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 ´etale 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–Kato 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].