[Paper Review] On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{\mathbb{Q}_p}$

Shen-Ning Tung|arXiv (Cornell University)|Mar 19, 2018

Finite Group Theory Research27 references3 citations

TL;DR

This paper proves that every irreducible component of the 2-dimensional potentially semi-stable deformation ring for $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ is automorphic by leveraging the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ and analyzing patched modules $M_\infty(\sigma)[1/p]$. It provides a new proof of the Breuil-Mézard conjecture for $p > 2$, including the novel case $p = 3$ when $\bar{r}$ is a twist of the trivial representation by the mod $p$ cyclotomic character, and removes a local restriction in the proof of the Fontaine-Mazur conjecture.

ABSTRACT

Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_p)$, we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_p$ when $p > 2$, which is new in the case $p = 3$ and $\bar{r}$ a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture by Kisin is removed.

Motivation & Objective

• To establish the automorphy of all irreducible components of the potentially semi-stable deformation ring for 2-dimensional Galois representations of $\mathbb{Q}_p$.
• To provide a new proof of the Breuil-Mézard conjecture for $p > 2$, particularly in the case $p = 3$ and $\bar{r}$ a twist of the trivial character by the mod $p$ cyclotomic character.
• To remove a local restriction in the proof of the Fontaine-Mazur conjecture by showing that automorphy of components implies modularity under standard conditions.
• To demonstrate that the patched module $M_\infty(\sigma)[1/p]$ meets every irreducible component of $R_{\bar{r}}^\Box(\sigma)[1/p]$ via the $p$-adic local Langlands correspondence.

Proposed method

• Using the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$, the paper relates the patched module $M_\infty(\sigma)[1/p]$ to admissible Banach space representations $\Pi_y$ via the Montreal functor $\check{\mathbf{V}}$.
• Applying Colmez's Montreal functor $\check{\mathbf{V}}$ to the fixed determinant quotient $M_\infty^\psi$ to show that $\check{\mathbf{V}}(M_\infty^\psi)$ is a finitely generated module over $R_\infty^\psi$, overcoming the non-finite generation of $M_\infty$ over $R_\infty$.
• Proving that the action of $R_\infty^\psi$ on $\check{\mathbf{V}}(M_\infty^\psi)$ is faithful using a result of Emerton and Paškūnas, which implies that the specialization at any $y \in \operatorname{m-Spec} R_\infty^\psi[1/p]$ is non-zero.
• Showing that $\check{\mathbf{V}}(M_\infty^\psi)$ is a finitely generated module over $R_\infty^\psi$ by factoring through the ideal $J$ generated by Cayley-Hamilton relations in $R_\infty^\psi\llbracket G_{\mathbb{Q}_p} \rrbracket$, which allows control over the module structure.
• Using the formalism of [Kis09, GK14, EG14, Paš15] to deduce automorphy of components from the non-vanishing of $\Pi_y$, and verifying the exceptional case (reducible, non-crystalline, semi-stable) via Hilbert-Samuel multiplicity computations.
• Combining results from [BLGG13] on ordinary components and [EG14] on cycle decompositions to show that all components are automorphic, including the previously unproven case for $p=3$.

Experimental results

Research questions

• RQ1 Does every irreducible component of the potentially semi-stable deformation ring $R_{\bar{r}}^\Box(\sigma)[1/p]$ for $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ arise from automorphic forms?
• RQ2 Can the Breuil-Mézard conjecture be proven independently of prior results in the case $p=3$ and $\bar{r}$ a twist of the trivial representation by the mod $p$ cyclotomic character?
• RQ3 Does the non-vanishing of the patched module $M_\infty(\sigma)[1/p]$ at every point in the deformation space imply that all components are automorphic?
• RQ4 Can the local restriction in the proof of the Fontaine-Mazur conjecture be removed using this automorphy result?
• RQ5 Is the cycle decomposition of $R_{\bar{r}}^\Box(\sigma)/\varpi$ compatible with the automorphic cycle structure predicted by the Breuil-Mézard conjecture?

Key findings

• The support of the patched module $M_\infty(\sigma)[1/p]$ meets every irreducible component of the potentially semi-stable deformation ring $R_{\bar{r}}^\Box(\sigma)[1/p]$, proving that all such components are automorphic.
• A new proof of the Breuil-Mézard conjecture is established for 2-dimensional representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ when $p > 2$, including the case $p = 3$ and $\bar{r}$ a twist of the trivial character by the mod $p$ cyclotomic character.
• The action of $R_\infty^\psi$ on $\check{\mathbf{V}}(M_\infty^\psi)$ is faithful and $\check{\mathbf{V}}(M_\infty^\psi)$ is a finitely generated $R_\infty^\psi$-module, which ensures that $\Pi_y \neq 0$ for all $y \in \operatorname{m-Spec} R_\infty^\psi[1/p]$.
• The exceptional case of reducible, non-crystalline, semi-stable components is verified directly by computing Hilbert-Samuel multiplicities, completing the proof of the Breuil-Mézard conjecture in full for $p > 2$.
• The Fontaine-Mazur conjecture is now proven without the local restriction in [Kis09], as the automorphy of all components implies modularity of odd, irreducible, potentially semi-stable representations with distinct Hodge-Tate weights.
• The cycle decomposition $\mathcal{Z}(R_{\bar{r}}^\Box(\sigma)/\varpi) = \sum m_{\bar{\sigma}}(\lambda,\tau) \mathfrak{C}_{\bar{\sigma}}(\bar{r}) \times \mathcal{Z}(k\llbracket x_1,\dots,x_d\rrbracket)$ holds, confirming compatibility with the automorphic cycle structure.