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Iwasawa Theory, Projective Modules, and Modular Representations
Ralph Greenberg, University of Washington, Seattle, WA
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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

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.

• 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