On the rationality of the Weil Representation and the local theta correspondence

TL;DR

The paper establishes that the local Weil representation over a non-archimedean local field can be realized over a number field, providing explicit Galois descent data and Schur-index calculations for both even and odd components, including modular and integral settings. It moreover shows that the local theta correspondence over a perfect field $R$ holds if and only if it holds over its algebraic closure, hence affirming the rationality of the classical local theta correspondence. The descent arguments are given both via explicit cohomological/norm obstructions (for $p eq 2$ and $p=2$) and through a Morita-theoretic interpretation of Galois descent, enabling realisation over coefficient rings such as rings of integers and finite fields. Together, these results provide a precise, constructive framework for realizing Weil representations and theta lifts rationally, with concrete descriptions of the requisite fields and Schur indices. The methods pave the way for applications to global theta correspondences and automorphic periods, and they extend the scope to modular coefficients and family settings.

Abstract

We prove that the Weil representation over a non-archimedean local field can be realised with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell$-modular coefficient fields, as well as coefficient rings such as rings of integers i.e. in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational.

Theorems & Definitions (67)

• Theorem A: Character fields and realisation fields when $p \neq 2$
• Theorem B: Theorem \ref{['thm:theta_lifts_and_galois_action']}
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• Remark 1.5
• Definition 1.6
• Definition 1.7
• Lemma 2.1