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44
Automorphic lifts of prescribed types
, 2006
"... Abstract. We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger ..."
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Cited by 24 (8 self)
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Abstract. We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of ndimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for ndimensional automorphic Galois representations. Contents
The weight in a Serretype conjecture for tame ndimensional Galois representations
, 2006
"... Abstract. We formulate a Serretype conjecture for ndimensional Galois representations that are tamely ramified at p. The weights are predicted using a representationtheoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. ..."
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Cited by 24 (2 self)
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Abstract. We formulate a Serretype conjecture for ndimensional Galois representations that are tamely ramified at p. The weights are predicted using a representationtheoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters. 1.
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.
Companion forms for unitary and symplectic groups
, 2009
"... Abstract. We prove a companion forms theorem for ordinary ndimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre ..."
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Cited by 15 (11 self)
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Abstract. We prove a companion forms theorem for ordinary ndimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp 4 over totally real fields.
On the torsion in the cohomology of arithmetic hyperbolic 3manifolds
 Duke Math. J
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Rigidity of padic cohomology classes of congruence subgroups of GL(n
"... The main result of [AshStevens 97] describes a framework for constructing padic analytic families of pordinary arithmetic Hecke eigenclasses in the cohomology of congruence subgroups of GL(n)/Q where the Hecke eigenvalues vary padic analytically as functions of the weight. Unanswered in that pap ..."
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Cited by 8 (4 self)
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The main result of [AshStevens 97] describes a framework for constructing padic analytic families of pordinary arithmetic Hecke eigenclasses in the cohomology of congruence subgroups of GL(n)/Q where the Hecke eigenvalues vary padic analytically as functions of the weight. Unanswered in that paper was the question of whether or not every pordinary arithmetic eigenclass can be “deformed ” in a “positive dimensional family ” of arithmetic eigenclasses. In this paper we make precise the notions of “deformation ” and “rigidity ” and investigate their properties. Rigidity corresponds to the nonexistence of positive dimensional deformations other than those coming from twisting by the powers of the determinant. Formal definitions are given in §3. When n = 3, we will give a necessary and sufficient condition for a given pordinary arithmetic eigenclass to be padically rigid and we will use this criterion to give examples of padically rigid eigenclasses on GL(3). More precisely, we define a “Hecke eigenpacket ” to be a map from the Hecke algebra to a coefficient ring which gives the Hecke eigenvalues attached to a Hecke eigenclass in the cohomology. We then consider padic analytic “deformations”
Weight cycling and Serretype conjectures for unitary groups
 In preparation
"... Abstract. We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predi ..."
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Cited by 7 (4 self)
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Abstract. We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by the third author. We do this by combining explicit computations in padic Hodge theory, based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop in the paper, with a technique that we call “weight cycling”. 1.
SL3(F2)extensions of Q and arithmetic cohomology modulo 2
 Experiment. Math
, 2004
"... We generate extensions of Q with Galois group SL3(F2) giving rise to threedimensional mod 2 Galois representations with sufficiently low level to allow the computational testing of a conjecture of Ash, Doud, Pollack, and Sinnott relating such representations to mod 2 arithmetic cohomology. We test t ..."
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Cited by 5 (0 self)
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We generate extensions of Q with Galois group SL3(F2) giving rise to threedimensional mod 2 Galois representations with sufficiently low level to allow the computational testing of a conjecture of Ash, Doud, Pollack, and Sinnott relating such representations to mod 2 arithmetic cohomology. We test the conjecture for these examples and offer a refinement of the conjecture that resolves ambiguities in the predicted weight. 2 Introduction and statement of the conjecture The purpose of this paper is to test the main conjecture of [2] in characteristic 2. This conjecture (which we will refer to as the AshDoudPollackSinnott or ADPS conjecture) asserts the existence of Hecke cohomology eigenclasses in the mod p cohomology of certain arithmetic subgroups of GLn attached to n
EVEN ICOSAHEDRAL GALOIS REPRESENTATIONS OF PRIME CONDUCTOR
, 2004
"... Abstract. In this paper, we use a series of targeted Hunter searches to prove that the minimal prime conductor of an even icosahedral Galois representation is 1951. In addition, we give a complete list of all even icosahedral Galois representations of prime conductor less than 10,000. 1. ..."
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Abstract. In this paper, we use a series of targeted Hunter searches to prove that the minimal prime conductor of an even icosahedral Galois representation is 1951. In addition, we give a complete list of all even icosahedral Galois representations of prime conductor less than 10,000. 1.