The arithmetic properties of modular forms and elliptic curves 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. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

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

About the Author
Daniel Delbourgo is Senior Lecturer in the School of Mathematical Sciences at Monash University in Australia.

Reviews
"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, Mathematical Reviews



Book Information
ISBN 9780521728669
Author Daniel Delbourgo
Format Paperback
Page Count 288
Imprint Cambridge University Press
Publisher Cambridge University Press
Weight(grams) 410g
Dimensions(mm) 227mm * 153mm * 15mm