## Existence of the Weil index

From now on, will be a non-Archimedean local field, with characteristic not equal to 2 as always, and a fixed nontrivial continuous character. For a finite-dimensional vector space over , we will denote by the space of locally constant, compactly supported -valued functions on . If we are given a nondegenerate symmetric bilinear form , there is a unique self-dual Haar measure with respect to the pairing . Concretely, given a subset , we can define

and then is the unique Haar measure on such that for some (equivalently, for any) lattice . So we obtain a canonical Fourier transform defined by the formula

Define the space of distributions on to be the dual vector space to . There is a natural injection , where denotes the space of all locally constant -valued functions on , which sends to the linear functional

Any invertible operator induces an adjoint operator defined by the formula . We will usually abuse notation and denote by also.

Let be the associated quadratic form and consider the function given by . Obviously does not lie in , since never vanishes. However, since is continuous and is locally constant it follows that is locally constant, so we can think of as a distribution and take its Fourier transform.

**Theorem** There exists a scalar , called the *Weil index of* , such that

and furthermore . In fact, for a sufficiently large lattice , we have

The proof of the theorem we will give is, unlike Weil’s original proof, completely independent from other results on the Weil representation and uses only the theory of Fourier transforms. The fact that takes values in complex numbers of modulus one will follow from an elementary result on Gauss sums.

Unfortunately, finishing the proof this time would probably make this post unreasonably long, so I’ll content myself with a lemma on Fourier transforms. Let us define translation and multiplication operators on by the formulas

for all . In particular, these operate on , so as discussed above we obtain adjoint operators and on .

Observe also that is a -module, so formally we see that also operates on by the formula .

**Lemma** We have the following equations of operators on :

Also, the translation and multiplication operators interact with the -module structure on via

*Proof.* These are trivial calculations.