Algebraic Cycles, Modular Forms and Euler Systems

Fix a squarefree integer N and let f be a newform of weight 2 for Γ0(N); we assume that f does not have complex multiplication. It was shown in [14] and [15] that for a set of primes l of density 1 the naive deformation theory of the mod l Galois representation associated to f is unobstructed (in the sense that the universal deformation ring is a power series ring over the Witt vectors). In [31] these methods were modified to obtain results on the deformation problems studied by Taylor-Wiles. In this paper we extend the results of Flach and Mazur to the case of newforms f of weight κ ≥ 2 for Γ1(N). We now state our results more precisely. Fix l > max{5, κ+1}, let f be as above and let H be the associated l-adic representation: H is a free module of rank 2 over a certain Hecke algebra A, which itself is a finite, flat, local, Gorenstein Zl-algebra. Let T be the Tate twist EndAH(1) of the module of trace zero endomorphisms of H. Using techniques of Flach we construct a collection of cohomology classes {c} in H(Q, T ) with tightly controlled ramification. With some mild additional hypotheses, applying the methods of Kolyvagin to these classes yields a certain annihilator η ∈ A of the Selmer group H f (Q, T ∗) of the Cartier dual of T . This Selmer group is dual to the differentials ΩR⊗RA, where R is the universal minimally ramified deformation ring of the residual representation of H. In the case that η is a unit this then implies that both R and A are isomorphic to the ring of Witt vectors over the residue field of A. In the general case, following Mazur we show that our construction yields a derivation from A to the Selmer group H f (Q, T/ηT ); it follows by a formal argument that the natural surjection R A induces an isomorphism ΩR⊗RA ∼= ΩA. Although not the strongest possible result, this does provide a great deal of information 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 ΩR⊗RA ∼= ΩA is characterized by the fact that ΩA ∼= ΩR⊗RA ∼= HomZl ( H f (Q, T ),Ql/Zl )