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Arithmetic level raising on triple product of Shimura curves and Gross--Kudla--Schoen Diagonal cycles II: Bipartite Euler system

Haining Wang

Abstract

In this article, we study the Gross--Kudla--Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction and prove an unramified arithmetic level raising theorem for the cohomology of this triple product. We deduce from it a reciprocity law which relates the image of the diagonal cycle under the Abel--Jacobi map to certain period integral of Gross--Kudla type. Combing this with the first reciprocity law we proved in a previous work, we show that the Gross--Kudla--Schoen diagonal cycles form a bipartite Euler system for the symmetric cube motive of a modular form. As an application we provide some evidence for the rank one case of the Bloch--Kato conjecture for the symmetric cube motive of a modular form.

Arithmetic level raising on triple product of Shimura curves and Gross--Kudla--Schoen Diagonal cycles II: Bipartite Euler system

Abstract

In this article, we study the Gross--Kudla--Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction and prove an unramified arithmetic level raising theorem for the cohomology of this triple product. We deduce from it a reciprocity law which relates the image of the diagonal cycle under the Abel--Jacobi map to certain period integral of Gross--Kudla type. Combing this with the first reciprocity law we proved in a previous work, we show that the Gross--Kudla--Schoen diagonal cycles form a bipartite Euler system for the symmetric cube motive of a modular form. As an application we provide some evidence for the rank one case of the Bloch--Kato conjecture for the symmetric cube motive of a modular form.

Paper Structure

This paper contains 18 sections, 25 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The symmetric cube motive
  4. Notations and conventions
  5. Arithmetic level raising for Shimura curves
  6. Shimura curves and Shimura sets
  7. Shimura curves with Iwahori level
  8. Unramified level raising for Shimura curves
  9. Arithmetic level raising for triple product of Shimura curves
  10. The triple tensor product Galois representation
  11. Unramified level raising for triple product of Shimura curves
  12. Reciprocity laws for Gross--Kudla--Schoen diagonal cycles
  13. Ramified arithmetic level raising for Shimura curves
  14. Ramified arithmetic level raising for triple product of Shimura curves
  15. Applications to the Bloch--Kato conjectures
  16. ...and 3 more sections

Key Result

Theorem 2

Let $p$ be an $n$-admissible prime for $\underline{\mathbf{f}}$. We assume that each maximal ideal in the triple $\mathfrak{m}_{\underline{\mathbf{f}}}=(\mathfrak{m}_{1}, \mathfrak{m}_{2}, \mathfrak{m}_{3})$ satisfies Assumption intro-ass. Then we have the following isomorphism between $\mathcal{O}_{\underline{\lambda}, n}$-modules.

Theorems & Definitions (54)

  • Definition 1.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Ihara's lemma
  • Definition 2.4
  • ...and 44 more