  Under relatively mild and natural conditions, we establish an integral period
relations for the (real or imaginary) quadratic base change of an elliptic cusp
form. This answers a conjecture of Hida regarding the {\it congruence number}
controlling the congruences between this base change and other eigenforms which
are not base change. As a corollary, we establish the Bloch-Kato conjecture for
adjoint modular Galois representations twisted by an even quadratic character.
In the odd case, we formulate a conjecture linking the degree two topological
period attached to the base change Bianchi modular form, the cotangent complex
of the corresponding Hecke algebra and the archimedean regulator attached to
some Beilinson-Flach element.
