**Luca Demangos**^{[1]}

*A few remarks on a Manin-Mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates
*

**Pages:** 205-216

**Received:** 29 December 2015

**Accepted in revised form:** 17 February 2016

**Mathematics Subject Classification (2010):** 11G09, 14G22.

**Keywords:** \(T-\)modules, rational points, Manin-Mumford conjecture, local fields.

**Author address:**

[1] : Universidad Nacional Autonoma de Mexico, Av. Universidad S/N, C.P. 62210, Cuernavaca, Morelos, MEXICO

**Abstract:**
We present here the natural extension of our Pila-Wilkie type estimates on the number of rational points
of the trascendent part of a compact analytic subset of \(\mathbb{F}_{q}((1/T))^{n}\) (see [D1])
to analogous subsets of \(K^{n}\), where \(K\) is a general local field of any characteristic.
That would integrate the analogous estimate provided by F. Loeser, G. Comte and R. Cluckers in [CCL, Theorem 4.1.6].
We remind in the first two sections the main ideas of our construction by correcting two minor mistakes we made in [D1].
We then generalize the strategy to any local field.

**References**

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*Non-archimedean Yomdin-Gromov parametrizations and points of bounded height*, Forum Math. Pi 3 (2015), e5, 60 pp.
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[D1] L. Demangos,
*\(T-\)modules and Pila-Wilkie estimates*, J. Number Theory 154 (2015), 201-277.
MR3339573

[D2] L. Demangos,
*Some examples toward a Manin-Mumford conjecture for abelian uniformizable \(T-\)modules*,
Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 1, 171-190.
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