An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters

Details An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters

2007-01-05

arxiv.org/abs/math/0701177v1

Compos. Math. 145 (2009), no. 3, 603--632

Mathematics

Number Theory

26 pages

Scientific paper

10.1112/S0010437X09003984

For certain algebraic Hecke characters chi 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,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-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,chi), coinciding with the value given by the Bloch-Kato conjecture.

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