On the dihedral Euler characteristics of Selmer groups of abelian varieties

Details On the dihedral Euler characteristics of Selmer groups of abelian varieties On the dihedral Euler characteristics of Selmer groups of abelian varieties

2011-12-16

arxiv.org/abs/1112.3825v1

Mathematics

Number Theory

32 pages

Scientific paper

We use the framework of Euler characteristics to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the usual formula predicted by the refined conjecture of Birch-Swinnerton-Dyer and Tate via standard techniques. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a conjectural description of the Euler characteristic of the cotorsion submodule following works of Perrin-Riou and Howard. Finally, assuming two-variable main conjectures or more general main conjectures of two-variable type, we deduce some special cases of this conjecture, as well as some results towards the refined conjecture of Birch-Swinnerton-Dyer and Tate.

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