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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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