Arithmetic level raising on triple product of Shimura curves and Gross–Schoen Diagonal cycles, I : Ramified case

In this article we study the Gross-Schoen diagonal cycle on a triple product of Shimura curves at place bad reduction. We relate image under Abel-Jacobi map to certain period integral that governs central critical value Garrett-Rankin type $L$-function via level raising congruences. As an application prove rank $0$ case Bloch-Kato conjecture for symmetric cube motive weight $2$ modular form.