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

Abstract

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.

26 pages

References in corpus (3)

• Automorphic forms and rational homology 3--spheres
• l-Adic representations associated to modular forms over imaginary quadratic fields
• Denominators of Eisenstein cohomology classes for GL_2 over imaginary quadratic fields

Cited by in corpus (4)

• A torsion Jacquet--Langlands correspondence
• Denominators of Eisenstein cohomology classes for GL_2 over imaginary quadratic fields
• On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields
• Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts