On the Eisenstein ideal for imaginary quadratic fields

For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F . By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0, χ). We further prove that its index is bounded from above by the p-valuation of the order of the Selmer group of the p-adic Galois character associated to χ−1. This uses the work of R. Taylor et al. on attaching Galois representations to cuspforms of GL2/F . Together these results imply a lower bound for the size of the Selmer group in terms of L(0, χ), coinciding with the value given by the Bloch-Kato conjecture.