Results 1  10
of
74
Arithmetic height functions over finitely generated fields
 Inventiones Mathematicae 140
, 2000
"... ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (ManinMumford’s conjecture). CONTENTS ..."
Abstract

Cited by 37 (10 self)
 Add to MetaCart
(Show Context)
ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (ManinMumford’s conjecture). CONTENTS
Intersections of polynomial orbits, and a dynamical MordellLang conjecture
 INVENT. MATH
, 2007
"... We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the MordellLang conjecture. ..."
Abstract

Cited by 29 (12 self)
 Add to MetaCart
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the MordellLang conjecture.
Independence of rational points on twists of a given curve, to appear
 in Compositio Math. arXiv: math.NT/0603557 School of Engineering and Science, International University Bremen, P.O.Box 750561, 28725
"... Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly g ..."
Abstract

Cited by 28 (13 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′ (K) and c is a constant depending on C. For the proof, we use a refinement of the method of ChabautyColeman; the main new ingredient is to use it for an extension field of Kv, where v is a place of bad reduction for C ′. 1.
Rational points in periodic analytic sets and the ManinMumford conjecture
 Atti Accad. Naz. Lincei
, 2008
"... Abstract. We present a new proof of the ManinMumford 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 ra ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We present a new proof of the ManinMumford 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 (BombieriPilaWilkie) 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 semialgebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation. The socalled ManinMumford 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 groupstructure). Raynaud actually considered the more general case when C is
Ominimality and the André–Oort conjecture for Cn
 Annals of Mathematics. Second Series
, 2011
"... We give an unconditional proof of the AndréOort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the ManinMumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang’s conjecture for torsion points in power ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
(Show Context)
We give an unconditional proof of the AndréOort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the ManinMumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang’s conjecture for torsion points in powers of the multiplicative group. The second includes the ManinMumford conjecture for abelian varieties defined over Q. Our approach uses the theory of ominimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the ManinMumford conjecture by PilaZannier; a proof of a special (but new) case of Pink’s relative ManinMumford conjecture by MasserZannier; and new proofs of certain known results of AndréOortManinMumford type by Pila. 1.
Towards a dynamical ManinMumford conjecture
 Int. Math. Res. Not
"... Abstract. We provide a family of counterexamples to a first formulation of the dynamical ManinMumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of abelian varieties under the action of endomorphisms of abelian varieties and for lines under the act ..."
Abstract

Cited by 17 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We provide a family of counterexamples to a first formulation of the dynamical ManinMumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of abelian varieties under the action of endomorphisms of abelian varieties and for lines under the action of diagonal endomorphisms of P 1 × P 1 .
On Hrushovski’s proof of the ManinMumford conjecture
 in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 539– 546, Higher Ed. Press, Beijing, 2002. TETSUSHI ITO
"... The ManinMumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
(Show Context)
The ManinMumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the ManinMumford conjecture using only classical algebraic geometry.
The ManinMumford Conjecture: A Brief Survey
 Bull. London Math. Soc
, 1999
"... This is a survey paper on the ManinMumford conjecture for number fields with some emphasis on effectivity. It is based on the author's lecture at the Arizona Winter School on Arithmetical Algebraic Geometry (March 1999). We discuss some of the history of this conjecture (and of related conject ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
This is a survey paper on the ManinMumford conjecture for number fields with some emphasis on effectivity. It is based on the author's lecture at the Arizona Winter School on Arithmetical Algebraic Geometry (March 1999). We discuss some of the history of this conjecture (and of related conjectures) and some recent explicit results.
DYNAMICS OF PROJECTIVE MORPHISMS HAVING IDENTICAL CANONICAL HEIGHTS
, 2007
"... Let ϕ, ψ: P N → P N be morphisms of degree at least 2 whose canonical heights ˆ hϕ and ˆ hψ are identical. We draw various conclusions about the Green functions, Julia sets, and canonical local heights of ϕ and ψ. We use this information to completely characterize ϕ and ψ in the following cases: (i) ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Let ϕ, ψ: P N → P N be morphisms of degree at least 2 whose canonical heights ˆ hϕ and ˆ hψ are identical. We draw various conclusions about the Green functions, Julia sets, and canonical local heights of ϕ and ψ. We use this information to completely characterize ϕ and ψ in the following cases: (i) ϕ and ψ are polynomial maps in one variable; (ii) ϕ is the dthpower map; (iii) ϕ is a Lattès map.
Common divisors of elliptic divisibility sequences over function fields
, 2004
"... Abstract. Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR = AR /D2 R with relatively prime polynomials AR(T), DR(T) ∈ k[T]. The sequence {DnR} n≥1 is called the elliptic divisibility se ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R ∈ E(k(T)), write xR = AR /D2 R with relatively prime polynomials AR(T), DR(T) ∈ k[T]. The sequence {DnR} n≥1 is called the elliptic divisibility sequence of R. Let P, Q ∈ E(k(T)) be independent points. We conjecture that and that deg ( gcd(DnP,DmQ) ) is bounded for m, n ≥ 1, gcd(DnP,DnQ) = gcd(DP,DQ) for infinitely many n ≥ 1. We prove these conjectures in the case that j(E) ∈ k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E) ∈ k, we show that deg ( gcd(DnP,DnQ)) is as large as n + O ( √ n) for infinitely many n ̸ ≡ 0 (mod p).