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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 K ¯ be its algebraic closure. Let X K be a proper, smooth, geometrically connected curve of genus g2 over K and let J be its jacobian. Let D 0 be a divisor of degree 1 on X K and let ϕ 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 xJ(K ¯).

Theorem 1. There exists ε>0 such that {PX(K ¯)|h NT (ϕ D 0 (P))ε} is finite.

This generalizes a theorem by M. Raynaud [ Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)] that the set of points PX K (K ¯) such that ϕ D 0 (P) is torsion is finite. Raynaud’s result is recaptured in theorem 1 that ϕ D 0 (P) being torsion is equivalent to h NT (ϕ 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 𝒳Spec(𝒪 𝒦 ) be a regular minimal model of a smooth geometrically connected curve X K over K of genus g2. If 𝒳 has a semi-stable reduction, then

(ω Ar ,ω Ar ) Ar (ω a ,ω a ) a >0·

Here (,) Ar denotes the Arakelov intersection pairing, ω Ar =ω 𝒳/𝒪 K ¯ is an element of Pic Ar (𝒳), and (,) a is the Zhang intersection pairing.

The proof is by indirect method, namely, assuming (ω a ,ω a )=0 and D 0 =Ω 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:
14G40Arithmetic varieties and schemes; Arakelov theory; heights
14H40Jacobians, Prym varieties
11G30Curves of arbitrary genus or genus 1 over global fields
14H25Arithmetic ground fields (curves)