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Localglobal compatibility in the padic Langlands programme for GL2/Q
, 2010
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CONGRUENCES BETWEEN HILBERT MODULAR FORMS: CONSTRUCTING ORDINARY LIFTS IN PARALLEL WEIGHT TWO
"... Abstract. Under mild hypotheses, we prove that if F is a totally real field, and ρ: GF → GL2(Fl) is irreducible and modular, then there is a finite solvable totally real extension F ′/F such that ρGF ′ has a modular lift which is ordinary at each place dividing l. We deduce a similar result for ρ i ..."
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Cited by 17 (8 self)
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Abstract. Under mild hypotheses, we prove that if F is a totally real field, and ρ: GF → GL2(Fl) is irreducible and modular, then there is a finite solvable totally real extension F ′/F such that ρGF ′ has a modular lift which is ordinary at each place dividing l. We deduce a similar result for ρ itself, under the assumption that at places vl the representation ρG has an ordinary Fv potentially BarsottiTate lift. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially BarsottiTate representations and the BuzzardDiamondJarvis conjecture. The proof makes use of a novel lifting technique, going via rank 4 unitary groups. Contents
Serre’s modularity conjecture (II)
, 2007
"... We provide proofs of Theorems 4.1 and 5.1 of [30]. ..."
Modularity of CalabiYau Varieties
 GLOBAL ASPECTS OF COMPLEX GEOMETRY
, 2006
"... In this paper we discuss recent progress on the modularity of CalabiYau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the Lfunction in connection with mirror symmetry. Finally, we address some questions and open problems. ..."
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Cited by 7 (1 self)
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In this paper we discuss recent progress on the modularity of CalabiYau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the Lfunction in connection with mirror symmetry. Finally, we address some questions and open problems.
Remarks on Serre’s modularity conjecture
"... In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any ..."
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Cited by 4 (2 self)
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In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any step of our proof). The key tool in the proof is Kisin’s recent modularity lifting result, which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for semistable abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved. 1
Proving Serre’s modularity conjecture via
, 2006
"... In this article we give a proof of Serre’s conjecture for the cases of odd conductor and even conductor semistable at 2, and arbitrary weight. Our proof in both cases will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not c ..."
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In this article we give a proof of Serre’s conjecture for the cases of odd conductor and even conductor semistable at 2, and arbitrary weight. Our proof in both cases will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any step of our proof). The new key ingredient is the use of Sophie Germain primes to perform an efficient weight reduction (“Sophie Germain’s weight reduction”), which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and myself. 1
MODULARITY LIFTING THEOREMS OUTLINE
"... The first aim of the course will be to explain modularity/automorphy lifting theorems for twodimensional padic representations, using wherever possible arguments that go over to the ndimensional case. In particular, I will incorporate Taylor’s arguments in [Tay08] that avoid the use of Ihara’s le ..."
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The first aim of the course will be to explain modularity/automorphy lifting theorems for twodimensional padic representations, using wherever possible arguments that go over to the ndimensional case. In particular, I will incorporate Taylor’s arguments in [Tay08] that avoid the use of Ihara’s lemma. For the most part I will ignore the issues which are local at p, focusing on representations which satisfy the Fontaine–Laffaille condition. The second aim is to explain the application of these theorems to questions of level raising and lowering for (Hilbert) modular forms, via the method of Khare– Wintenberger. 2. Project The basic aim of the project will be to use the Khare–Wintenberger method to generalise various results on level raising and lowering from the case of modular forms to Hilbert modular forms (or automorphic forms on a quaternion algebra), under the assumption that p> 2. That such generalisations are possible is wellknown to the experts, and various versions of them are scattered across the literature,
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, 2009
"... Galois deformation theory for norm fields and its arithmetic applications by ..."
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Galois deformation theory for norm fields and its arithmetic applications by