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Elliptic curves and big Galois representations

Author: Daniel Delbourgo
Publisher: Cambridge, UK ; New York : Cambridge University Press, 2008.
Series: London Mathematical Society lecture note series, 356.
Edition/Format:   Print book : EnglishView all editions and formats
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 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  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Daniel Delbourgo
ISBN: 9780521728669 0521728665
OCLC Number: 227275650
Description: ix, 281 pages : illustrations ; 23 cm.
Contents: Introduction; List of notations; 1. Background; 2. p-adic L-functions and Zeta-elements; 3. Cyclotomic deformations of modular symbols; 4. A user's guide to Hida theory; 5. Crystalline weight deformations; 6. Super Zeta-elements; 7. Vertical and half-twisted arithmetic; 8. Diamond-Euler characteristics: the local case; 9. Diamond-Euler characteristics: the global case; 10. Two-variable Iwasawa theory of elliptic curves; A. The primitivity of Zeta elements; B. Specialising the universal path vector; C. The weight-variable control theorem; Bibliography.
Series Title: London Mathematical Society lecture note series, 356.
Responsibility: Daniel Delbourgo.
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Abstract:

Describes the arithmetic of modular forms and elliptic curves; self-contained and ideal for both graduate students and professional number theorists.  Read more...

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"This research monograph contains much that has not been published elsewhere, and will be useful for specialists in the field who want to catch up on the author's work." Neil P. Dummigan, Read more...

 
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    schema:reviewBody ""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." ;
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