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The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order

Citation

Colwell, Jason (2004) The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-04012004-151307

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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:complex multiplication; elliptic curve; equivariant Tamagawa number conjecture; L-function; Tate-Shafarevich group
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Flach, Matthias
Thesis Committee:
  • Flach, Matthias (chair)
  • Ramakrishnan, Dinakar
  • Goins, Edray
  • Aschbacher, Michael
Defense Date:18 November 2003
Record Number:CaltechETD:etd-04012004-151307
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-04012004-151307
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1239
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:02 Apr 2004
Last Modified:26 Dec 2012 02:36

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