Electronic ISBN:  9781470406097 
Product Code:  MEMO/211/992.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 211; 2011; 185 ppMSC: 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 nontrivial relationships between these invariants under certain hypotheses.

Table of Contents

Chapters

1. Introduction.

2. Projective and quasiprojective modules.

3. Projectivity or quasiprojectivity 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. Selfdual representations.

10. A duality theorem.

11. $p$modular functions.

12. Parity.

13. More arithmetic illustrations.


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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 nontrivial relationships between these invariants under certain hypotheses.

Chapters

1. Introduction.

2. Projective and quasiprojective modules.

3. Projectivity or quasiprojectivity 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. Selfdual representations.

10. A duality theorem.

11. $p$modular functions.

12. Parity.

13. More arithmetic illustrations.