Results 1  10
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154
Chebotarëv and his density theorem
, 1995
"... The Russian mathematician Nikolaĭ Grigor′evich Chebotarëv (1894–1947) is famous for his density theorem in algebraic number theory. His centenary was commemorated on June 15, 1994, at the University of Amsterdam. The present paper is based on two lectures that were delivered on that occasion, and ..."
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Cited by 67 (3 self)
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The Russian mathematician Nikolaĭ Grigor′evich Chebotarëv (1894–1947) is famous for his density theorem in algebraic number theory. His centenary was commemorated on June 15, 1994, at the University of Amsterdam. The present paper is based on two lectures that were delivered on that occasion, and its content is summarized by the titles of those lectures: ‘Life and work of Chebotarev’, and ‘Chebotarev’s density theorem for the layman’. An appendix to the paper provides a modern proof of the theorem.
Compatibility of local and global Langlands correspondences
 J. Amer. Math. Soc
, 2007
"... Abstract. We prove the compatibility of local and global Langlands correspondences for GLn, which was proved up to semisimplification in [HT]. More precisely, for the ndimensional ladic representation Rl(Π) of the Galois group of a CMfield L attached to a conjugate selfdual regular algebraic cusp ..."
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Cited by 50 (12 self)
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Abstract. We prove the compatibility of local and global Langlands correspondences for GLn, which was proved up to semisimplification in [HT]. More precisely, for the ndimensional ladic representation Rl(Π) of the Galois group of a CMfield L attached to a conjugate selfdual regular algebraic cuspidal automorphic representation Π, which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of Rl(Π) to the decomposition group of a prime v of L not dividing l corresponds to Πv by the local Langlands correspondence.
Congruence properties of Zariskidense subgroups
 Proc. London Math. Soc
, 1984
"... This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of Krational points of G. Then what should it mean for F to be a 'large ' subgroup? We might require F to be a lattice in G, to be ari ..."
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Cited by 48 (2 self)
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This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of Krational points of G. Then what should it mean for F to be a 'large ' subgroup? We might require F to be a lattice in G, to be arithmetic, to contain many elements of a specific
On the Meromorphic Continuation of Degree Two LFunctions
 DOCUMENTA MATH.
, 2006
"... We prove that the Lfunction of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation. ..."
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Cited by 37 (2 self)
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We prove that the Lfunction of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.
POTENTIAL AUTOMORPHY AND CHANGE OF WEIGHT.
"... Abstract. We show that a strongly irreducible, odd, essentially selfdual, regular, weakly compatible system of ladic representations of the absolute Galois group of a totally real field is potentially automorphic. Along the way we prove a new automorphy lifting theorem for ladic representations w ..."
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Cited by 36 (11 self)
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Abstract. We show that a strongly irreducible, odd, essentially selfdual, regular, weakly compatible system of ladic representations of the absolute Galois group of a totally real field is potentially automorphic. Along the way we prove a new automorphy lifting theorem for ladic representations where we impose a new condition at l, which we call ‘potential diagonalizability’. This seems to be a more flexible condition than has been previously considered, and allows for substantial ‘change of weight ’ in our automorphy lifting result.
The Iwasawa main conjectures for GL2
, 2010
"... In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we ..."
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Cited by 29 (1 self)
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In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 26 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
The SatoTate Conjecture for Hilbert Modular Forms
"... We prove the SatoTate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the SatoTate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF), F a totally real field, which are not of CM type. The argument is based on the potentia ..."
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Cited by 24 (8 self)
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We prove the SatoTate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the SatoTate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential
REGULATOR CONSTANTS AND THE PARITY CONJECTURE
, 2009
"... The pparity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ∞Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K ∞ is its maximal prop ex ..."
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Cited by 23 (6 self)
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The pparity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ∞Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K ∞ is its maximal prop extension, then the pparity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K ∞ /Q). We also give analogous results when K/Q is nonabelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.
Decidability of the isomorphism problem for stationary AFalgebras
, 1999
"... The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C∗isomorphism induces an equivalence relation on these matrices, call ..."
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Cited by 16 (4 self)
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The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C∗isomorphism induces an equivalence relation on these matrices, called C ∗equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e., there is an algorithm that can be used to check in a finite number of steps whether two given primitive nonsingular matrices are C∗equivalent or not.