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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:
 [1] Denis, L., Canonical heights and Drinfeld modules, Math. ann., 294, 2, 213-223, (1992), (in French) · Zbl 0764.11027 [2] D. Ghioca, The arithmetic of Drinfeld modules, PhD thesis, UC Berkeley, May 2005 · Zbl 1158.11030 [3] Ghioca, D., The mordell – lang theorem for Drinfeld modules, Int. math. res. not., 53, 3273-3307, (2005) · Zbl 1158.11030 [4] Ghioca, D., The local Lehmer inequality for Drinfeld modules, J. number theory, 123, 2, 426-455, (2007) · Zbl 1173.11035 [5] Ghioca, D., Equidistribution for torsion points of a Drinfeld module, Math. ann., 336, 1, 841-865, (2006) · Zbl 1171.11038 [6] Goss, D., Basic structures of function field arithmetic, Ergeb. math. grenzgeb. (3), vol. 35, (1996), Springer Berlin · Zbl 0874.11004 [7] Mattuck, A., Abelian varieties over p-adic ground field, Ann. of math. (2), 62, 92-119, (1955) · Zbl 0066.02802 [8] Poonen, B., Local height functions and the mordell – weil theorem for Drinfeld modules, Compos. math., 97, 349-368, (1995) · Zbl 0839.11024 [9] Scanlon, T., p-adic distance from torsion points to semi-abelian varieties, J. reine angew. math., 499, 225-236, (1998) · Zbl 0932.11041 [10] Scanlon, T., The conjecture of Tate and voloch on p-adic proximity to torsion, Int. math. res. not., 17, 909-914, (1999) · Zbl 0986.11038 [11] Scanlon, T., Diophantine geometry of the torsion of a Drinfeld module, J. number theory, 97, 1, 10-25, (2002) · Zbl 1055.11037 [12] Serre, J.-P., Lectures on the mordell – weil theorem, Aspects math., vol. E15, (1989), Friedr. Vieweg & Sohn Braunschweig, x+218 pp. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt
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