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


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