Let be a number field and let be its algebraic closure. Let be a proper, smooth, geometrically connected curve of genus over and let be its jacobian. Let be a divisor of degree 1 on and let be the embedding of into defined by . Let be the Néron-Tate height of a point .
Theorem 1. There exists such that is finite.
This generalizes a theorem by M. Raynaud [ Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)] that the set of points such that is torsion is finite. Raynaud’s result is recaptured in theorem 1 that being torsion is equivalent to . 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 be a regular minimal model of a smooth geometrically connected curve over of genus . If has a semi-stable reduction, then
Here denotes the Arakelov intersection pairing, is an element of , and is the Zhang intersection pairing.
The proof is by indirect method, namely, assuming and , 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)].