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Integral period relations and the Blochâ Kato formula for quadratic twists of the adjoint L-function Tilouine, Jacques

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In a joint work with E. Urban, we prove under mild assumptions Integral Period Re- lations for the quadratic base change of a modular form and we compute the relative congruence number in terms of the value at s = 1 of a quadratic twist of the adjoint L-function. This proves a conjecture of Hida (1999). It also implies, under standard Taylor-Wiles type assumptions, in the real quadratic case, resp. in the imaginary quadratic case, the Bloch-Kato formula, resp. an analogue of it. This analogue would be the exact conjectural Bloch-Kato formula if a Bianchi period defined by E. Urban in his thesis could be related to a Bloch-Kato-Beilinson regulator.

Related references:
S. Bloch, K. Kato: Grothendieck Festschrift vol I, Birkhauser pp 333-400, 1990

H. Hida: Non-critical values of adjoint L-functions for SL(2), Proc. Symp. Pure Math. 66 (1999) Part I, 123-175

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