iK and jK,
i.e., classifying quadratic forms over F which become hyperbolic over K (resp. bilinear forms
over F which become metabolic over K). In general, for an arbitrary eld extension K, it is
di¢ cult to compute the kernels of iK and jK. The case of an extension given by the function
eld of a quadric arouses a lot of interest, and many results are proven in this case, but we are
far from a complete computation of the kernel of iF( ) for an arbitrary quadratic (bilinear) form
.
In this talk, we consider the kernel for graded Witt groups, namely the kernel of the homo-
morphism fnK
: In
q F=In+1
q F �����! In
q F( )=In+1
q F( ), where In
q F = InF Wq(F) and InF is the
nth power of the fundamental ideal IF of W(F), for a quadratic form of small dimension.
So the rst open case to treat is when is a nonsingular quadratic form of dimension 4 and
nontrivial Arf invariant, namely, = [1; a] ? b[1; c] such that a + c 62 }(F) := fx2 ����� x j x 2 Fg.
References
[A-B] R. Aravire, R. Baeza, The behavior of quadratic and di¤erential forms under function eld
extensions in characteristic two, J. Algebra 259 (2003), 361 414.
[A-J] R. Aravire, B. Jacob, H1(X; ) of conics and Witt kernels in characteristic 2, Contemp. Math.
493 (2009), 1 19.
[H-L] D. Ho¤mann, A. Laghribi, Quadratic forms and P ster neighbors in characteristic 2, Trans.
Amer. Math. Soc. 356 (2004), 4019 4053.
[L] A. Laghribi, Witt kernels of function eld extensions in characteristic 2. J. Pure Appl. Algebra
199 (2005), 167-182. | |
