I will start from the situation of a cuspidal Hecke eigenform f of real quadratic character, congruent to its complex conjugate modulo a prime P ramified in the coefficient field.

Following Hida, P then divides an appropriately normalised near-central critical value of the adjoint L-function of f. In fact the same is true of the other critical values. The Bloch-Kato conjecture then predicts non-zero elements of P torsion in Selmer groups, which can be constructed using experimentally-observed congruences involving non-parallel weight, level one Hilbert modular forms. Using p-adic deformation in Hida's nearly ordinary family, these appear to be equivalent to congruences between certain parallel-weight (but level p, non-trivial character) Hilbert modular cusp forms and Eisenstein series.

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