Work

Rational points in periodic analytic sets and the Manin–Mumford conjecture

Year: 2008

Type: article

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.

Source: Rendiconti Lincei Matematica e Applicazioni

DOI: 10.4171/rlm/514

Authors: Jonathan Pila, Umberto Zannier

Institutions: Scuola Normale Superiore, University of Bristol, University of Oxford

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: Scuola Normale Superiore, University of Bristol

OpenAlex ID: W2963123722

FWCI: 5.306

Cites: 27

Cited by: 116

Related to: 10