Rational points in periodic analytic sets and the Manin-Mumford conjecture
Number Theory
2008-02-28 v1 Algebraic Geometry
Abstract
We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation.
Cite
@article{arxiv.0802.4016,
title = {Rational points in periodic analytic sets and the Manin-Mumford conjecture},
author = {Jonathan Pila and Umberto Zannier},
journal= {arXiv preprint arXiv:0802.4016},
year = {2008}
}
Comments
12 pages