Elliptic curves and big Galois representations /
"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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge, UK ; New York :
Cambridge University Press,
2008.
|
| Series: | London Mathematical Society lecture note series ;
356. |
| Subjects: | |
| Online Access: | CONNECT |
Table of Contents:
- Cover; 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.
- 3.2 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.
- 6.2 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.
- 10.6 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.
