# zbMATH — the first resource for mathematics

The conjecture of Tate and Voloch on $$p$$-adic proximity to torsion. (English) Zbl 0986.11038
The author extends his earlier result [J. Reine Angew. Math. 499, 225-236 (1998; Zbl 0932.11041)] on the conjecture of Tate and Voloch, which asserts that for a subvariety $$X$$ in a semiabelian variety $$G$$ over $${\mathbb{C}}_p$$, the torsion points in $$G({\mathbb{C}}_p) \setminus X$$ stay $$p$$-adically away from $$X$$. The result is that this holds whenever $$G$$ (but not necessarily $$X$$) is defined over a finite extension of $${\mathbb{Q}}_p$$. The improvement is that the result is now also proved for torsion points of order a multiple of $$p$$.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 14K15 Arithmetic ground fields for abelian varieties 14G20 Local ground fields in algebraic geometry
Full Text: