THE CONJECTURE OF TATE AND VOLOCH ON p-ADIC PROXIMITY TO TORSION

Tate and Voloch have conjectured that the p-adic distance from torsion points of semi-abelian varieties over Cp to subvarieties may be uniformly bounded. We prove this conjecture for torsion points on semi-abelian varieties over Qp using methods of algebraic model theory and a result of Sen on Galois representation of Hodge-Tate type. As a generalization of their theorem on linear forms in p-adic roots of unity, Tate and Voloch conjectured: Conjecture: (Tate, Voloch) Let G be a semi-abelian variety over Cp. Let X ⊆ G be a subvariety defined over Cp. Then there is a constant N ∈ N such that for any torsion point ζ ∈ G(Cp)tor either ζ ∈ X or λ(ζ,X) ≤ N . In the above statement, Cp denotes the completion of the algebraic closure of Qp and λ(·, X) is the p-adic proximity to X function. In [Sc] this conjecture was proved under the assumptions that G is defined over Qp and that ζ is a torsion point of order prime-to-p. It was suggested that the same method of proof would work without the latter restriction. This suggestion is carried out in this note. The main theorem is the following. Main Theorem: Let K be a finite extension of Qp. Let G be a semi-abelian variety defined over K. Let X ⊆ G be a closed subvariety defined over Cp. There is a constant N ∈ N depending only on X such that for any torsion point ζ ∈ G(Cp)tor either ζ ∈ X or λ(ζ,X) ≤ N . In the above statement, λ(·, X) is the p-adic proximity to X function. In [Sc], this function was denoted by “d(·, X)” and called a “distance function,” but to match the notation common in the literature, we revert to “λ” instead of “d.” The proof of the Main Theorem passes through an analysis (due to Sen) of the action of the inertia group on the Tate module of the p-divisible group of G and is completed by combining this argument with the main theorem of [Sc]. Date: 4 February 1999, revised 20 March 1999. 1991 Mathematics Subject Classification. Primary: 11D88; Secondary: 03C60, 11U09. Supported by an NSF MSPRF. The research for this paper took place at MSRI during the Spring 1998 Model Theory of Fields program.