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16
PERIODIC POINTS ON VEECH SURFACES AND THE MORDELLWEIL GROUP OVER A TEICHMÜLLER CURVE
, 2005
"... Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two ..."
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Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main tool is the Hodgetheoretic characterisation of Teichmüller curves. We deduce from it a finiteness result for the MordellWeil group of the family of Jacobians over a Teichmüller curve.
A Combination of the Conjectures by MordellLang and AndréOort
 In: Geometric Methods in Algebra and Number Theory (Bogomolov, F., Tschinkel, Y., Eds.) Progress in Math. 235
, 2005
"... Summary. We propose a conjecture combining the Mordell–Lang conjecture with an important special case of the André–Oort conjecture, and explain how existing results imply evidence for it. 1 ..."
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Summary. We propose a conjecture combining the Mordell–Lang conjecture with an important special case of the André–Oort conjecture, and explain how existing results imply evidence for it. 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 ..."
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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
Local AndréOort conjecture for the universal abelian variety
 Invent. Math
"... Abstract. We prove a padic analogue of the AndréOort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let g and n be integers with n ≥ 3 and p a prime], the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli ..."
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Abstract. We prove a padic analogue of the AndréOort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let g and n be integers with n ≥ 3 and p a prime], the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space A = Ag,1,n of gdimensional principally polarized abelian varieties with full level nstructure as well as the universal abelian variety π: X → A over A may be defined over R. We call a point ξ ∈ X(R) Rspecial if Xπ(ξ) is a canonical lift and ξ is a torsion point of its fibre. We show that an irreducible subvariety of XR containing a dense set of pspecial points must be a special subvariety in the sense of mixed Shimura varieties. Our proof employs the model theory of difference fields. number not dividing n. Let R be a finite extension of W[F alg p 1.
On the ManinMumford conjecture
 Proc. NATOASI Conf. Model Theory of Fields (Fields Institute
, 2003
"... We present an elementary algebraic proof of the ManinMumford conjecture, concerning the the intersection of a subvariety X of a semiabelian variety A with the group of torsion points of A, when all data are defined over a number field. The proof is inspired by Hrushovski's work [2], uses i ..."
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We present an elementary algebraic proof of the ManinMumford conjecture, concerning the the intersection of a subvariety X of a semiabelian variety A with the group of torsion points of A, when all data are defined over a number field. The proof is inspired by Hrushovski's work [2], uses ideas originating in work of the author and Martin Ziegler [5], and is closely related to recent papers of Pink and Roessler [6],[7].
On algebraic σgroups
, 2006
"... We introduce the categories of algebraic σvarieties and σgroups over a difference field (K, σ). Under a “linearly σclosed” assumption on (K, σ) we prove an isotriviality theorem for σgroups. This theorem yields immediately the key lemma in a proof of the ManinMumford conjecture. The present pap ..."
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We introduce the categories of algebraic σvarieties and σgroups over a difference field (K, σ). Under a “linearly σclosed” assumption on (K, σ) we prove an isotriviality theorem for σgroups. This theorem yields immediately the key lemma in a proof of the ManinMumford conjecture. The present paper uses crucially ideas from [10] but in a model theory free manner. The applications to ManinMumford are inspired by Hrushovski’s work [5] and are also closely related to papers of Pink and Roessler ([11] and [12]).
Conjecture of ManinMumford.
"... 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 ..."
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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.
Diophantine Geometry”, Arizona Winter School, 2003
, 2003
"... These notes are for the first two lectures of the lecture series ”Model Theory and Diophantine Geometry ” by Thomas Scanlon and myself. The full series of lectures will be on the ManinMumford conjecture and variants. As is wellknown, modeltheoretic ideas, specifically definability theory in diffe ..."
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These notes are for the first two lectures of the lecture series ”Model Theory and Diophantine Geometry ” by Thomas Scanlon and myself. The full series of lectures will be on the ManinMumford conjecture and variants. As is wellknown, modeltheoretic ideas, specifically definability theory in difference
3. Lecture: A combination of the MordellLang and the AndréOort conjecture.
"... conjecture in positive characteristic and related spaces. ..."
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