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Diagonal classes and the Bloch-Kato conjecture. (English) Zbl 07243784
Summary: The aim of this note is twofold. Firstly, we prove an explicit reciprocity law for certain diagonal classes in the étale cohomology of the triple product of a modular curve, stated by M. Bertolini et al. [“Reciprocity laws for balanced diagonal classes” (2018), preprint] and used there as a crucial ingredient in the proof of the main results. Secondly, we apply the aforementioned reciprocity law to address the rank-zero case of the equivariant Bloch-Kato conjecture for the self-dual motive of an elliptic newform of weight $$k\ge 2$$. In the special case $$k=2$$, our result gives a self-contained and simpler proof of the main result by H. Darmon and V. Rotger [J. Am. Math. Soc. 30, No. 3, 601–672 (2017; Zbl 1397.11090)].
##### MSC:
 11 Number theory 33 Special functions
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