The Birch and Swinnerton-Dyer conjecture for Q-curves

by Yu Zhao

Institution: McGill University
Department: Department of Mathematics and Statistics
Degree: PhD
Year: 2011
Keywords: Pure Sciences - Mathematics
Record ID: 1895787
Full text PDF: http://digitool.library.mcgill.ca/thesisfile103574.pdf


The main result of this thesis is the proof of a Kolyvagin-like result for Q-curves defined over Q=(square root N) of perfect square conductor (including trivial conductor) over that field. Such a setting lies beyond the scope of the general results of Zhang [Zh1] because of the absence of a Shimura curve parameterization for E. This thesis also describes an explicit construction of Heegner points on E in a setting which so far has not yet studied in the literature and provides numerical examples. In turn, these computations yield numerical evidence for a conjectural connection, which we propose in this thesis, between the Heegner points we construct and the ATR points obtained by Darmon-Logan in [DL].