Serre’s modularity conjecture (II)

By Chandrashekhar B. Khare, Jean-Pierre Wintenberger. 2009

244 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3cc1dab4f0415f76f2383376e5a18b50bf412fe9

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In Geometry and Topology

1. J. Thorne
2. David Geraghty
3. Frank Calegari
4. James Newton
5. T. Gee
6. Nuno Freitas
7. Bao V. Le Hung
8. F. Herzig

In Mathematical Physics

1. J. Thorne
2. David Geraghty
3. Frank Calegari
4. James Newton
5. T. Gee
6. F. Herzig
7. Bao V. Le Hung
8. Samir Siksek

Citations in the corpus

• Rankin--Eisenstein classes and explicit reciprocity laws (2015, unknown): “f the “big image” hypothesis (iv)). So it suffices to show that the former group is finite. One knows that both Eand ρ∗ are modular, associated to modular forms fof weight 2 ([BCDT01]) and gof weight 1 ([KW09]). Then we have an isomorphism of G Q-representations Ep∞ ⊗ρ∗ ∼=T∨(1), where T= MO(f⊗g)∗. By N”
• The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations (2008, background): “up Ip at pusing an essentially combinatorial recipe [Ser87]. After important results of Mazur, Ribet, Gross, Taylor, and many others, the conjecture was finally proved by Khare and Wintenberger [KWa], [KWb] (and Kisin [Kis]). In this paper we consider n-dimensional irreducible, odd Galois representat”
• Symmetric power functoriality for holomorphic modular forms, II (2020, methodology): “ric power L-function Λ(Symn E,s) as defined in e.g. [DMW09], admits an analytic continuation to the entire complex plane. Our strategy to prove Theorem A is inspired by the proof of Serre’s conjecture [KW09a]. There one takes as given Serre’s conjecture in the level 1 case (i.e. for every prime numbe”
• Elliptic curves over real quadratic fields are modular (2013, background): “Q-curves are modular. Indeed, Ribet [60, Corollary 6.2] had shown that modularity of Q-curves is a consequence of Serre’s modularity conjecture, which is now a theorem of Khare and Wintenberger [43], [44]. 2Indeed, the absence of (c) is the reason why we do not yet have a parametrization of quadrati”
• Modularity lifting beyond the Taylor–Wiles method (2012, methodology): “ the multiplicity of the corresponding Hecke modules will either be the same or twice as expected — depending on whether ρ(Frobq) has distinct eigenvalues or not — by Lemma 3.6. By Serre’s conjecture [KW09] and by the companion form result of Gross [Gro90] and Coleman–Voloch [CV92], there exists a m”