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Lower bounds for heights on abelian varieties. (Minorations de hauteurs sur les variétés abéliennes.) (French) Zbl 0803.11031
J. H. Silverman [Duke Math. J. 51, 395-403 (1984; Zbl 0579.14035)] has conjectured that, given a number field \(k\) and an integer \(g>0\) there exists a constant \(c= c(k,g) >0\) such that for all points \(\tau\) in Siegel space \(S_ g\) corresponding to an abelian variety \(A= A(\tau)\) defined over \(k\) and all points \(P\in A(k)\) which are of infinite order modulo every proper abelian subvariety \(B\subseteq A\), the canonical height \(\widehat {h} (P)\) satisfies the inequality \(\widehat {h}(P)\geq c(k,g) h(A(\tau))\). Here \(h(A (\tau))\) is a logarithmic height of the abelian variety \(A\). This generalizes a conjecture of Serge Lang on elliptic curves.
The present paper proves a partial result in this direction. Let \(g\), \(k\), and \(\tau\) be as above and let \(D= \max (2, [k: \mathbb{Q}])\), \(h= \max (1, h(A (\tau)))\), and \(\rho= D(h+\log D)/ \| \text{Im } \tau\|+ D^{1/( g+2)}\). Then there exist constants \(c_ 1= c_ 1(g) >0\) and \(c_ 2= c_ 2 (g)\) such that all \(P\in A(k)\) satisfy either (1) there exists a subvariety \(B\neq A\) of \(A\) of degree \(\leq c_ 2 (\rho\log \rho)^ g\) such that \(P\) is of order \(\leq c_ 2 (\rho\log \rho)^ g\) modulo \(B\); or (2) \(\widehat {h} (P)\geq c_ 1 \rho^{-4g-2} (\log 2\rho)^{-4g -1} h\).
This is the first lower bound that can go to infinity with the height of the variety. The proof uses transcendence theory, via an associated semiabelian variety of dimension \(g+1\) associated to \(A\) and \(P\).
Reviewer: P.Vojta (Berkeley)

MSC:
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
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