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On the Eisenstein ideal for imaginary quadratic fields

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 May 2009

Tobias Berger
Tobias Berger*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WB, UK (email: t.berger@dpmms.cam.ac.uk)
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Abstract

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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.

Keywords

Selmer groupsEisenstein cohomologycongruences of modular formsBloch–Kato conjecture

MSC classification

Secondary: 11F80: Galois representations 11F33: Congruences for modular and $p$-adic modular forms 11F75: Cohomology of arithmetic groups
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 603 - 632
Copyright
Copyright © Foundation Compositio Mathematica 2009

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