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

TL;DR

This paper investigates the behavior of Gross-Schoen diagonal cycles on triple products of Shimura curves with bad reduction, linking their Abel-Jacobi images to period integrals and L-functions, and applies this to a case of the Bloch-Kato conjecture.

Contribution

It establishes a connection between diagonal cycles, period integrals, and L-values in the ramified setting, advancing the understanding of level raising and the Bloch-Kato conjecture.

Findings

Proves a relation between the Abel-Jacobi image of the diagonal cycle and period integrals.

Links the diagonal cycle to the central critical value of a triple product L-function.

Verifies a rank 0 case of the Bloch-Kato conjecture for a symmetric cube motive.

Abstract

In this article 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 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 certain rank $0$ case of the Bloch-Kato conjecture for the symmetric cube motive of a weight $2$ modular form.