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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 F with characteristic not equal to 2. This post will probably consist mostly of setup to put us in a natural context for the proofs.

For now let k be any field of characteristic not equal to 2.

Definition By a quadratic space over k we mean a finite-dimensional vector space W over k equipped with a nondegenerate quadratic form q : W \to k.

Since the characteristic of k is not 2, quadratic forms correspond bijectively with symmetric bilinear forms. An isometry between quadratic spaces (W_1,q_1) and (W_2,q_2) is a linear isomorphism T : W_1 \to W_2 satifying q_2(T(w)) = q_1(w) for all w \in W_1. Clearly isometry is an equivalence relation on quadratic spaces.

Example For any a \in k^{\times}, the map x \mapsto ax^2 is a nondegenerate quadratic form on k which we denote by \langle a \rangle. It is not hard to see that any one-dimensional quadratic space has this form, and \langle a \rangle is isometric to \langle b \rangle if and only if a \equiv b \ (\text{mod } k^{\times 2}).

Given two quadratic spaces (W_1,q_1) and (W_2,q_2), we can construct their sum in the obvious way: its underlying space is W_1 \oplus W_2, and the quadratic form is defined by the formula (q_1 \oplus q_2)(w_1,w_2) = q_1(w_1) + q_2(w_2).

Example The quadratic space \mathfrak{H} = \langle 1 \rangle \oplus \langle -1 \rangle is called the hyperbolic plane. It is isometric to the form on k^2 which sends (x,y) \mapsto xy.

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

Proposition Any quadratic space (V,q) is isometric to one of the form \langle a_1 \rangle \oplus \cdots \oplus \langle a_n \rangle for some a_1,\cdots,a_n \in k^{\times}.

Proof. We induct on the dimension n of V: the case n = 0 is trivial. For the inductive step, certainly there exists v \in V with q(v) \neq 0, and if we write W for the subspace of V spanned by v then we have W \cong \langle q(v) \rangle. Denote by W^{\perp} the subspace of V orthogonal to W with respect to q, so we need only show V = W \oplus W^{\perp} and we will be done by the inductive hypothesis.

Clearly W \cap W^{\perp} = 0 since q is nondegerate when restricted to W, so it remains to prove V = W + W^{\perp}. For this, one observes that the isomorphism V \to V^* induced by q restricts to an isomorphism W^{\perp} \to (V/W)^*.

\Box

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 \mathfrak{H} is a group, called the Witt group of k and denoted by W(k).

Proof. The content of the statement is that for any quadratic space (V_1,q_1), there exists another (V_2,q_2) such that (V_1 \oplus V_2,q_1 \oplus q_2) is isometric to a direct sum of copies of \mathfrak{H}. So choose a diagonalization \langle a_1 \rangle \oplus \cdots \oplus \langle a_n \rangle of (V_1,q_1) and take (V_2,q_2) to be \langle -a_1 \rangle \oplus \cdots \oplus \langle -a_n \rangle. Then the resulting direct sum (V_1 \oplus V_2,q_1 \oplus q_2) is isometric to \langle a_1 \rangle \oplus \langle -a_1 \rangle \oplus \cdots \oplus \langle a_n \rangle \oplus \langle -a_n \rangle, so it suffices to show that for any a \in k^{\times} we have \langle a \rangle \oplus \langle -a \rangle \cong \mathfrak{H} = \langle 1 \rangle \oplus \langle -1 \rangle. This isometry is given by (x,y) \mapsto \tfrac{a+1}{2}(x,y).

\Box

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 W(F) for F a non-Archimedean local field of characteristic not equal to 2 using the theory of distributions and their Fourier transforms.

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