When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications, emphasizing in particular the case of codimension two. We also state some related conjectures about the boundedness of absolute height on such intersections as well as their finiteness.

Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed elsewhere.

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. §1. IntroductionThe so-called Manin-Mumford conjecture was raised independently by Manin and Mumford and first proved by Raynaud [R] in 1983; its original form stated that a curve C (over C) of genus ≥ 2, embedded in its Jacobian J, can contain only finitely many torsion points (relative of course to the Jacobian group-structure). Raynaud actually considered the more general case when C is embedded in any abelian variety. Soon afterwards, Raynaud [R2] produced a further significant generalization, replacing C and J respectively by a subvariety X in an abelian variety A; in this situation he proved that if X contains a Zariski-dense set of torsion points, then X is a translate of an abelian subvariety of A by a torsion point. Other proofs (sometimes only for the case of curves) later appeared, due to Serre, to Coleman, to Hindry, to Buium, to Hrushovski (see [Py]), to Pink & Roessler [PR], to M. Baker & Ribet [BR]. We also remark that a less deep precedent of this problem was an analogous question for multiplicative algebraic groups, raised by Lang already in the '60s. (See [L]; Lang started the matter by asking to describe the plane curves f (x, y) = 0 with infinitely many points (ζ, η), with ζ, η roots of unity.)In the meantime, the statement was put into a broader perspective, by viewing it as a special case of the general Mordell-Lang conjecture and also, from another viewpoint, of the Bogomolov conjecture on points of small canonical height on (semi)abelian varieties (we recall that torsion points are those of zero height). These conjectures have later been proved and unified (by Faltings, Vojta, Ullmo, Szpiro, Zhang, Poonen, David, Philippon,...) by means of different approaches providing, as a byproduct, several further proofs of the Manin-Mumford statement (we refer to the survey papers [Py] and [T] for a history of the topic and for references.) Recent work of Klingler, Ullmo and Yafaev proving (under GRH) the André-Oort conjecture, an analogue of the ManinMumford conjecture for Shimura varieties, has inspired another proof of Manin-Mumford due to Ratazzi & Ullmo [RU]. All of these approaches are rather sophisticated and depend on tools of various nature.It is the purpose of this paper to present a completely different proof compared to the existing ones. Our approach too relies on certain auxiliary results, having however anot...

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Abstract. -Let A be a commutative algebraic group defined over a number field k. We consider the following question: Let r be a positive integer and let P ∈ A(k). Suppose that for all but a finite number of primes v of k, we have P = rDv for some Dv ∈ A(kv). Can one conclude that there exists D ∈ A(k) such that P = rD? A complete answer for the case of the multiplicative group m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and A is either an elliptic curve or a torus of small dimension with respect to r. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime. Résumé (Divisibilité locale-globale des points rationnels en certains groupes algé-briques commutatifs)Pour un groupe algébrique commutatif A, défini sur un corps de nombres k, on se pose la question suivante :étant donnés un entier r strictement positif et unélément