Equidistribution of small points.
(Equirépartition des petits points.)

*(French)*Zbl 0991.11035Let \(X\) be an arithmetic variety (i.e. a proper flat scheme over the ring of integers in a number field \(K\) whose generic fiber is smooth) of dimension \(d\). Let \(\overline L\) be a positive Hermitian line bundle on \(X\), i.e. a line bundle on \(X\) endowed with a smooth Hermitian metric on \(X(\mathbb{C})\) whose curvature form \(c_1(\overline L)\) is strictly positive. Using Arakelov geometry as developed by Gillet and Soulé, these data allow one to define height functions on the set of subvarieties of \(X_K\) which are actual functions, as opposed to functions defined up to a bounded one.

The main result of the paper is Theorem 3.1, which states: Suppose the height \(h_{\overline L}(X)=0\) and choose an embedding \(\sigma\) of \(K\) into the complex numbers. Then for any sequence \((x_n)\) of points in \(X(\overline K)\) such that (a) \(h_{\overline L}(x_n)\) converges to \(0\) and (b) any subsequence of \((x_n)\) is dense in \(X\) with respect to the Zariski topology, the sequence of probability measures \((\mu_n)\) on \(X_\sigma(\mathbb{C})\) given by \[ \mu_n = {1\over \#O(x_n)} \sum_{x\in O(x_n)} \delta_{\sigma(x)} \] (\(O(x_n)\) being the orbit of \(x_n\) under the Galois group of \(\overline K/K\)) converges vaguely to the measure \(\mu= c_1(\overline L)^{d-1} / c_1(L)^{d-1}\).

Note that the result of S. Zhang on arithmetic ampleness [J. Am. Math. Soc. 8, No. 1, 187-221 (1995; Zbl 0861.14018)] implies under hypotheses (a) and (b) that the height \(h_{\overline L}(X)\leq 0\). Granted the equality, the proof of the theorem goes by twisting the Hermitian metric by any small enough smooth function.

The authors also show (Proposition 4.1) how the theory of adelic metrics, as developed by S. Zhang [J. Algebr. Geom. 4, No. 2, 281-300 (1995; Zbl 0861.14019)], allows one to extend the result stated to quite more general situations in the most interesting example of abelian varieties and Neron-Tate heights. The limit measure is in this case the normalized Haar measure on the complex tori underlying the abelian variety.

Under Bogomolov’s conjecture and in the case of abelian varieties, Hypothesis (b) of Theorem 4.1 may be replaced by supposing that no subsequence of \((x_n)\) is contained in a strict algebraic subgroup of \(X\). Note, however, that this theorem is a cornerstone to the recent proof of Bogomolov’s conjecture by E. Ullmo [Ann. Math. (2) 147, 167-179 (1998; Zbl 0934.14013)] and S. Zhang [Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11030)]. Note also that an analogue of this uniform distribution theorem has been given recently by Y. Bilu [Duke Math. J. 89, 465-476 (1997; Zbl 0918.11035)] in the case of tori using entirely different methods.

The main result of the paper is Theorem 3.1, which states: Suppose the height \(h_{\overline L}(X)=0\) and choose an embedding \(\sigma\) of \(K\) into the complex numbers. Then for any sequence \((x_n)\) of points in \(X(\overline K)\) such that (a) \(h_{\overline L}(x_n)\) converges to \(0\) and (b) any subsequence of \((x_n)\) is dense in \(X\) with respect to the Zariski topology, the sequence of probability measures \((\mu_n)\) on \(X_\sigma(\mathbb{C})\) given by \[ \mu_n = {1\over \#O(x_n)} \sum_{x\in O(x_n)} \delta_{\sigma(x)} \] (\(O(x_n)\) being the orbit of \(x_n\) under the Galois group of \(\overline K/K\)) converges vaguely to the measure \(\mu= c_1(\overline L)^{d-1} / c_1(L)^{d-1}\).

Note that the result of S. Zhang on arithmetic ampleness [J. Am. Math. Soc. 8, No. 1, 187-221 (1995; Zbl 0861.14018)] implies under hypotheses (a) and (b) that the height \(h_{\overline L}(X)\leq 0\). Granted the equality, the proof of the theorem goes by twisting the Hermitian metric by any small enough smooth function.

The authors also show (Proposition 4.1) how the theory of adelic metrics, as developed by S. Zhang [J. Algebr. Geom. 4, No. 2, 281-300 (1995; Zbl 0861.14019)], allows one to extend the result stated to quite more general situations in the most interesting example of abelian varieties and Neron-Tate heights. The limit measure is in this case the normalized Haar measure on the complex tori underlying the abelian variety.

Under Bogomolov’s conjecture and in the case of abelian varieties, Hypothesis (b) of Theorem 4.1 may be replaced by supposing that no subsequence of \((x_n)\) is contained in a strict algebraic subgroup of \(X\). Note, however, that this theorem is a cornerstone to the recent proof of Bogomolov’s conjecture by E. Ullmo [Ann. Math. (2) 147, 167-179 (1998; Zbl 0934.14013)] and S. Zhang [Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11030)]. Note also that an analogue of this uniform distribution theorem has been given recently by Y. Bilu [Duke Math. J. 89, 465-476 (1997; Zbl 0918.11035)] in the case of tori using entirely different methods.

Reviewer: Antoine Chambert-Loir (MR 98i;14027)