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Modular Weil representation and compatibility of cuspidals with congruences

Justin Trias

TL;DR

The paper extends the local Weil representation to the $\ell$-modular setting over non-archimedean fields, generalising the Stone–von Neumann theorem to representations over a field $R$ of characteristic $\ell\neq p$. It develops the $R$-metaplectic group, multiple Weil-models (Schrödinger, lattice, mixed), and a non-normalised Weil factor that avoids square-root choices, enabling a modular theta correspondence for dual pairs and a robust study of congruences for cuspidal representations via integral lattices. It proves modular analogues of theta-lifting phenomena, showing irreducibility in the cuspidal regime under banal characteristic and providing an integral framework to compare modular lifts with their characteristic-zero counterparts. Finally, it connects these constructions to congruence phenomena for supercuspidal blocks, establishing block decompositions, reductions modulo $\ell$, and the transfer of formal degrees, thereby enabling precise control of modular theta lifts and their congruences in a purely local setting.

Abstract

Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field $R$ of characteristic $\ell \neq p$. We obtain along the way a generalisation of the Stone-von Neumann theorem to the $\ell$-modular setting, together with the Weil representation with coefficients in $R$ on the $R$-metaplectic group. Surprisingly enough, the latter $R$-metaplectic group happens to be split over the symplectic group if $\ell = 2$. The theory also makes sense when $F$ is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as $\ell$ does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift.

Modular Weil representation and compatibility of cuspidals with congruences

TL;DR

The paper extends the local Weil representation to the -modular setting over non-archimedean fields, generalising the Stone–von Neumann theorem to representations over a field of characteristic . It develops the -metaplectic group, multiple Weil-models (Schrödinger, lattice, mixed), and a non-normalised Weil factor that avoids square-root choices, enabling a modular theta correspondence for dual pairs and a robust study of congruences for cuspidal representations via integral lattices. It proves modular analogues of theta-lifting phenomena, showing irreducibility in the cuspidal regime under banal characteristic and providing an integral framework to compare modular lifts with their characteristic-zero counterparts. Finally, it connects these constructions to congruence phenomena for supercuspidal blocks, establishing block decompositions, reductions modulo , and the transfer of formal degrees, thereby enabling precise control of modular theta lifts and their congruences in a purely local setting.

Abstract

Let be a non-archimedean local field of characteristic different from and of residual characteristic . We generalise the theory of the Weil representation over with complex coefficients to -modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field of characteristic . We obtain along the way a generalisation of the Stone-von Neumann theorem to the -modular setting, together with the Weil representation with coefficients in on the -metaplectic group. Surprisingly enough, the latter -metaplectic group happens to be split over the symplectic group if . The theory also makes sense when is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift.
Paper Structure (14 sections, 38 theorems, 138 equations)

This paper contains 14 sections, 38 theorems, 138 equations.

Table of Contents

  1. The Heisenberg representation
  2. Schrödinger model
  3. Lattice model
  4. The $R$-metaplectic group and models of the Weil representation
  5. Schrödinger model
  6. Mixed Schrödinger model
  7. Lattice model
  8. Another model
  9. Properties of the $R$-metaplectic group and the Weil representation
  10. Non-normalised Weil factor
  11. The metaplectic cocycle
  12. Around a modular theta correspondence
  13. Supercuspidal blocks in banal characteristic
  14. Congruences of supercuspidal theta lifts in banal characteristic

Key Result

Theorem A

Let $\psi : F \to R^\times$ be a non-trivial character. Up to isomorphism, there exists a unique irreducible representation $(\rho_\psi,S) \in \textup{Rep}_R(H)$ with central character $\psi$.

Theorems & Definitions (73)

  • Theorem A
  • Theorem B
  • Theorem 1.1
  • proof
  • Lemma 1.2
  • Lemma 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • ...and 63 more