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

Payman L Kassaei,Yichao Tian,Shu Sasaki

doi:10.1017/fms.2014.12

Abstract

AbstractWe extend the modularity lifting result of P. Kassaei (‘Modularity lifting in parallel weight one’,J. Amer. Math. Soc.26 (1) (2013), 199–225) to allow Galois representations with some ramification at $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$. We also prove modularity mod 5 of certain Galois representations. We use these results to prove 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 P. Kassaei (‘Modularity lifting in parallel weight one’, J. Amer. Math. Soc.26 (1) (2013), 199–225).

Highlights

The work of Buzzard and Taylor [6] beautifully combined methods of Wiles and Taylor with a geometric analysis of overconvergent p-adic modular forms to prove a modularity lifting result for geometric representations of Gal(Q/Q) which are split and unramified at p
We prove that every finite slope overconvergent Hilbert modular form extends to a region Σ which is vastly larger than R, a region only slightly smaller than the tube of the ‘generic’ locus of the special fibre Y, defined in terms of the stratification of Y studied in [15]
Preliminaries on the geometry of Y we prove some results on the geometry of the special fibre of Y, which will be useful in the analytic continuation process of later sections

Summary

Introduction

The work of Buzzard and Taylor [6] beautifully combined methods of Wiles and Taylor (ala Diamond [11]) with a geometric analysis of overconvergent p-adic modular forms to prove a modularity lifting result for geometric representations of Gal(Q/Q) which are split and unramified at p. Let Q = (A, H ) be a rigid point of Yrig defined over a finite extension K /K. They defined a locally closed subset Wφ,η equidimensional of codimension |φ| + |η| − g in Y They proved that, if Zφ,η denotes the closure of Wφ,η in Y , . We apply this to each fT as in Theorem 4.1, and show that, after analytic continuation, the various Atkin–Lehren involutions of fT can be glued together to a form defined over a very large area. We prove in Theorem 7.1 that the glued form extends automatically to the whole Hilbert modular variety; it is classical

Analytic continuation

Gluing

Hida theory and Λ-adic companion forms