Description 
1 online resource (ix, 281 pages) : illustrations. 
Series 
London Mathematical Society lecture note series ; 356


London Mathematical Society lecture note series ; 356.

Bibliography 
Includes bibliographical references (pages 275279) and index. 
Summary 
"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 twovariable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and SwinnertonDyer (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. 
Contents 
Cover; Title; Copyright; Dedication; Contents; Introduction; List of Notations; Chapter I Background; 1.1 Elliptic curves; 1.2 HasseWeil Lfunctions; 1.3 Structure of the MordellWeil group; 1.4 The conjectures of Birch and SwinnertonDyer; 1.5 Modular forms and Hecke algebras; Chapter II pAdic Lfunctions and Zeta Elements; 2.1 The padic Birch and SwinnertonDyer conjecture; 2.2 PerrinRiou's local Iwasawa theory; 2.3 Integrality and (<U+007a>, <U+0044>)modules; 2.4 Norm relations in Ktheory; 2.5 Kato's padic zetaelements; Chapter III Cyclotomic Deformations of Modular Symbols; 3.1 Qcontinuity. 

3.2 Cohomological subspaces of Euler systems3.3 The onevariable 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 <U+004e>adic modular forms; 4.3 Multiplicity one for Iadic modular symbols; 4.4 Twovariable padic Lfunctions; Chapter V Crystalline Weight Deformations; 5.1 Cohomologies over deformation rings; 5.2 pOrdinary deformations of Bcris and Dcris; 5.3 Constructing big dual exponentials; 5.4 Local dualities; Chapter VI Super ZetaElements; 6.1 The Radic version of Kato's theorem. 

6.2 A twovariable interpolation6.3 Applications to Iwasawa theory; 6.4 The Coleman exact sequence; 6.5 Computing the R[[<U+0044>]]torsion; Chapter VII Vertical and HalfTwisted Arithmetic; 7.1 Big Selmer groups; 7.2 The fundamental commutative diagrams; 7.3 Control theory for Selmer coranks; Chapter VIII DiamondEuler 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 halftwisted case); 8.5 Evaluating the covolumes. 

10.6 Numerical examples, open problemsAppendices; A: The Primitivity of Zeta Elements; B: Specialising the Universal Path Vector; C: The WeightVariable 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 Extpairings; C.6 Controlling the Selmer groups; Bibliography; Index. 
Local Note 
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection  North America 
Subject 
Curves, Elliptic.


Galois theory.

Genre/Form 
Electronic books.

Other Form: 
Print version: Delbourgo, Daniel. Elliptic curves and big Galois representations. Cambridge, UK ; New York : Cambridge University Press, 2008 9780521728669 (DLC) 2008021192 (OCoLC)227275650 
ISBN 
9781107363069 (electronic bk.) 

1107363063 (electronic bk.) 

9780511894046 (ebook) 

051189404X (ebook) 

9780511721281 (ebook) 

0511721285 (ebook) 

9781107367975 

1107367972 

9780521728669 

0521728665 
