## Small values of heights on families of abelian varieties.(English)Zbl 0639.14025

Diophantine approximation and transcendence theory, Semin., Bonn/FRG 1985, Lect. Notes Math. 1290, 109-148 (1987).
[For the entire collection see Zbl 0621.00008.]
Let V be an algebraic variety defined over a number field k and A be an abelian variety over V. Fix an ample divisor D on A which is ample on all the fibers $$A_ v$$, $$v\in V$$. Denote by $$q_ v$$ the corresponding Néron-Tate height on the group $$A_ v(\bar k)$$. Let $$\nu_ v$$ be the distance-function on the $$A_ v(\bar k)$$ introduced by D. Bertrand [in Diophantine analysis, Proc. Number Theory Sect. Aust. Math. Soc. Conv., Univ. New South Wales 1985, Lond. Math. Soc. Lect. Note Ser. 109, 21-35 (1986; Zbl 0597.10032)]. Then there exists a non-empty open subset $$V_ 0\subset V$$ with the following property: For each $$d\geq 1$$ there exists $$c>0$$, depending only on k, A, D, d and the embedding of V such that for any extension $$k\supset k$$ of degree $$\leq d$$ and any $$v\in V_ 0(K)$$ we have $$\max (q_ v(P),\nu_ v(P))>c^{-1}(\max (1,h(v)))^{- 1}$$ for all $$P\neq 0$$ from $$A_ v(K)$$. Here h(v) is the Weil height corresponding to the embedding of V. This gives a lower bound for values of Néron-Tate height $$q_ v(P)$$, $$v\in V_ 0(K)$$ for non-torsion points P in terms of the height h(v). And also the upper bounds for #A$${}_ v(K)_{tors}.\quad In$$ both cases the bounds are polynomial on h(v).
Reviewer: A.Parshin

### MSC:

 14K15 Arithmetic ground fields for abelian varieties 14K10 Algebraic moduli of abelian varieties, classification

### Citations:

Zbl 0621.00008; Zbl 0597.10032