An error was encountered while trying to add the item to the cart. Please try again.
The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below.
Copy To Clipboard
Successfully Copied!
Iwasawa Theory, Projective Modules, and Modular Representations

Ralph Greenberg University of Washington, Seattle, WA
Available Formats:
Electronic ISBN: 978-1-4704-0609-7
Product Code: MEMO/211/992.E
185 pp
List Price: $88.00 MAA Member Price:$79.20
AMS Member Price: $52.80 Click above image for expanded view Iwasawa Theory, Projective Modules, and Modular Representations Ralph Greenberg University of Washington, Seattle, WA Available Formats:  Electronic ISBN: 978-1-4704-0609-7 Product Code: MEMO/211/992.E 185 pp  List Price:$88.00 MAA Member Price: $79.20 AMS Member Price:$52.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 2112011
MSC: Primary 11; Secondary 20;

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.

• Chapters
• 1. Introduction.
• 2. Projective and quasi-projective modules.
• 3. Projectivity or quasi-projectivity of $X_{E}^{\Sigma _{\mbox {\tiny {0}}}}(K_{\infty })$.
• 4. Selmer atoms.
• 5. The structure of ${{\mathcal H}}_v(K_{\infty }, E)$.
• 6. The case where $\Delta$ is a $p$-group.
• 7. Other specific groups.
• 8. Some arithmetic illustrations.
• 9. Self-dual representations.
• 10. A duality theorem.
• 11. $p$-modular functions.
• 12. Parity.
• 13. More arithmetic illustrations.
• Request Review Copy
• Get Permissions
Volume: 2112011
MSC: Primary 11; Secondary 20;

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.

• Chapters
• 1. Introduction.
• 2. Projective and quasi-projective modules.
• 3. Projectivity or quasi-projectivity of $X_{E}^{\Sigma _{\mbox {\tiny {0}}}}(K_{\infty })$.
• 4. Selmer atoms.
• 5. The structure of ${{\mathcal H}}_v(K_{\infty }, E)$.
• 6. The case where $\Delta$ is a $p$-group.
• 7. Other specific groups.
• 8. Some arithmetic illustrations.
• 9. Self-dual representations.
• 10. A duality theorem.
• 11. $p$-modular functions.
• 12. Parity.
• 13. More arithmetic illustrations.
Please select which format for which you are requesting permissions.