×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI