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A. In the general case, following Mazur we show that our construction yields a derivation from A to the Selmer group H1 f (Q;T=T ); it follows by a formal argu- ment that the natural surjection R A induces an isomorphism R RA = A. Although not the strongest possible result, this does provide a great deal of infor- mation on the structure of the ring R. (It is possible that any such map R A must be an isomorphism, although as far as I know this question remains open.) We also show that the isomorphism R RA = A is characterized by the fact that A = R RA = HomZl H1 f (Q;T ); Ql=Zl identies

Algebraic cycles, modular forms and Euler systems