New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

Kolyvagin Systems
Barry Mazur, Harvard University, Cambridge, MA, and Karl Rubin, Stanford University, CA
 SEARCH THIS BOOK:
Memoirs of the American Mathematical Society
2004; 96 pp; softcover
Volume: 168
ISBN-10: 0-8218-3512-2
ISBN-13: 978-0-8218-3512-8
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/168/799

Since their introduction by Kolyvagin, Euler systems have been used in several important applications in arithmetic algebraic geometry. For a $$p$$-adic Galois module $$T$$, Kolyvagin's machinery is designed to provide an upper bound for the size of the Selmer group associated to the Cartier dual $$T^*$$.

Given an Euler system, Kolyvagin produces a collection of cohomology classes which he calls "derivative" classes. It is these derivative classes which are used to bound the dual Selmer group.

The starting point of the present memoir is the observation that Kolyvagin's systems of derivative classes satisfy stronger interrelations than have previously been recognized. We call a system of cohomology classes satisfying these stronger interrelations a Kolyvagin system. We show that the extra interrelations give Kolyvagin systems an interesting rigid structure which in many ways resembles (an enriched version of) the "leading term" of an $$L$$-function.

By making use of the extra rigidity we also prove that Kolyvagin systems exist for many interesting representations for which no Euler system is known, and further that there are Kolyvagin systems for these representations which give rise to exact formulas for the size of the dual Selmer group, rather than just upper bounds.

Graduate students and research mathematicians interested in number theory.

• Introduction
• Local cohomology groups
• Global cohomology groups and Selmer structures
• Kolyvagin systems
• Kolyvagin systems over principal artinian rings
• Kolyvagin systems over integral domains
• Examples
• Appendix A. Proof of Theorem 3.2.4
• Appendix B. Proof of Theorem 4.3.3, by Benjamin Howard
• Bibliography