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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 \(p\).

11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
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