Positivity and discretion of algebraic points of curves. (Positivité et discrétion des points algébriques des courbes.)(French)Zbl 0934.14013

Let $$K$$ be a number field and let $$\overline K$$ be its algebraic closure. Let $$X_K$$ be a proper, smooth, geometrically connected curve of genus $$g\geq 2$$ over $$K$$ and let $$J$$ be its jacobian. Let $$D_0$$ be a divisor of degree $$1$$ on $$X_K$$ and let $$\varphi_{D_0}$$ be the embedding of $$X_K$$ into $$J$$ defined by $$D_0$$. Let $$h_{NT}(x)$$ be the Néron-Tate height of a point $$x\in J(\overline K)$$.
Theorem 1. There exists $$\varepsilon>0$$ such that $$\{P\in X(\overline K) |h_{NT}(\varphi_{D_0}(P))\leq \varepsilon\}$$ is finite.
This generalizes a theorem by M. Raynaud [ Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)] that the set of points $$P\in X_K(\overline K)$$ such that $$\varphi_{D_0}(P)$$ is torsion is finite. Raynaud’s result is recaptured in theorem 1 that $$\varphi_{D_0}(P)$$ being torsion is equivalent to $$h_{NT}(\varphi_{D_0}(P))=0$$. Also this generalizes the works of L. Szpiro [The Grothendieck Festschrift. III, Prog. Math. 88, 229-246 (1990; Zbl 0759.14018)] and S. Zhang [Invent. Math. 112, No. 1, 171-193 (1993; Zbl 0795.14015)].
Theorem 2. Let $${\mathcal X}\to \text{Spec}({\mathcal O}_{{\mathcal K}})$$ be a regular minimal model of a smooth geometrically connected curve $$X_K$$ over $$K$$ of genus $$g\geq 2$$. If $$\mathcal X$$ has a semi-stable reduction, then $(\omega_{Ar}, \omega_{Ar})_{Ar}\geq (\omega_a, \omega_a)_a>0.$ Here $$(\;,\;)_{Ar}$$ denotes the Arakelov intersection pairing, $$\omega_{Ar}=\overline{\omega_{\mathcal{X}/\mathcal{O}_K}}$$ is an element of $$\text{Pic}_{Ar}(\mathcal X)$$, and $$(\;,\;)_a$$ is the Zhang intersection pairing.
The proof is by indirect method, namely, assuming $$(\omega_a, \omega_a)=0$$ and $$D_0={\Omega_X^1}/{2g-2}$$, one leads to a contraction.
This generalizes the results of S. Zhang (loc. cit.), J.-F. Burnol [Invent. Math. 107, No. 2, 421-432 (1992; Zbl 0723.14019)], and S. Zhang [J. Algebr. Geom. 4, No. 2, 281-300 (1995; Zbl 0861.14019)].

MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14H40 Jacobians, Prym varieties 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14H25 Arithmetic ground fields for curves
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