The $\ell$-modular local theta correspondence

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.