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Ordinary parts of admissible representations of padic reductive groups II. Derived functors
, 2010
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On Serre’s conjecture for mod ℓ Galois representations over totally real fields
, 2009
"... In 1987 Serre conjectured that any mod ℓ twodimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2dimensional representations of the absolute Galois group of a totally ..."
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Cited by 19 (2 self)
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In 1987 Serre conjectured that any mod ℓ twodimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the “weight ” part of Serre’s conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod ℓ Langlands philosophy”. Using ideas of Emerton and Vigneras, we formulate a mod ℓ localglobal principle for the group D ∗ , where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and show how it implies the conjecture.
The local Langlands correspondence for GLn in families, preprint
, 2011
"... 2. Representation theory — general background 5 3. Representation theory — the case of GLn 11 4. The local Langlands correspondence in characteristic zero 24 ..."
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2. Representation theory — general background 5 3. Representation theory — the case of GLn 11 4. The local Langlands correspondence in characteristic zero 24
Relative padic Hodge theory, I: Foundations
, 2011
"... We initiate a new approach to relative padic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry. In this paper, we give a thorough development of ϕmodules over a relative Robba ring associated to a perfect Banach ring of characteristic p, includi ..."
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We initiate a new approach to relative padic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry. In this paper, we give a thorough development of ϕmodules over a relative Robba ring associated to a perfect Banach ring of characteristic p, including the relationship between these objects and étale Zplocal systems and Qplocal systems on the algebraic and analytic spaces associated to the base ring. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite étale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite étale algebras over a corresponding Qpalgebra. This recovers the homeomorphism between the absolute Galois groups of Fp((π)) and Qp(µp∞) given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, and Scholl. Applications to the description of étale local systems on nonarchimedean analytic spaces will be described in subsequent papers. Contents 0
CRITICAL SLOPE pADIC LFUNCTIONS
"... Abstract. Let g be an eigenform of weight k +2 on Γ0(p)∩Γ1(N) with p ∤ N. If g is noncritical (i.e. of slope less than k + 1), using the methods of [1, 20], one can attach a padic Lfunction to g which is uniquely determined by its interpolation property together with a bound on its growth. Howeve ..."
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Abstract. Let g be an eigenform of weight k +2 on Γ0(p)∩Γ1(N) with p ∤ N. If g is noncritical (i.e. of slope less than k + 1), using the methods of [1, 20], one can attach a padic Lfunction to g which is uniquely determined by its interpolation property together with a bound on its growth. However, in the critical slope case, the corresponding growth bound is too large to uniquely determine the padic Lfunction with its standard interpolation property. In this paper, using the theory of overconvergent modular symbols, we give a natural definition of padic Lfunctions in this critical slope case. If, moreover, the modular form is not in the image of theta then the padic Lfunction satisfies the standard interpolation property. 1.
Density of crystalline points on unitary Shimura varieties
 Int. J. Number Theory
, 2013
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