Minimal heights and polarizations on group varieties. (English) Zbl 0847.11036

Bertrand, Daniel

Duke Math. J. 80, No. 1, 223-250 (1995).

Let \(G\) be a commutative algebraic group defined over a number field \(k\). In this paper, the author defines a notion of canonical height \(h_L\) attached to an admissible line bundle \(L\) on a compactification \(\overline G/k\) of \(G\) and gives lower bounds for the height \(h_L(P)\) of a point \(P\in G(k)\) in terms of \(L\), when \(G\) is isogenous to the product of an Abelian variety by a torus. For a point \(P\in G(k)\), denote by \(G(P)\) the smallest algebraic subgroup of \(G\) which contains \(P\), and put \(g(P)= \dim G(P)\). It is shown that, if \(G\) is isogenous to the product of an Abelian variety by a torus, then there is a constant \(c= c(G, \overline G, k)\) such that for any admissible line bundle \(L\) and any point \(p\in G(k)\), we have \[ h_L(P)^{g(P)}\geq c(L^{q(P)}\cdot G(P)),\tag{\(*\)} \] where the right-hand side is an intersection number. In particular, when \(G(P)= G\), we get \(h_L(P)\geq c(L^g)\), where \(g= \dim G\). For a general group \(G\), it is shown that such a constant does not exist. When \(G\) is a split torus/\(k\) or an isotypic Abelian variety, a stronger inequality is proved in the form \[ h_L(P)^{g(P)}\geq R(G(P)/k, P)(L^{g(P)}\cdot G(P)), \] where \(R(G(P)/k, P)\) denotes an intrinsic regulator. An appendix explains how bounds of the type \((*)\) answer a classical problem on linear dependence relations [ see D. Masser, New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 248-262 (1988; Zbl 0656.10031)].

Reviewer: D.Roy (Ottawa)

MSC:

11J81	Transcendence (general theory)
14L10	Group varieties
11G35	Varieties over global fields

Keywords:

group variety; Lehmer’s problem; Lang-Silverman conjecture; lower bounds for the height of a point; canonical height; admissible line bundle; Abelian variety; torus

Citations:

Zbl 0656.10031

References:

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