## Fonctions thêta et points de torsion des variétés abéliennes. (Theta functions and torsion points of abelian varieties).(French)Zbl 0741.14025

Let $$k$$ be an arithmetic field of finite degree over $$\mathbb{Q}$$ and $$A$$ an abelian variety of dimension $$g\geq 1$$ defined over $$k$$. Given a torsion point of $$A$$, $$e(\neq 0)$$, let $$n(e)$$ be the order of $$e$$ and $$d(e)$$ the degree, over $$k$$, of the field of definition of $$e$$. The main result of this paper is a lower bound for $$d(e)$$ in terms of $$n(e)$$, $$g$$, the degree of $$k$$ over $$\mathbb{Q}$$, and the Faltings height of $$A$$. With precision: There exists a constant $$c_ 1>0$$ (depending on $$g$$) such that for any integer number $$\delta\geq 2$$, any real number $$h\geq 1$$, any field $$k$$ of degree $$\leq\delta$$ over $$\mathbb{Q}$$ and any abelian variety $$A$$ over $$k$$ of dimension $$g$$ with Faltings height $$h(A/k)\leq h$$ — $$A$$ simple and principally polarized —, the degree of the torsion points $$e\neq 0$$ verify: $$d(e)\geq c_ 1h^{-3/2}\delta^{- 3/2}n(e)^{1/g}/(\log(\delta)\cdot\log(n(e))).$$ The proof is transcendental and it is based on the study of the growth of the theta functions and the works of D. W. Masser [Compos. Math. 53, 153-170 (1984; Zbl 0551.14015) and in Diophantine approximations and transcendence theory, Semin., Bonn/FRG 1985, Lect. Notes Math. 1290, 109- 148 (1987; Zbl 0639.14025)].

### MSC:

 14K15 Arithmetic ground fields for abelian varieties 14K25 Theta functions and abelian varieties 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G99 Arithmetic problems in algebraic geometry; Diophantine geometry 11G10 Abelian varieties of dimension $$> 1$$

### Citations:

Zbl 0551.14015; Zbl 0639.14025
Full Text:

### References:

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