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

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