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\(p\)-adic distance from torsion points of semi-abelian varieties. (English) Zbl 0932.11041
The main result of the paper under review (and part of the author’s Harvard Ph.D. thesis) is the following theorem.
Let \(K\) be a finite extension of \({\mathbb Q}_p\). Let \(G\) be a semi-abelian variety over \(K\), and let \(X \subset G\) be a closed subvariety defined over \({\mathbb C}_p\). Then the prime-to-\(p\) torsion points in \(G({\mathbb C}_p) \setminus X\) stay away from \(X\) in the sense of \(p\)-adic distance.
This theorem is a special case of a conjecture of J. Tate and J. F. Voloch [Int. Math. Res. Not. 1996, 12, 589-601 (1996; Zbl 0893.11015)], which deals with all torsion points and semi-abelian varieties defined over \({\mathbb C}_p\) (instead of over \(K\)). The proof makes use of methods of algebraic model theory.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
11U09 Model theory (number-theoretic aspects)
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