[Paper Review] Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

Payman L Kassaei, Shu Sasaki+1 more|Research Portal (King's College London)|Apr 3, 2012

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper establishes new modularity lifting theorems for Galois representations with tame ramification at $p$ in parallel weight one, extending prior results over totally real fields. By proving a powerful automatic analytic continuation for finite slope overconvergent Hilbert modular forms to a region nearly covering the generic locus of the special fiber, the authors establish new cases of the strong Artin conjecture for $\mathrm{Gal}(\overline{\mathbb{Q}}/F)$-representations with projective image $A_5$ and $5$ unramified in $F$. The key contribution is a refined analytic continuation result that enables modularity lifting in the tamely ramified setting.

ABSTRACT

We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.

Motivation & Objective

• To generalize modularity lifting results to Galois representations with tame ramification at $p$ in parallel weight one over totally real fields.
• To establish new cases of the strong Artin conjecture for $\mathrm{Gal}(\overline{\mathbb{Q}}/F)$-representations with projective image $A_5$ and $5$ unramified in $F$.
• To prove a sharp analytic continuation result for finite slope overconvergent Hilbert modular forms, extending beyond previous regions.
• To overcome geometric obstructions in the mod $p$ special fiber by analyzing strata and their intersections in the Hilbert modular variety.
• To construct a compatible system of overconvergent eigenforms and characters that descend to classical forms via automorphic descent.

Proposed method

• Prove that every finite slope overconvergent Hilbert modular form extends analytically to a region $\Sigma$ defined as the union of tubes over generic loci of codimension 0 and 1 strata in the special fiber of the Hilbert modular variety.
• Use a gluing argument involving the Atkin-Lehner involution $w$ to extend forms from $\Sigma$ to $\Sigma \cup w^{-1}(\Sigma)$, controlling the connected components via the geometry of strata intersections.
• Analyze the stratification of the special fiber $\overline{Y}$ using results from [15] to ensure that the complement of $\Sigma \cup w^{-1}(\Sigma)$ has codimension at least 2, enabling application of the Koecher principle.
• Apply a rigid-analytic Koecher principle to extend the glued form to the entire Hilbert modular variety $\mathfrak{Y}_{\rm rig}$, proving classicality.
• Construct a system of overconvergent eigenforms $\{f_T\}$ and associated characters $\chi_T$ such that $\rho_{f_T} = \rho_\iota|_{\mathrm{Gal}(\overline{\mathbb{Q}}/L)} \otimes \chi_T^{-1}$ for a finite soluble extension $L/F$, using the modularity lifting theorems.
• Use automorphic descent to show that the original Galois representation $\rho_\iota$ arises from a classical Hilbert modular form of parallel weight one on $\mathrm{Res}_{L/\mathbb{Q}}GL_{2,L}$, hence on $\mathrm{Res}_{F/\mathbb{Q}}GL_{2,F}$.

Experimental results

Research questions

• RQ1 Can modularity lifting theorems in parallel weight one be extended to Galois representations with tame ramification at $p$ over totally real fields, beyond the unramified case?
• RQ2 What is the optimal region to which finite slope overconvergent Hilbert modular forms can be analytically continued, and how does this compare to previous results?
• RQ3 Can the analytic continuation and gluing techniques be adapted to handle the non-trivial action of totally positive units in the totally real field $F$?
• RQ4 Does the existence of a $p$-adic Galois representation with projective image $A_5$ and tame ramification at $p=5$ imply modularity over a totally real field where $5$ is unramified?
• RQ5 Can the strong Artin conjecture be verified for such representations via $p$-adic analytic continuation and descent techniques?

Key findings

• The paper establishes a new automatic analytic continuation result: every finite slope overconvergent Hilbert modular form extends to a region $\Sigma$ that is the union of tubes over the generic loci of codimension 0 and 1 strata in the special fiber of the Hilbert modular variety.
• This region $\Sigma$ is substantially larger than previous continuation regions and is defined using the stratification of the special fiber studied in [15], enabling stronger lifting theorems.
• The authors prove that the complement of $\Sigma \cup w^{-1}(\Sigma)$ in $\mathfrak{Y}_{\rm rig}$ has codimension at least 2 in reduction mod $p$, allowing the application of the Koecher principle to extend forms to the entire space.
• Using this analytic continuation, the authors prove that certain Galois representations with projective image isomorphic to $A_5$ and tame ramification at $p=5$ are modular, thus proving new cases of the strong Artin conjecture over totally real fields where $5$ is unramified.
• The modularity lifting result is applied to show that the residual representation $\overline{\rho}_\iota$ mod 5 arises from a classical Hilbert modular form of parallel weight one, via a finite soluble base change and descent.
• The paper confirms that the Artin $L$-function associated to such representations is entire, as predicted by the strong Artin conjecture, by realizing the representation as arising from a holomorphic Hilbert cusp form of parallel weight one.