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
[CCL] R. Cluckers, G. Comte and F. Loeser,
Non-archimedean Yomdin-Gromov parametrizations and points of bounded height, Forum Math. Pi 3 (2015), e5, 60 pp.
MR3406825
[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.
Article
[Goss] D. Goss,
Basic structures of Function Field Arithmetic, Springer-Verlag, Berlin 1996.
MR1423131
[PW] J. Pila and A. J. Wilkie,
The rational points of a definable set,
Duke Math. J. 133 (2006), no. 3, 591-616.
MR2228464
[PZ] J. Pila and U. Zannier,
Rational points in periodic analytic sets and the Manin-Mumford conjecture,
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), no. 2, 149-162.
MR2411018
[Yu] J. Yu,
Analytic homomorphisms into Drinfeld modules,
Ann. of Math. (2) 145 (1997), 215-233.
MR1441876