## The Witt group and its Weil character

For the next couple of posts I’ll be proving the existence and some properties of the Weil index, which is a numerical invariant of a quadratic space over a local field, and is an important ingredient in the construction of the metaplectic group and its (linear) Weil representation. I will treat only the case of a non-Archimedean local field with characteristic not equal to . This post will probably consist mostly of setup to put us in a natural context for the proofs.

For now let be any field of characteristic not equal to .

**Definition **By a *quadratic space over * we mean a finite-dimensional vector space over equipped with a nondegenerate quadratic form .

Since the characteristic of is not , quadratic forms correspond bijectively with symmetric bilinear forms. An *isometry* between quadratic spaces and is a linear isomorphism satifying for all . Clearly isometry is an equivalence relation on quadratic spaces.

**Example** For any , the map is a nondegenerate quadratic form on which we denote by . It is not hard to see that any one-dimensional quadratic space has this form, and is isometric to if and only if .

Given two quadratic spaces and , we can construct their *sum* in the obvious way: its underlying space is , and the quadratic form is defined by the formula .

**Example** The quadratic space is called the *hyperbolic plane*. It is isometric to the form on which sends .

Thankfully, all quadratic spaces decompose (noncanonically) into a direct sum of spaces of the form , so understanding arbitrary quadratic forms reduces to understanding .

**Proposition** Any quadratic space is isometric to one of the form for some .

*Proof.* We induct on the dimension of : the case is trivial. For the inductive step, certainly there exists with , and if we write for the subspace of spanned by then we have . Denote by the subspace of orthogonal to with respect to , so we need only show and we will be done by the inductive hypothesis.

Clearly since is nondegerate when restricted to , so it remains to prove . For this, one observes that the isomorphism induced by restricts to an isomorphism .

Finally we can introduce the group of which the Weil index is a character.

**Definition-Proposition** The quotient of the commutative monoid formed by isometry classes of quadratic spaces by the submonoid generated by the hyperbolic plane is a group, called the *Witt group of* and denoted by .

*Proof.* The content of the statement is that for any quadratic space , there exists another such that is isometric to a direct sum of copies of . So choose a diagonalization of and take to be . Then the resulting direct sum is isometric to , so it suffices to show that for any we have . This isometry is given by .

In fact, the Witt group can be given a ring structure using the tensor product of quadratic spaces, but we will not need this.

Next time we will prove the existence of the Weil character of for a non-Archimedean local field of characteristic not equal to using the theory of distributions and their Fourier transforms.