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Zbl 0934.14013
Ullmo, Emmanuel
Positivity and discretion of algebraic points of curves. (Positivité et discrétion des points algébriques des courbes.) (French)
[J] Ann. Math. (2) 147, No. 1, 167-179 (1998). ISSN 0003-486X; ISSN 1939-0980/e

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 $x\in J(\overline K)$. \par Theorem 1. There exists $\varepsilon>0$ such that $\{P\in X(\overline K) |h_{NT}(\varphi_{D_0}(P))\leq \varepsilon\}$ is finite. \par This generalizes a theorem by {\it 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 {\it L. Szpiro} [The Grothendieck Festschrift. III, Prog. Math. 88, 229-246 (1990; Zbl 0759.14018)] and {\it S. Zhang} [Invent. Math. 112, No. 1, 171-193 (1993; Zbl 0795.14015)]. \par Theorem 2. Let ${\cal X}\to \text{Spec}({\cal O}_{{\cal K}})$ be a regular minimal model of a smooth geometrically connected curve $X_K$ over $K$ of genus $g\geq 2$. If $\cal X$ 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, $\omega_{Ar}=\overline{\omega_{\cal{X}/\cal{O}_K}}$ is an element of $\text{Pic}_{Ar}(\cal X)$, and $(\ ,\ )_a$ is the Zhang intersection pairing. \par 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. \par This generalizes the results of {\it S. Zhang} (loc. cit.), {\it J.-F. Burnol} [Invent. Math. 107, No. 2, 421-432 (1992; Zbl 0723.14019)], and {\it S. Zhang} [J. Algebr. Geom. 4, No. 2, 281-300 (1995; Zbl 0861.14019)].
[N.Yui (Kingston/Ontario)]

MSC 2000:: *14G40 Arithmetic varieties and schemes
14H40 Jacobians
11G30 Curves of arbitrary genus
14H25 Arithmetic ground fields (curves)

Keywords: curves; jacobian; Néron-Tate height; Arakelov intersection pairing; Zhang intersection pairing

Citations: Zbl 0564.14020; Zbl 0795.14015; Zbl 0759.14018; Zbl 0861.14019; Zbl 0741.14011; Zbl 0723.14019

Cited in: Zbl 1196.11092 Zbl 1163.11049 Zbl 1171.11038 Zbl 1007.11042 Zbl 0991.11034 Zbl 0991.11037 Zbl 1014.11042 Zbl 0991.11035

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