| Type of Document |
Dissertation |
| Author |
Colwell, Jason
|
| URN |
etd-04012004-151307 |
| Persistent URL |
http://resolver.caltech.edu/CaltechETD:etd-04012004-151307 |
| Title |
The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order |
| Degree |
PhD |
| Option |
Mathematics |
| Advisory Committee |
| Advisor Name |
Title |
| Matthias Flach |
Committee Chair |
| Dinakar Ramakrishnan |
Committee Member |
| Edray Goins |
Committee Member |
| Michael Aschbacher |
Committee Member |
|
| Keywords |
- L-function
- Tate-Shafarevich group
- equivariant Tamagawa number conjecture
- elliptic curve
- complex multiplication
|
| Date of Defense |
2003-11-18 |
| Availability |
unrestricted |
Abstract
The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve -- the value of the L-function, to an algebraic invariant of the curve -- the order of the Tate--Shafarevich group. Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary field. It is this version which interests us here. Gross' Conjecture has been reformulated, by Fontaine and Perrin-Riou, in the language of derived categories and determinants of perfect complexes. Burns and Flach then realized that this immediately leads to a refined conjecture for elliptic curves with complex multiplication by a nonmaximal order. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. We prove this conjecture of Burns and Flach.
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