Integral period relations and congruences

@article{Tilouine2022IntegralPR,
  title={Integral period relations and congruences},
  author={Jacques Tilouine and Eric Urban},
  journal={Algebra \& Number Theory},
  year={2022}
}
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… 
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