On the Eisenstein ideal for imaginary quadratic fields

Abstract

Abstract 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 GL 2/ 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 GL 2/ 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.