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

Abstract: 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.