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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

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.

level raising, Shimura varieties, diagonal cycles
Mathematical Subject Classification
Primary: 11G40
Secondary: 11G18
Received: 2 November 2020
Revised: 3 August 2021
Accepted: 17 September 2021
Published: 28 January 2023
Haining Wang
Shanghai Center for Mathematical Sciences
Fudan University