[Paper Review] Automorphic lifts of prescribed types

Toby Gee|arXiv (Cornell University)|Oct 10, 2008

Advanced Algebra and Geometry14 references4 citations

TL;DR

This paper establishes the existence of automorphic Galois representations lifting residual mod $p$ Galois representations of prescribed local types at places away from $p$, using deformation ring structure results and modularity lifting techniques. It proves level-raising and level-lowering theorems for $n$-dimensional automorphic Galois representations and provides a framework for generalizing Serre weight conjectures.

ABSTRACT

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for $n$-dimensional automorphic Galois representations.

Motivation & Objective

• To establish the existence of automorphic Galois representations that lift a given residual mod $p$ Galois representation with specified local types at places not dividing $p$.
• To extend the method of Khare and Wintenberger to Hilbert modular forms of parallel weight 2, proving level-lowering results in full generality under standard hypotheses.
• To generalize Serre weight conjectures for $n$-dimensional mod $p$ Galois representations, particularly in the context of Hilbert modular forms.
• To prove level-raising and level-lowering results for $n$-dimensional automorphic Galois representations using recent work of Taylor.
• To provide a framework for future $R=T$ theorems in the context of $n$-dimensional Galois representations over number fields unramified at $p$.

Proposed method

• Analyzes the structure of local deformation rings for mod $p$ Galois representations at places $l \neq p$, adapting techniques from Kisin's work on the $l=p$ case.
• Applies the method of Khare and Wintenberger to prove the existence of modular lifts of specified type for Hilbert modular forms of parallel weight 2.
• Uses the universal deformation ring $R_{ au}^{univ}$ and its reduction to show finiteness over the ring of integers $\mathcal{O}$, relying on trace generation and Nakayama’s lemma.
• Applies results from Gouvêa and Hida on Hecke algebras to establish that $R_{ au}^{univ}$ is finite over $\mathcal{O}$, ensuring the existence of automorphic lifts.
• Leverages Taylor’s recent work on $R=T$ theorems to deduce level-raising and level-lowering results for $n$-dimensional Galois representations.
• Constructs a deformation problem $\mathcal{S}$ with specified local types $\tau_v$ at places $v \in Y$, and uses the universal deformation ring to lift $\overline{\rho}$ to an automorphic representation $\pi$.

Experimental results

Research questions

• RQ1 Under what conditions does a residual Galois representation $\overline{\rho}$ of $\mathrm{GL}_n$ over $\overline{\mathbb{F}}_l$ admit a modular lift of prescribed local type at places $v \notin \{p\}$?
• RQ2 Can level-lowering results be established in full generality for Hilbert modular forms of parallel weight 2, assuming the existence of ordinary lifts?
• RQ3 What are the possible Serre weights for $n$-dimensional mod $p$ Galois representations, and how can existing conjectures be generalized?
• RQ4 To what extent can level-raising and level-lowering theorems be extended to $n$-dimensional automorphic Galois representations beyond the 2-dimensional case?
• RQ5 What conditions ensure that the universal deformation ring $R_{ au}^{univ}$ is finite over $\mathcal{O}$, enabling the construction of automorphic lifts?

Key findings

• The universal deformation ring $R_{ au}^{univ}$ is a finite $\mathcal{O}$-module of rank at least 1, ensuring the existence of automorphic lifts with prescribed local types.
• For Hilbert modular forms of parallel weight 2, under the Taylor-Wiles hypothesis and the existence of ordinary lifts, level-lowering holds at all places not dividing $p$.
• In the case where $p$ splits completely and the Taylor-Wiles hypothesis holds, the conjectures of Buzzard, Diamond, and Jarvis on Serre weights are proven in full generality.
• The paper proves that $\dim(R_{ au}^{univ}) \geq 1$, which follows from the congruence $\mu_\pi \equiv n \pmod{2}$ and the structure of the deformation ring.
• Level-raising and level-lowering results are established for $n$-dimensional automorphic Galois representations, relying on Taylor’s recent $R=T$ theorems.
• The reduction $(R_{ au}^{univ})^{\text{red}}$ is finite over $\mathcal{O}$, as it is finite over $(R_{\mathcal{S}}^{\text{univ}})^{\text{red}}$, which is isomorphic to a Hecke algebra $\mathbb{T}$ finite over $\mathcal{O}$.