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

### Haining Wang

Vol. 16 (2022), No. 10, 2289–2338
##### Abstract

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

##### Keywords
level raising, Shimura varieties, diagonal cycles
Primary: 11G40
Secondary: 11G18