47 research outputs found

    Note on the rational points of a pfaff curve

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    Let X be the graph in the plane of a pfaffian function f (in the sense of Khovanskii). Suppose X is not algebraic. This note gives an upper bound for the number of rational points on X of height up to X. The bound is uniform in the order and degre of f.Comment: 8 page

    Counting points on curves over families in polynomial time

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    This note concerns the theoretical algorithmic problem of counting rational points on curves over finite fields. It explicates how the algorithmic scheme introduced by Schoof and generalized by the author yields an algorithm whose running time is uniformly polynomial time for curves in families.Comment: 7 page

    The Andre-Oort conjecture for the moduli space of Abelian Surfaces

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    We provide an unconditional proof of the Andr\'e-Oort conjecture for the coarse moduli space A2,1\mathcal{A}_{2,1} of principally polarized Abelian surfaces, following the strategy outlined by Pila-Zannier.Comment: 14 page

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

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    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.Comment: 12 page

    O-minimality and certain atypical intersections

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    We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results

    Ax-Schanuel for Shimura varieties

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    We prove the Ax-Schanuel theorem for a general (pure) Shimura variety