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21
Equirépartition des petits points
, 1997
"... Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont mêm ..."
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Cited by 42 (4 self)
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Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont même équidistribués
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 ..."
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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.
Subvarieties of Semiabelian Varieties
, 1995
"... this paper can be summarized by the following theorem: ..."
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Cited by 14 (1 self)
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this paper can be summarized by the following theorem:
Toward A Proof Of The MordellLang Conjecture In Characteristic p
 in characteristic p, International Mathematics Research Notices
, 1995
"... This paper is concerned with an analogue in positive characteristic of the conjecture known as the Mordell  Lang conjecture. ..."
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Cited by 13 (0 self)
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This paper is concerned with an analogue in positive characteristic of the conjecture known as the Mordell  Lang conjecture.
The geometric Bogomolov conjecture for small genus curves
"... Abstract. The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 over a function field of characteristic zero. We recov ..."
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Abstract. The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved. 1.
Families of absolutely simple hyperelliptic jacobians
 Ellenberg Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53705 USA ellenber@math·wisc·edu Chris Hall Department of Mathematics University of Michigan Ann Arbor MI 48109 USA hallcj@umich·edu Christian Elsholtz Department of
"... Abstract. We prove that the jacobian of a hyperelliptic curve y 2 = (x−t)h(x) has no nontrivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t ∈ K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and ..."
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Abstract. We prove that the jacobian of a hyperelliptic curve y 2 = (x−t)h(x) has no nontrivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t ∈ K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number> 8. (The case of odd deg(h)> 3 follows easily from previous results of the author.) 1. Statements As usual, Z, Q and C stand for the ring of integers, the field of rational numbers and the field of complex numbers respectively. If ℓ is a prime then we write Fℓ, Zℓ and Qℓ for the ℓelement (finite) field, the ring of ℓadic integers and field of ℓadic numbers respectively. If A is a finite set then we write #(A) for the number of its elements. Let K be a field of characteristic different from 2, let ¯ K be its algebraic closure and Gal(K) = Aut ( ¯ K/K) its absolute Galois group. Let n ≥ 5 be an integer, f(x) ∈ K[x] a degree n polynomial without multiple roots, Rf ⊂ ¯ K the nelement
ORDINARY REDUCTION OF K3 SURFACES
, 2009
"... Let K be a number field and A an abelian variety of positive dimension over K. It is well known that A has good reduction at all but finitely many (nonarchimedean) places of K. It is natural to ask whether among those reductions there is ordinary one. In the most optimistic form the precise question ..."
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Let K be a number field and A an abelian variety of positive dimension over K. It is well known that A has good reduction at all but finitely many (nonarchimedean) places of K. It is natural to ask whether among those reductions there is ordinary one. In the most optimistic form the precise question sounds as follows. Is it true that there exists a finite algebraic field extension L/K and a density 1 set S of places of L such that A × L has ordinary good reduction at every place from S? The positive answer is known for elliptic curves (Serre [21]), abelian surfaces (Ogus [19]) and certain abelian fourfolds and threefolds [15, 16, 25]. One may ask a similar question for other classes of (smooth projective) algebraic varieties. The aim of this note is to settle this question for K3 surfaces. Recall that an (absolutely) irreducible smooth projective surface X over an algebraically closed field is called a K3 surface if the canonical sheaf Ω 2 X structure sheaf OX and H 1 (X, OX) = {0}. Our main result is the following statement. is isomorphic to the Theorem 0.1. Let X be a K3 surface that is defined over a number field K. Then there exists a finite algebraic field extension L/K and a density 1 set Σ(L, X) of (nonarchimedean) places of L such that X ×K L has ordinary good reduction at every place v ∈ Σ(L, X). Remark 0.2. The case of Kummer surfaces follows from the result of Ogus concerning the existence of ordinary reductions of abelian surfaces. When the endomorphism field E of X ×K C [28, Th. 1.6] is totally real (e.g., the Picard number is odd), the assertion of Theorem 0.1 was proven by Tankeev [25]. Acknowledgements. The first named author (F.B.) would like to thank the Clay Institute for financial support and Centre Di Giorgi in Pisa for its hospitality during the work on this paper. The second named author (Y.Z.) would like to thank Courant Institute of Mathematical Sciences for its hospitality during his several short visits in the years 2006–2009. After this paper had appeared on arXiv and was submitted, we received a letter from professor K. Joshi who brought to our attention his joint preprint with C.Rajan,
0 ENDOMORPHISMS OF ABELIAN VARIETIES, CYCLOTOMIC EXTENSIONS AND LIE ALGEBRAS
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