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Iwasawa Theory, Projective Modules, and Modular Representations
 
Ralph Greenberg University of Washington, Seattle, WA
Iwasawa Theory, Projective Modules, and Modular Representations
eBook ISBN:  978-1-4704-0609-7
Product Code:  MEMO/211/992.E
List Price: $88.00
MAA Member Price: $79.20
AMS Member Price: $52.80
Iwasawa Theory, Projective Modules, and Modular Representations
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Iwasawa Theory, Projective Modules, and Modular Representations
Ralph Greenberg University of Washington, Seattle, WA
eBook ISBN:  978-1-4704-0609-7
Product Code:  MEMO/211/992.E
List Price: $88.00
MAA Member Price: $79.20
AMS Member Price: $52.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2112011; 185 pp
    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.
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2112011; 185 pp
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.
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.