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The Tate-Voloch conjecture for Drinfeld modules. (English) Zbl 1113.11035
The Tate-Voloch conjecture [J. Tate and J. F. Voloch, Int. Math. Res. Not. 1996, No. 12, 589–601 (1996; Zbl 0893.11015)] conjectures that a torsion point of a semi-abelian variety \(G\) over \({\mathbb C}_p\) that is too close to a subvariety \(X\) must actually lie on \(X\).
The author proves two similar results in the context of Drinfeld modules. Let \(\varphi:A\to K\{\tau\}\) be a Drinfeld module. For \(g\geq 1\) consider the diagonal action of \(\varphi\) on \({\mathbb G}_a^g\). A point \((x_1,\dots,x_g)\in{\mathbb G}_a^g(K^{\text{alg}})\) is called a torsion point if every \(x_i\) is a torsion point of \(\varphi\).
If \(L\) is an extension of \(K\) and \(w\) a valuation of \(L\), one can define a \(w\)-adic distance from \(P\in{\mathbb G}_a^g(L)\) to an affine subvariety \(X\) using \(w(f(P))\) for \(w\)-integral polynomials \(f\) in the vanishing ideal of \(X\).
The main theorem of the paper assumes that the valuation \(v\) of \(K\) satisfies certain conditions. It asserts the existence of a bound (depending on \(\varphi\), \(X\) and \(v\)) with the following property: If \(L\) is a finite extension of \(K\) and \(P\in{\mathbb G}_a^g(L)\) a torsion point whose \(w\)-adic distance to \(X\) is below this bound for every valuation \(w\) of \(L\) lying above \(v\), then \(P\in X(L)\).
A second result, involving only one \(w\)-adic distance, is also given for the special case where \(X\) is degenerate to a point.
Moreover, it is mentioned that the bounds for the \(w\)-adic distances can be made effective.

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G50 Heights
11G18 Arithmetic aspects of modular and Shimura varieties
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References:
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