Serre’s modularity conjecture (II)

@article{Khare2009SerresMC,
  title={Serre’s modularity conjecture (II)},
  author={Chandrashekhar Khare and Jean-Pierre Wintenberger},
  journal={Inventiones mathematicae},
  year={2009},
  volume={178},
  pages={505-586}
}
We provide proofs of Theorems 4.1 and 5.1 of Khare and Wintenberger (Invent. Math., doi:10.1007/s00222-009-0205-7, 2009). 
Symmetric power functoriality for holomorphic modular forms, II
<jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$f$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> Expand
Potentially diagonalizable modular lifts of large weight
We prove that for a Hecke cuspform $f\in S_k(\Gamma_0(N),\chi)$ and a prime $l>\max\{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ such that forExpand
On Fermat’s equation over some quadratic imaginary number fields
TLDR
Fermat’s Last Theorem over Q (i) is proved and it is proved that, for all prime exponents, Fermat's equation does not have non-trivial solutions over Q(-2) and Q(-7). Expand
Automorphy of $\mathrm{GL}_2\otimes \mathrm{GL}_n$ in the self-dual case
In this paper we establish a new case of Langlands functoriality. More precisely, we prove that the tensor product of the compatible system of Galois representations attached to a level-1 classicalExpand
On 2-dimensional 2-adic Galois representations of local and global fields
We describe the generic blocks in the category of smooth locally admissible mod $2$ representations of $\mathrm{GL}_2(\mathbb{Q}_2)$. As an application we obtain new cases of Breuil--M\'ezard andExpand
FORMES MODULAIRES DE HILBERT MODULO p ET VALEURS D'EXTENSIONS GALOISIENNES
Resume. — Soit F un corps totalement reel, v une place de F non ramifiee di- visant p et ρ : Gal(Q/F) → GL2(Fp) une representation continue irreductible dont la restriction ρ| Gal(Fv/Fv) estExpand
Serre’s conjecture and its consequences
Abstract.After some generalities about the absolute Galois group of $$\mathbb Q$$, we present the historical context in which Serre made his modularity conjecture. This was recently proved byExpand
ON THE $p$-ADIC VARIATION OF HEEGNER POINTS
In this paper, we prove an ‘explicit reciprocity law’ relating Howard’s system of big Heegner points to a two-variable $p$-adic $L$-function (constructed here) interpolating the $p$-adic RankinExpand
Proof of de Smit’s conjecture: a freeness criterion
Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$ -flat $B$ -module is $B$ -flat. This freeness criterion was conjectured by de Smit inExpand
On Bhargava's heuristics for $\mathbf{GL}_2(\mathbb{F}_p)$-number fields and the number of elliptic curves of bounded conductor
We propose a new model for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields having the same local properties as the splitting field of the mod $p$-Galois representation associated with anExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 133 REFERENCES
Presentations of universal deformation rings. preprint
  • Presentations of universal deformation rings. preprint
Moduli of finite flat group schemes, and modularity
We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptualExpand
Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)
When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. ThisExpand
Ring-Theoretic Properties of Certain Hecke Algebras
The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a methodExpand
Modularity of 2-adic Barsotti-Tate representations
We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’sExpand
On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q)
We prove the existence in many cases of minimally ramied p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations of the absolute Galois group of Q. It isExpand
On p-Adic Geometric Representations of G Q To
A conjecture of Fontaine and Mazur states that a geometric odd irreducible p-adic representation ρ of the Galois group of Q comes from a modular form ([10]). Dieulefait proved that, under certainExpand
On the Meromorphic Continuation of Degree Two L-Functions
We prove that the L-function of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational num- bers has meromorphic continuation to the whole complex plane and satisfiesExpand
Serre's modularity conjecture: The level one case
Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture
Abstract In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensionalExpand
...
1
2
3
4
5
...