Number Theory Days in Lille

Quadratic and Differential Forms over fields of characteristic 2

10 juil. 2019, 11:00

1h

Salle de réunion (Université de Lille)

Quadratic forms

Orateur

Roberto Aravire (Universidad Arturo Prat)

Description

In this talk $F$ denotes a field of characteristic $2$, $W_q(F)$ the Witt of nonsingular quadratic forms over $F$, $W(F)$ the Witt ring of regular symmetric bilinear forms over $F$. For any integer $m\ge 0$, we denote by $I_q^{m+1}(F)$ the group $I^{m}F\otimes W_q(F)$, where $I^{m}F$ \ is the $m$-th power of the fundamental ideal $IF$ of $W(F)$, and $\otimes$ is the module action of $W(F)$ on\ $W_q(F)$.

Given a field extension $K$ of $F$, we have two homomorphisms $i_K:W_q(F)⟶W_q(K)$ and $j_K:W(F)⟶W(K)$ induced by the inclusion $F\subset K$. A natural question that arises is to compute the kernels of these homomorphisms. This is an outstanding problem in the theory of quadratic forms.

A way to study the above problems is based on differential forms where we use a celebrated result of K. Kato which gives connections between quadratic forms and differential forms, and, by this way, obtain results in terms of graded Witt groups.

Following this approach we will provide examples who illustrate the use of tools.