Abstract
(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.
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