Abstract
AbstractConsider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.
Highlights
The study of diagonal cycles on triple products of Shimura curves has its origins in the work of Gross, Kudla, and Schoen [11, 12]
They introduced a null-homologous modification of the diagonal embedding of the curve in its triple product, referred to as the modified diagonal cycle, or more commonly today as the Gross–Kudla–Schoen cycle
Given three cuspidal newforms of weight 2 and square-free level N such that the sign of the functional equation of the associated triple product L-function is −1, Gross and Kudla [11] conjectured that the central value at s = 2 of the derivative of this L-function is given by a complex period times the Beilinson–Bloch height of the corresponding Hecke isotypic component of the modified diagonal cycle on the triple product of an indefinite Shimura curve determined by the local triple product root numbers
Summary
The study of diagonal cycles on triple products of Shimura curves has its origins in the work of Gross, Kudla, and Schoen [11, 12]. Given three cuspidal newforms of weight 2 and square-free level N such that the sign of the functional equation of the associated triple product L-function is −1, Gross and Kudla [11] conjectured that the central value at s = 2 of the derivative of this L-function is given by a complex period times the Beilinson–Bloch height of the corresponding Hecke isotypic component of the modified diagonal cycle on the triple product of an indefinite Shimura curve determined by the local triple product root numbers. Theorem 1.1 Let f1, f2, and f3 be three normalised eigenforms of weight 2 and level Γ0(p), denote by F = f1 ⊗ f2 ⊗ f3 their triple product, and suppose that the global root number of.
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