## 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

### Keywords:

abelian variety; canonical height; lower bound

Zbl 0579.14035
Full Text:

### References:

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