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049 _aMAIN
100 1 _aDelbourgo, Daniel.
_9937359
245 1 0 _aElliptic curves and big Galois representations /
_cDaniel Delbourgo.
260 _aCambridge, UK ;
_aNew York :
_bCambridge University Press,
_c2008.
300 _a1 online resource (ix, 281 pages) :
_billustrations
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
490 1 _aLondon Mathematical Society lecture note series ;
_v356
504 _aIncludes bibliographical references (pages 275-279) and index.
520 1 _a"The mysterious properties of modular forms lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula." "Three main steps are outlined. The first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. One can then establish finiteness results for big Selmer groups. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture." "This is the first book on the subject, and the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction to the subject. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases."--Jacket
588 0 _aPrint version record.
505 0 _aCover; Title; Copyright; Dedication; Contents; Introduction; List of Notations; Chapter I Background; 1.1 Elliptic curves; 1.2 Hasse-Weil L-functions; 1.3 Structure of the Mordell-Weil group; 1.4 The conjectures of Birch and Swinnerton-Dyer; 1.5 Modular forms and Hecke algebras; Chapter II p-Adic L-functions and Zeta Elements; 2.1 The p-adic Birch and Swinnerton-Dyer conjecture; 2.2 Perrin-Riou's local Iwasawa theory; 2.3 Integrality and (z, D)-modules; 2.4 Norm relations in K-theory; 2.5 Kato's p-adic zeta-elements; Chapter III Cyclotomic Deformations of Modular Symbols; 3.1 Q-continuity.
505 8 _a3.2 Cohomological subspaces of Euler systems3.3 The one-variable interpolation; 3.4 Local freeness of the image; Chapter IV A User's Guide to Hida Theory; 4.1 The universal ordinary Galois representation; 4.2 N-adic modular forms; 4.3 Multiplicity one for I-adic modular symbols; 4.4 Two-variable p-adic L-functions; Chapter V Crystalline Weight Deformations; 5.1 Cohomologies over deformation rings; 5.2 p-Ordinary deformations of Bcris and Dcris; 5.3 Constructing big dual exponentials; 5.4 Local dualities; Chapter VI Super Zeta-Elements; 6.1 The R-adic version of Kato's theorem.
505 8 _a6.2 A two-variable interpolation6.3 Applications to Iwasawa theory; 6.4 The Coleman exact sequence; 6.5 Computing the R[[D]]-torsion; Chapter VII Vertical and Half-Twisted Arithmetic; 7.1 Big Selmer groups; 7.2 The fundamental commutative diagrams; 7.3 Control theory for Selmer coranks; Chapter VIII Diamond-Euler Characteristics: the Local Case; 8.1 Analytic rank zero; 8.2 The Tamagawa factors away from p; 8.3 The Tamagawa factors above p (the vertical case); 8.4 The Tamagawa factors above p (the half-twisted case); 8.5 Evaluating the covolumes.
505 8 _a10.6 Numerical examples, open problemsAppendices; A: The Primitivity of Zeta Elements; B: Specialising the Universal Path Vector; C: The Weight-Variable Control Theorem (by Paul A. Smith); C.1 Notation and assumptions; C.2 Properties of affinoids; C.3 The cohomology of a lattice L; C.4 Local conditions; C.5 Dualities via the Ext-pairings; C.6 Controlling the Selmer groups; Bibliography; Index.
590 _aeBooks on EBSCOhost
_bEBSCO eBook Subscription Academic Collection - Worldwide
650 0 _aCurves, Elliptic.
_9162527
650 0 _aGalois theory.
_9288332
650 6 _aCourbes elliptiques.
_9869417
650 6 _aThéorie de Galois.
_9937360
650 7 _aMATHEMATICS
_xGeometry
_xAlgebraic.
_2bisacsh
_9869419
650 7 _aCurves, Elliptic.
_2fast
_0(OCoLC)fst00885455
_9162527
650 7 _aGalois theory.
_2fast
_0(OCoLC)fst00937326
_9288332
650 7 _aElliptische Kurve
_2gnd
_9937361
650 7 _aGalois-Darstellung
_2gnd
_9937362
655 4 _aElectronic books.
776 0 8 _iPrint version:
_aDelbourgo, Daniel.
_tElliptic curves and big Galois representations.
_dCambridge, UK ; New York : Cambridge University Press, 2008
_z9780521728669
_w(DLC) 2008021192
_w(OCoLC)227275650
830 0 _aLondon Mathematical Society lecture note series ;
_v356.
_9144795
856 4 0 _uhttps://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=552371
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