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Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.

4 votes
1 answer
247 views

Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$

There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\m …
IMeasy's user avatar
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0 votes
1 answer
116 views

Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector ...

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation. I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of dimen …
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6 votes
1 answer
1k views

orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for sho …
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21 votes
1 answer
2k views

Rationality of intersection of quadrics

Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking …
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