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

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