Realizing the local Weil representation over a number field

TL;DR

This paper demonstrates that the Weil representation of the symplectic group over a non-archimedean local field can be realized over a specific number field obtained by adjoining square roots of p and -p, with p odd.

Contribution

It provides a new realization of the Weil representation over a particular number field, extending understanding of its algebraic structure over local fields.

Findings

Weil representation can be realized over the field obtained by adjoining square roots of p and -p.

The result applies to non-archimedean local fields with odd residue characteristic.

This realization clarifies the algebraic nature of the Weil representation over certain number fields.

Abstract

We show that the Weil representation of the symplectic group Sp(2n,F), where F is a non-archimedian local field, can be realized over the field obtained from the rationals by adjoining the square roots of p and -p, where p is the residue characteristic of F; p is assumed to be odd.