Diagonal classes and the Bloch–Kato conjecture

The aim of this note is twofold. Firstly, we prove an explicit reciprocity law for certain diagonal classes in the \ub4etale cohomology of the triple product of a modular curve, stated in [8] 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\u2013Kato conjecture for the self-dual motive of an elliptic newform of weight k > 2. In the special case k = 2, our result gives a self-contained and simpler proof of the main result of [15]