Prescribed lifts of 2-dimensional representations

Based on the provided text, here is a detailed technical summary of the paper "Prescribed Lifts of 2-Dimensional Representations" by Emerton, Gee, Pan, and Zhu.

Problem Statement

The paper addresses a local-to-global lifting problem for 2-dimensional Galois representations. Let $F$ be a totally real field and $p$ a prime. Given a continuous, absolutely irreducible, totally odd representation $\rho: \text{Gal}_F \to \text{GL}_2(\mathbb{F})$ (where $\mathbb{F}$ is a finite field of characteristic $p$), the authors investigate the existence and finiteness of lifts $\tilde{\rho}: \text{Gal}_F \to \text{GL}_2(\mathcal{O})$ (where $\mathcal{O}$ is the ring of integers of a finite extension of $\mathbb{Q}_p$) that satisfy specific local conditions.

The prescribed conditions are defined by a global deformation problem $S$, which includes:

1. A fixed determinant character $\psi$ lifting $\det \rho$.
2. Local conditions at a finite set of places $\Sigma$ (containing all places above $p$).
   • Away from $p$: The lift must lie on a prescribed non-empty union of irreducible components of the fixed-determinant local deformation ring.
   • Above $p$: The lift must lie on a prescribed non-empty union of irreducible components of a potentially semistable deformation ring with arbitrary regular Hodge–Tate weights.

The central question is whether the global deformation ring $R_S$ associated with these conditions is a finite $\mathcal{O}$-algebra of positive rank. If so, this implies the set of such lifts is finite and non-empty.

Methodology

The authors employ a strategy that combines Galois cohomology with the geometry of moduli stacks of étale $(\phi, \Gamma)$-modules, avoiding the need for new modularity lifting theorems for arbitrary weights.

1. Galois Cohomological Lower Bound:
   The authors first establish a lower bound on the dimension of the global deformation ring $R_S$. Under standard Taylor–Wiles hypotheses (specifically that $\rho|_{\text{Gal}_F(\zeta_p)}$ is absolutely irreducible, and non-solvable conditions for $p=2$ or non-exceptional conditions for $p=5$), they show via tangent-obstruction calculations that $\dim R_S \geq 1$. This ensures that if $R_S$ is finite over $\mathcal{O}$, it must have positive rank, guaranteeing the existence of characteristic-zero lifts.

2. Finiteness via Special Fibre Comparison:
   The core technical challenge is proving that $R_S$ is finite over $\mathcal{O}$. In previous work, this was often achieved by comparing the deformation ring to a Hecke algebra via modularity lifting theorems. However, such theorems are generally only known for "parallel weight 2" (Hodge–Tate weights $\{0, 1\}$ at every embedding).

   • The authors utilize the moduli stack $X$ of étale $(\phi, \Gamma)$-modules constructed in [EG23].
   • They invoke a "topological" form of the Breuil–Mézard philosophy: the underlying reduced substacks of potentially semistable deformation rings for arbitrary regular Hodge–Tate weights are equal to suitable unions of components of the underlying reduced substack $X_{\text{red}}$.
   • Crucially, they show (Proposition 3.1.2) that every residual 2-dimensional local representation (with the required determinant) is contained in the special fibre of a tame, potentially semistable weight 2 substack.
   • This implies that, modulo the uniformizer $\varpi$ and up to nilpotent thickening, any potentially semistable local deformation ring is a quotient of a weight 2 local deformation ring.

3. Global Reduction:
   By tensoring these local comparisons, the authors construct a surjection from a global deformation ring $R_{S'}$ (where $S'$ imposes weight 2 conditions) to a quotient of the original ring $R_S$ (modulo $\varpi$ and a nilpotent ideal). Since $R_{S'}$ is known to be finite over $\mathcal{O}$ (via Snowden's results [Sno09] and Kisin's theorems), the finiteness of $R_S$ follows.

Key Contributions

• Generalization of Weight Conditions: The paper extends the known finiteness of global deformation rings from the "parallel weight 2" case to the case of arbitrary regular Hodge–Tate weights.
• Independence from New Modularity Theorems: The authors achieve this generalization without establishing the potential modularity of the lifts or proving new automorphy lifting theorems for higher weights. Instead, they rely on the geometric structure of the deformation stacks.
• Arbitrary Local Components: The result allows for the prescription of arbitrary non-empty unions of irreducible components of local deformation rings, covering a wide range of level and weight conditions simultaneously.

Main Results

Theorem 1.1.1 (Theorem 4.2.1):
Let $F$ be a totally real field and $\rho: \text{Gal}_F \to \text{GL}_2(\mathbb{F})$ be an irreducible, totally odd representation. Assume:

• $\rho|_{\text{Gal}_F(\zeta_p)}$ is absolutely irreducible.
• If $p=2$, $\rho$ has non-solvable image.
• If $p=5$, $\rho$ is not exceptional (in the sense of Definition 4.1.1).

Then, for any global deformation problem $S$ (prescribing local conditions as described above), the universal deformation ring $R_S$ is a finite $\mathcal{O}$-algebra of positive $\mathcal{O}$-rank.

Corollary: The set of isomorphism classes of lifts of $\rho$ to $\text{GL}_2(\mathcal{O})$ satisfying the prescribed local conditions is finite and non-empty.

Significance and Scope

The paper claims to provide a Galois-theoretic analogue of theorems on level raising, level lowering, and weight changing for Hilbert modular forms. By proving that the global deformation ring is finite and non-empty, the authors demonstrate that arbitrary prescribed non-empty sets of local components occur globally.

The authors note that their hypotheses are those required for the standard Taylor–Wiles argument in the 2-dimensional setting. They explicitly state that they do not address the residually reducible case or generalizations to higher-dimensional representations or other reductive groups in this work, though they discuss the theoretical requirements (such as the "numerical coincidence" in the Taylor–Wiles method) that would be necessary for such extensions. The result is presented as a structural theorem on the existence of lifts with specific local properties, rather than a construction of specific automorphic forms.