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$K_1$-invariants in the mod $p$ cohomology of $U(3)$ arithmetic manifolds

Daniel Le, Bao Viet Le Hung, Stefano Morra

Abstract

Let $F/F^+$ be a CM extension and $H_{/F^+}$ a definite unitary group in three variables that splits over $F$. We describe Hecke isotypic components of mod $p$ algebraic modular forms on $H$ at first principal congruence level at $p$ and "minimal" level away from $p$ in terms of the restrictions of the associated Galois representation to decomposition groups at $p$ when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod $p$ Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support.

$K_1$-invariants in the mod $p$ cohomology of $U(3)$ arithmetic manifolds

Abstract

Let be a CM extension and a definite unitary group in three variables that splits over . We describe Hecke isotypic components of mod algebraic modular forms on at first principal congruence level at and "minimal" level away from in terms of the restrictions of the associated Galois representation to decomposition groups at when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support.
Paper Structure (52 sections, 84 theorems, 275 equations, 6 figures, 9 tables)

This paper contains 52 sections, 84 theorems, 275 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The main result
  3. Methods
  4. Outline
  5. Acknowledgements
  6. Notation
  7. Preliminaries
  8. Tame inertial types, inertial local Langlands, Serre weights
  9. Tame inertial types, Deligne--Lusztig representations and inertial local Langlands
  10. Serre weights
  11. Combinatorics of types and weights
  12. $L$-groups and $L$-parameters
  13. Geometry of multi-type deformation rings
  14. Multi-type deformation rings
  15. Presentations of single type deformation rings
  16. ...and 37 more sections

Key Result

Theorem 1.1

Suppose that $n=3$. Moreover, suppose that Then the $\mathrm{GL}_3(k_w)$-representation $\pi(\overline{r})^{U(p)}$ is uniquely determined by $\overline{r}|_{G_{F_w}}$.

Figures (6)

  • Figure 1: Labelling of alcoves for $\mathrm{GL}_3$
  • Figure 2: Weyl and dual Weyl filtrations for $Q_1(\lambda)$, case $\lambda\in B$
  • Figure 3: Weyl and dual Weyl filtrations for $Q_1(\lambda)$, case $\lambda\in A$
  • Figure 4: In this picture we represent the elements in the $\pi(j)$-th coordinate of $(\omega, (\widetilde{w}^0_{X_j}(A))_{j\in\mathcal{J}})$ for $\omega\in L(\sum_{j\in \mathcal{J}} \omega_{X_j})$, where $X_j\in\{A,B,C,D,E,F,G,J,I,H\}$. (Informally speaking, these elements pictures the $\pi(j)$-th component of the restriction of $L(\sum_{j\in \mathcal{J}} \omega_{X_j})$ to $\mathrm{G}$.) On the left we consider the case where $X_j\in\{A,B,C,D,E,F\}$. The gold (resp. red, resp. blue) circles represent the case $X_j=E$ (resp. $X_j=F$, resp. $X_j=B$). The green (resp. orange, resp. purple) dots represent the case $X_j=C$ (resp. $X_j=D$, resp. $X_j=A$). On the right we consider the case where $X_j\in\{G,J,I,H\}$. The green circles represent the case where $X_j=J$. The gold (resp. red, resp. blue) dots represent the case $X_j=G$ (resp. $X_j=H$, resp. $X_j=I$).
  • Figure 5: $\sigma^0$ and $\sigma^1$ are nonadjacent weights in $W^?(\overline{\rho})$. The blue circles represent $\mathrm{JH}(M^1|_{\mathrm{G}})$. The weight $\sigma$ linked to $\sigma^0$ is in $\mathrm{JH}(M^1|_{\mathrm{G}}) \cap W^?(\overline{\rho})$.
  • ...and 1 more figures

Theorems & Definitions (186)

  • Conjecture 1.1
  • Theorem 1.1: Theorem \ref{['main:glob:app']}
  • Remark 1.2
  • Remark 1.4
  • Definition 3.1
  • Definition 3.3
  • Proposition 3.4
  • Remark 3.5
  • proof
  • Definition 3.6
  • ...and 176 more