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

Wang, Haining

doi:10.2140/ant.2022.16.2289

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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

DOI: 10.2140/ant.2022.16.2289

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.

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Keywords

level raising, Shimura varieties, diagonal cycles

Mathematical Subject Classification

Primary: 11G40

Secondary: 11G18

Milestones

Received: 2 November 2020

Revised: 3 August 2021

Accepted: 17 September 2021

Published: 28 January 2023

Authors

Haining Wang
Shanghai Center for Mathematical Sciences Fudan University Shanghai China

Vol. 16, No. 10, 2022