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Iwasawa Theory, Projective Modules, and Modular Representations
Ralph Greenberg, University of Washington, Seattle, WA
cover
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Memoirs of the American Mathematical Society
2011; 185 pp; softcover
Volume: 211
ISBN-10: 0-8218-4931-X
ISBN-13: 978-0-8218-4931-6
List Price: US$83
Individual Members: US$49.80
Institutional Members: US$66.40
Order Code: MEMO/211/992
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This paper shows that properties of projective modules over a group ring \(\mathbf{Z}_p[\Delta]\), where \(\Delta\) is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve \(E\). Modular representation theory for the group \(\Delta\) plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a \(\mu\)-invariant. The author then studies \(\lambda\)-invariants \(\lambda_E(\sigma)\), where \(\sigma\) varies over the absolutely irreducible representations of \(\Delta\). He shows that there are non-trivial relationships between these invariants under certain hypotheses.

Table of Contents

  • Introduction
  • Projective and quasi-projective modules
  • Projectivity or quasi-projectivity of \(X_{E}^{\Sigma_0}(K_{\infty})\)
  • Selmer atoms
  • The structure of \(\mathcal{H}_v(K_{\infty}, E)\)
  • The case where \(\Delta\) is a \(p\)-group
  • Other specific groups
  • Some arithmetic illustrations
  • Self-dual representations
  • A duality theorem
  • \(p\)-modular functions
  • Parity
  • More arithmetic illustrations
  • Bibliography
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