**Memoirs of the American Mathematical Society**

2011;
185 pp;
Softcover

MSC: Primary 11;
Secondary 20

Print ISBN: 978-0-8218-4931-6

Product Code: MEMO/211/992

List Price: $88.00

AMS Member Price: $52.80

MAA member Price: $79.20

**Electronic ISBN: 978-1-4704-0609-7
Product Code: MEMO/211/992.E**

List Price: $88.00

AMS Member Price: $52.80

MAA member Price: $79.20

# Iwasawa Theory, Projective Modules, and Modular Representations

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*Ralph Greenberg*

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

# Table of Contents

## Iwasawa Theory, Projective Modules, and Modular Representations

- Chapter 1. Introduction. 18 free
- Chapter 2. Projective and quasi-projective modules. 1724
- Chapter 3. Projectivity or quasi-projectivity of XE0(K). 3138
- Chapter 4. Selmer atoms. 4754
- Chapter 5. The structure of Hv(K, E). 6976
- Chapter 6. The case where is a p-group. 7784
- Chapter 7. Other specific groups. 8188
- Chapter 8. Some arithmetic illustrations. 105112
- Chapter 9. Self-dual representations. 131138
- Chapter 10. A duality theorem. 141148
- Chapter 11. p-modular functions. 151158
- Chapter 12. Parity. 159166
- Chapter 13. More arithmetic illustrations. 169176
- Bibliography 183190