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99
Faltings, Degeneration of abelian varieties
, 1990
"... An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We fo ..."
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Cited by 139 (8 self)
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An abelian variety A defined over a finite field Fq admits sufficiently many complex multiplications, as Tate showed in [27]. For some details about complex multiplication, see §1.1. Is A the reduction of an abelian variety with sufficiently many complex multiplications in characteristic zero? We formulate several versions of this “CMlifting problem ” in §1.2. Honda
Modularity Of The RankinSelberg LSeries, And Multiplicity One For SL(2)
"... Contents 1. Introduction 1 2. Notations and Preliminaries 5 3. Construction of # : A(GL(2)) A(GL(2)) # A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product Lfunctions: local factors and holomorphy 15 3.4. Boundednes ..."
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Cited by 91 (15 self)
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Contents 1. Introduction 1 2. Notations and Preliminaries 5 3. Construction of # : A(GL(2)) A(GL(2)) # A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product Lfunctions: local factors and holomorphy 15 3.4. Boundedness in vertical strips 18 3.5. Modularity in the good case 30 3.6. A descent criterion 32 3.7. Modularity in the general case 35 4. Applications 37 4.1. A multiplicity one theorem for SL(2) 37 4.2. Some new functional equations 40 4.3. Root numbers and representations of orthogonal type 42 4.4. Triple product Lfunctions revisited 44 4.5. The Tate conjecture for 4fold products of modular curves 47 Bibliography 52 1. Introduction Let f, g be primitive cusp forms, holomorphic or otherwise, on
Valuation Theory on Finite Dimensional Division Algebras
 DEPARTMENT OF PURE MATHEMATICS, QUEEN’S UNIVERSITY, BELFAST BT7 1NN, UNITED KINGDOM EMAIL ADDRESS: R.HAZRAT@QUB.AC.UK DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA AT SAN DIEGO, LA JOLLA, CALIFORNIA 920930112, U.S.A. EMAIL ADDRESS: ARWADSWORTH@UC
, 1997
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Galois module structure of pthpower classes of extensions of degree p
 Israel J. Math
"... Abstract. In the mid1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pthpower classes of cyclic extensions K/F of pthpower degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois m ..."
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Cited by 24 (16 self)
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Abstract. In the mid1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pthpower classes of cyclic extensions K/F of pthpower degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F. In 1947 ˇ Safarevič initiated the study of Galois groups of maximal pextensions of fields with the case of local fields [12], and this study has grown into what is both an elegant theory as well as an efficient tool in the arithmetic of fields. From the very beginning it became clear that the groups of pthpower classes of the various field extensions of a base field encode basic information about the structure of the Galois groups of maximal pextensions. (See [7] and [13].) Such groups of pthpower classes arise naturally in studies in arithmetic algebraic geometry, for example in the study of elliptic curves. In 1960 Faddeev began to study the Galois module structure of pthpower classes of cyclic pextensions, again in the case of local fields, and during the mid1960s he and Borevič established the structure of these Galois modules using basic arithmetic invariants attached to Galois extensions. (See [6] and [4].) In 2003 two of the authors ascertained the Galois module structure of pthpower classes in the case of cyclic extensions of degree p over all base fields F containing a primitive pth root of unity [9]. Very recently, this work paved the way for the determination of the entire Galois cohomology as a Galois module in
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
Representations of abelian algebraic groups
 Pacific J. Math
, 1997
"... groupoveralocalfieldandtherepresentationsoftheWeilgroupofthelocalfieldinacertain associatedcomplexgroup. Thereshouldalsobearelation,althoughitwillnotbesoclose, betweentherepresentationsoftheglobalWeilgroupintheassociatedcomplexgroupand therepresentationsoftheadèlegroupthatoccurinthespaceofautomorphi ..."
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Cited by 19 (0 self)
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groupoveralocalfieldandtherepresentationsoftheWeilgroupofthelocalfieldinacertain associatedcomplexgroup. Thereshouldalsobearelation,althoughitwillnotbesoclose, betweentherepresentationsoftheglobalWeilgroupintheassociatedcomplexgroupand therepresentationsoftheadèlegroupthatoccurinthespaceofautomorphicforms. The natureoftheserelationswillbeexplainedelsewhere.FornowallIwanttodoisexplainand provetherelationswhenthegroupisabelian.Ishouldpointoutthatthiscaseisnottypical. Forexample,ingeneraltherewillberepresentationsofthealgebraicgroupnotassociatedto representationsoftheWeilgroup. Theproofsthemselvesaremerelyexercisesinclassfieldtheory.Iamwritingthemdown becauseitisdesirabletoconfirmimmediatelythegeneralprinciple,whichisverystriking,ina fewsimplecases.Moreover,itisprobablyimpossibletoattacktheproblemingeneralwithout havingfirstsolveditforabeliangroups.Iftheproofsseemclumsyandtooinsistentonsimple thingsrememberthattheauthor,toborrowametaphor,hasnotcocycledbeforeandhasonly minimumcontrolofhisvehicle. Itiswellknownthatthereisaonetoonecorrespondencebetweenisomorphismclassesof
Generalizing the GHS attack on the elliptic curve discrete logarithm problem
 Journal of Computation and Mathematics
, 2004
"... Abstract. We generalise the Weil descent construction of the GHS attack on the elliptic curve discrete logarithm problem (ECDLP) to arbitrary ArtinSchreier extensions. We give a formula for the characteristic polynomial of Frobenius of the obtained curves and prove that the large cyclic factor of ..."
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Cited by 18 (0 self)
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Abstract. We generalise the Weil descent construction of the GHS attack on the elliptic curve discrete logarithm problem (ECDLP) to arbitrary ArtinSchreier extensions. We give a formula for the characteristic polynomial of Frobenius of the obtained curves and prove that the large cyclic factor of the input elliptic curve is not contained in the kernel of the composition of the conorm and norm maps. As an application we considerably increase the number of elliptic curves which succumb to the basic GHS attack, thereby weakening curves over F2155 further. We also discuss other possible extensions or variations of the GHS attack and conclude that they are not likely to yield further improvements. 1
Class Field Theory in Characteristic p, its Origin and Development
 the Proceedings of the International Conference on Class Field Theory (Tokyo
, 2002
"... Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1 ..."
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Cited by 17 (5 self)
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Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function elds. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.