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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 $g\ge 2$ 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 ${h}_{NT}\left(x\right)$ be the Néron-Tate height of a point $x\in J\left(\overline{K}\right)$.

Theorem 1. There exists $\epsilon >0$ such that $\left\{P\in X\left(\overline{K}\right)|{h}_{NT}\left({\varphi }_{{D}_{0}}\left(P\right)\right)\le \epsilon \right\}$ is finite.

This generalizes a theorem by M. Raynaud [ Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)] that the set of points $P\in {X}_{K}\left(\overline{K}\right)$ such that ${\varphi }_{{D}_{0}}\left(P\right)$ is torsion is finite. Raynaud’s result is recaptured in theorem 1 that ${\varphi }_{{D}_{0}}\left(P\right)$ being torsion is equivalent to ${h}_{NT}\left({\varphi }_{{D}_{0}}\left(P\right)\right)=0$. 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 $𝒳\to \text{Spec}\left({𝒪}_{𝒦}\right)$ be a regular minimal model of a smooth geometrically connected curve ${X}_{K}$ over $K$ of genus $g\ge 2$. If $𝒳$ has a semi-stable reduction, then

${\left({\omega }_{Ar},{\omega }_{Ar}\right)}_{Ar}\ge {\left({\omega }_{a},{\omega }_{a}\right)}_{a}>0·$

Here ${\left(\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\right)}_{Ar}$ denotes the Arakelov intersection pairing, ${\omega }_{Ar}=\overline{{\omega }_{𝒳/{𝒪}_{K}}}$ is an element of ${\text{Pic}}_{Ar}\left(𝒳\right)$, and ${\left(\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\right)}_{a}$ is the Zhang intersection pairing.

The proof is by indirect method, namely, assuming $\left({\omega }_{a},{\omega }_{a}\right)=0$ 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 (curves)