Arithmetic level raising on triple product of Shimura curves and Gross--Kudla--Schoen Diagonal cycles II: Bipartite Euler system

In this article, we study the Gross–Kudla–Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction and prove an unramified arithmetic level raising theorem for the cohomology of this triple product. We deduce from it a reciprocity law which relates the image of the diagonal cycle under the Abel–Jacobi map to certain period integral of Gross–Kudla type. Combing this with the first reciprocity law we proved in a previous work, we show that the Gross–Kudla–Schoen diagonal cycles form a bipartite Euler system for the symmetric cube motive of a modular form. As an application we provide some evidence for the rank one case of the Bloch–Kato conjecture for the symmetric cube motive of a modular form.