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

By Jonathan Pila, Umberto Zannier. 2008

174 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6ad6ffb8a2ae6349edf5f5d1a0bace3712c7dd85

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Citations in the corpus

Equidistribution in Families of Abelian Varieties and Uniformity (2021, unknown): “Note that the uniformity in the genus has not been achieved by any of the other methods proposed up to the present day [41, 72, 74, 87].”
Counting Lattice Points and O-Minimal Structures (2012, methodology): “These results in turn have been applied to problems in Diophantine geometry (see [24], [22], [19], [20] and [16]).”
Ax-Lindemann for $\mathcal{A}_g$ (2014, unknown): “Manin-Mumford conjecture [30].”
The hyperbolic Ax-Lindemann-Weierstraß conjecture (2015, unknown): “Recently Pila and Zannier [27] came up with a new proof of the Manin-Mumford conjecture for abelian varieties using the flat Ax-Lindemann-Weierstraß theorem.”
Some unlikely intersections beyond André–Oort (2011, methodology): “His approach relies on a counting result due to Wilkie and himself [38], and adopts a basic strategy proposed by Zannier in the context of the Manin-Mumford conjecture (see [39]); this strategy also plays a central role in the current article.”
On the algebraic points of a definable set (2009, background): “The present author and Zannier [ 17 ] give a new proof of the Manin{Mumford conjecture, and Masser and Zannier [9] prove a special case (but of a new kind) of the Zilber{Pink conjecture.”
Uniform bounds for the number of rational points on curves of small Mordell–Weil rank (2015, background): “Many additional proofs, with an assortment of techniques and generalizations, were given later by [19], [27], [34], [42], [53], and others.”
Torsion points on families of squares of elliptic curves (2012, background): “When A and V are defined over the field Q of all algebraic numbers, our proof follows the general strategy of [16,17] and [20].”
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$ (2011, methodology): “1 is a special case of the Manin-Mumford conjecture (MM) for subvarieties of abelian varieties and our proof is a variant of the one in [71] for abelian varieties over Q.”
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, methodology): “Theorem 1.2 was proved for (semi-)abelian varieties by Ax [6], and then refound and reproved by Pila-Zannier [52] and Peterzil-Starchenko [46], where proofs of Manin-Mumford via Ax-Lindemann-Weierstraß can be found.”
O-minimality and certain atypical intersections (2014, methodology): “Of course in this special case, our argument does not differ significantly from the Pila-Zannier approach [35].”
On the distribution of the Hodge locus (2021, background): “Starting from Pila and Zannier [50], a link between functional transcendence and the Zilber–Pink conjecture has been observed in [60, 31, 19], at least for subvarieties of abelian and pure Shimura varieties.”