The Conjecture of Birch and Swinnerton-Dyer for Elliptic Curves with Complex Multiplication by a Nonmaximal Order

Citation

Colwell, Jason Andrew (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. doi:10.7907/G40X-ST27. https://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.

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