The $\ell$-modular local theta correspondence

TL;DR

This paper investigates the modular local theta correspondence over non-archimedean local fields, establishing a bijective relationship for certain dual pairs when the characteristic is sufficiently large.

Contribution

It introduces the modular local theta correspondence for representations over fields of positive characteristic, extending classical results to the modular setting.

Findings

Establishes bijection for symplectic-orthogonal and unitary-unitary dual pairs

Requires the characteristic of the coefficient field to be large enough

Extends classical theta correspondence to modular representations

Abstract

We study the validity of the local theta correspondence over a non-archimedean local field in the context of modular representation theory \textit{i.e.} for representations with coefficient fields of positive characteristic. For a symplectic-orthogonal or a unitary-unitary dual pair over a $p$-adic field, we obtain a bijective correspondence, as long as the characteristic of the coefficient field is large enough compared to the size of the dual pair, and call it the modular local theta correspondence.