Eisenstein congruences for split reductive groups

We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain $L$-functions. We examine the consequences in several special cases, and use the Bloch-Kato conjecture to further motivate a belief in the congruences.

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