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Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2

Karol Koziol, Stefano Morra

TL;DR

The paper analyzes mod p representations of p-adic rank-2 unitary groups in the isotypic components of automorphic cohomology. It combines GL_2-transfer techniques, polarized Kisin-module deformation theory, and Taylor–Wiles–Kisin patching to bound and determine Gelfand–Kirillov dimension, proving GK-dim = [K:Q_p] for the relevant representations. Central contributions include transferring minimal multiplicity phenomena from GL_2 to unitary groups, describing deformation rings for essentially conjugate self-dual L-parameters with polarized structures, and establishing freeness and component-matching results for patched modules in the global setting. The work advances mod p Langlands for unitary groups by connecting local deformation geometry, extension combinatorics of Serre weights, and global automorphic patching, with potential implications for supersingular phenomena and weight-part predictions in the mod p Langlands program.

Abstract

Let $p$ be a prime number and $F/F^+$ a CM extension of a totally real field such that every place of $F^+$ above $p$ is unramified and inert in $F$. We fix a finite place $v$ of $F^+$ above $p$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\textrm{U}_{1,1}(\overline{\mathbb{F}}_p)$ be a modular $L$-parameter valued in the $C$-group of a rank 2 unitary group associated to $F/F^+$. We assume $\overline{r}$ is semisimple and sufficiently generic at $v$. Using recent results of Breuil--Herzig--Hu--Morra--Schraen along with our previous work, we prove that certain admissible smooth $\overline{\mathbb{F}}_p$-representations of the $p$-adic unitary group $\textrm{U}_{1,1}(F^+_v)$ associated to $\overline{r}$ in spaces of mod $p$ automorphic forms have Gelfand--Kirillov dimension $[F^+_v:\mathbb{Q}_p]$.

Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2

TL;DR

The paper analyzes mod p representations of p-adic rank-2 unitary groups in the isotypic components of automorphic cohomology. It combines GL_2-transfer techniques, polarized Kisin-module deformation theory, and Taylor–Wiles–Kisin patching to bound and determine Gelfand–Kirillov dimension, proving GK-dim = [K:Q_p] for the relevant representations. Central contributions include transferring minimal multiplicity phenomena from GL_2 to unitary groups, describing deformation rings for essentially conjugate self-dual L-parameters with polarized structures, and establishing freeness and component-matching results for patched modules in the global setting. The work advances mod p Langlands for unitary groups by connecting local deformation geometry, extension combinatorics of Serre weights, and global automorphic patching, with potential implications for supersingular phenomena and weight-part predictions in the mod p Langlands program.

Abstract

Let be a prime number and a CM extension of a totally real field such that every place of above is unramified and inert in . We fix a finite place of above , and let be a modular -parameter valued in the -group of a rank 2 unitary group associated to . We assume is semisimple and sufficiently generic at . Using recent results of Breuil--Herzig--Hu--Morra--Schraen along with our previous work, we prove that certain admissible smooth -representations of the -adic unitary group associated to in spaces of mod automorphic forms have Gelfand--Kirillov dimension .
Paper Structure (51 sections, 51 theorems, 269 equations, 3 tables)

This paper contains 51 sections, 51 theorems, 269 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Context and main result
  3. Strategy
  4. Structure of the paper
  5. Notation
  6. Fields
  7. Representations
  8. Galois theory
  9. Miscellany
  10. Group theory
  11. Unitary groups over ${\mathcal{O}}_K$
  12. Unitary groups over ${\mathbb{Z}}_p$
  13. Base change
  14. Weyl modules
  15. Dual groups
  16. ...and 36 more sections

Key Result

Theorem 1.1

We maintain the above global setup: $F/F^+$ is a CM field extension of $F^+$ which is unramified at all finite places, and such that $p$ is unramified and inert in $F$. Let ${\mathbb{G}}$ be the unitary group defined above, and let ${\overline{r}}: \textnormal{Gal}(\overline{{\mathbb{Q}}}/F^+) \long Then $\dim_{{\textnormal{U}}_{1,1}(K)}(\pi({\overline{r}})) = [K:{\mathbb{Q}}_p]$, where $\dim_{{\t

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 4.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Corollary 5.3
  • proof
  • Lemma 5.4
  • ...and 88 more