Pairings in the Arithmetic of Elliptic Curves
The recent work of Cornut, Vatsal, Bertolini, Darmon, Nekovár and others on the Mordell-Weil and Shafarevich-Tate groups of elliptic curves over anticyclotomic towers has made it timely to organize, and possibly sharpen, the collection of yet unresolved conjectures regarding the finer structure we expect to be true about this piece of arithmetic. That was the general aim of two of our recent preprints ([MR1], [MR2]) the former being the text of an address given at the International Congress of Mathematicians in Beijing in August 2002 by the second author. It was also the aim of the lecture delivered by the first author at the conference in Barcelona in July 2002. Since there was some overlap in our two lectures (Beijing, Barcelona), we felt it reasonable not to repeat things, but rather, in this write-up for the proceedings of the Barcelona conference, to concentrate only on the following part of the story which was not really covered in any of the other accounts.
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