ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 14 Zhang, Shou-Wu, Columbia University, USA Small points, Heegner points, and Arakelov theory

In this talk, I will explain the recent applications of Arakelov theory to the Bogomolov conjecture on small points, and to the Gross-Zagier formula for Heegner points. Theorem A (Bogomolov conjecture). Let Y be a subvariety of an abelian variety A defined over ℚ̄. Let h : A(ℚ̄) → ℝ be a Néron-Tate height function (with respect to a symmetric projective embedding of A). Assume that Y is not a translation of an abelian subvariety by a torsion point. Then there is a positive number ϵ such that the set

{x ∈ Y (ℚ̄) : h(x) < ϵ}

is not Zariski dense in Y. Remarks:

1. As the above set contains all torsion points in Y, the above theorem implies a theorem of Raynaud on Lang’s conjecture.

2. When Y is the image of a map X → A = Jac(X) of a curve to its Jacobian defined by the canonical divisor of X, this theorem is due to E. Ullmo. Actually one can prove the following stronger statement about probability measure rather than Zariski topology: Theorem B. Let (x_{n}, n ∈ ℕ) be a sequence of points in A(ℚ̄) such that the following conditions are verified:

1. There is no subsequence of (x_{n}, n ∈ ℕ) contained in a translation of proper abelian subvariety by a torsion point;

2. lim_{n→∞} h(x_{n}) = 0. Then the probability measures of the Galois orbits of x_{n} converge to the Haar measure of A(ℂ). One can use this to show that the set of torsion points on A over the maximal totally real fields is finite. Theorem C (Gross-Zagier formula). Let E be a modular elliptic curve defined over a totally real field F with the conductor 𝔫. Let K/F be an imaginary quadratic extension which is unramified over places dividing 𝔫 and such that the functional equation of L(E/K, s) has the negative sign. Then

L′(E/K, 1) = (positive constant) · h(y_{K})

where y_{K} ∈ E(K) is the Heegner point, and h is the Néron-Tate height on E. Remarks:

1. Here we say E/F is modular if L(E, s) = L(f, s) with f a Hilbert modular form over F. In our case, E has a uniformization by certain Shimura curve (by the Jacquet-Langlands theory and the Eisenstein divisor class). The points y_{K} is the image of certain CM-points on Shimura curves.

2. When F = ℚ and every factor of 𝔫 is split in K, this theorem is due to Gross and Zagier. Modulo the construction of Eisenstein divisor class, the general statement was first conjectured by Gross. Assume that ord_{s=1}L(E, s) ≤ 1, and that either g is odd or 𝔫 is not a square. Then one may show that there is a K as in the theorem such that L′(E/K, 1) ≠ 0. This implies that y_{K} is nontorsion. By Kolyvagin’s theory of Euler system, this in turn implies the following statement concerning the Birch and Swinnerton-Dyer conjecture: Theorem D. Assume that ord_{s=1}L(E, s) ≤ 1, and that either g is odd or 𝔫 is not a square. Then

1. The Mordell-Weil group E(F) has the rank equal to ord_{s=1}L(E, s).

2. The Shafarevich group of E over F is finite. In the proofs of both Theorem A and Theorem C, Arakelov theory (developed by Deligne, Faltings, Gillet, Soulé, Szpiro, etc.) plays the crucial rule in the computation of Néron-Tate heights. By Hodge index theorem in this theory, one can either express the Néron-Tate heights as the intersections of arithmetic line bundles on the integral models of abelian varieties, or on the integral models (i. e., arithmetic surfaces) of curves.