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

complex multiplication;elliptic curve;equivariant Tamagawa number conjecture;L-function;Tate-Shafarevich group

Colwell, Jason ; Flach, Matthias

University:California Institute of Technology

Department:Physics, Mathematics and Astronomy

关键词: complex multiplication;    elliptic curve;    equivariant Tamagawa number conjecture;    L-function;    Tate-Shafarevich group;   

Others  :  https://thesis.library.caltech.edu/1239/1/abstract.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

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.