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Iwasawa Theory, Projective Modules, and Modular Representations
 
Ralph Greenberg University of Washington, Seattle, WA
Front Cover for Iwasawa Theory, Projective Modules, and Modular Representations
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
Front Cover for Iwasawa Theory, Projective Modules, and Modular Representations
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  • Front Cover for Iwasawa Theory, Projective Modules, and Modular Representations
  • Back Cover for Iwasawa Theory, Projective Modules, and Modular Representations
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.

  • Table of Contents
     
     
    • 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.
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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.
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