An Eisenstein ideal for imaginary quadratic fields.

Abstract

For certain algebraic Hecke characters chi of an imaginary quadratic field <italic>F</italic> we define an Eisenstein ideal in a Hecke algebra acting on cuspidal automorphic forms on GL₂(<bold>A</bold><italic>_F</italic>) and prove a lower bound for its index in terms of the special <italic> L</italic>-value <italic>L</italic><super>alg</super>(0, chi). From this we obtain a lower bound for the size of the Selmer group of a <italic>p</italic>-adic Galois character associated to chi. The method we use is to show that p-divisibility of Lalg(0, chi) implies a congruence mod <italic>p</italic> between a multiple of an Eisenstein cohomology class associated to chi (in the sense of G. Harder) and a cuspidal cohomology class in the cohomology of a hyperbolic 3-orbifold. Implementing this requires bounding the denominator of the Eisenstein cohomology class, which we do by analytic methods, and using the geometry of the Borel-Serre compactification of these spaces to control torsion in the compactly supported cohomology of degree 2. We then use the work of R. Taylor <italic>et al</italic>. on associating Galois representations to cuspidal automorphic representations of GL₂(<bold>A</bold><italic>_F</italic>) to construct elements in Selmer groups.