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arXiv · 1110.6197

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

Jeanine Van Order
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Abstract

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate-Shafarevich group of an elliptic curve in the Z_p^2-extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.

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Jeanine Van Order. 2011-10-27. Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication. https://doi.org/10.1016/j.jalgebra.2011.10.031

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