Heegner Cycles Research Articles

Heights of Heegner cycles and derivatives of L-series

0. Introduction 99 1. Intersections and heights 103 2. Kuga-Sato varieties and CM-cycles 117 3. Heights of CM-cycles 124 4. Heights of Heegner cycles 137 5. Proof of the main identity and the consequences 147 References 151

On thep-adic height of Heegner cycles

(0.1) In [Gr Za], Gross and Zagier proved a remarkable formula, which relates the first derivative of the L-function of a modular form f of weight 2 on Γ0(N) (over a suitable imaginary quadratic field) and the Neron-Tate height of a “Heegner point” (over the same quadratic field) on the f -part of the Jacobian of the modular curve X0(N). Later, Perrin-Riou [PR 2] proved a p-adic version of this formula. Kolyvagin’s method of Euler systems [Ko], combined with the formula of Gross-Zagier, proves the conjecture of Birch and Swinnerton-Dyer (up to a controlled rational factor) for all modular elliptic curves over Q with analytic rank ≤ 1.

Heegner points, cycles and Maass forms

We derive for Hecke-Maass cusp forms on the full modular group a relation between the sum of the form at Heegner points (and integrals over Heegner cycles) and the product of two Fourier coefficients of a corresponding form of half-integral weight. Specializing to certain cycles we obtain the nonnegativity of theL-function of such a form at the center of the critical strip. These results generalize similar formulae known for holomorphic forms.

Heegner cycles, modular forms and jacobi forms

Nous présentons une interprétation géométrique d'une loi arithmétique pour déduire des formules explicites pour les coefficients des formes modulaires elliptiques et des formes de Jacobi. Nous discutons des applications de ces formules et comme exemple nous dérivons de manière algorithmique un critère analogue au critère de Tunnell concernant des nombres congruentes.

Gross-Kohnen-Zagier theorem for higher weight forms

We calculate the height pairing between higher weight Heegner cycles associated to distinct imaginary quadratic fields. Under the positive definiteness hypothesis of the height pairing we prove a Gross-Kohnen-Zagier theorem for higher weight modular forms.