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Sur certains aspects géométriques et arithmétiques des variétés de Shimura orthogonales

Abstract : This thesis deals with some arithmetical and geometrical aspects of orthogonal Shimura varieties. These varieties appear naturally as moduli spaces of Hodge structures of K3 type. In some cases, they parametrize geometric objects as K3 surfaces and their analogous in higher dimensions, the hyperkähler varieties. This modular point of view will be our guiding principle throughout this dissertation. In the first part, we prove an equidistribution result of the Hodge locus in variations of Hodge structures of K3 type above complex quasi-projective curves. In the second part, we study analogous results in the arithemtic setting. An example of statements we get is the following: given a K3 surface having everywhere good reduction and satisfying an approximation hypothesis, there exists a specialization with strictly increasing geometric Picard rank. In both cases, our methods take advantage of the rich arithmetic, automorphic and geometric structure of orthogonal Shimura varieties as well as the Kuga-Satake construction that links them to moduli spaces of abelian varieties. Finally, we extend a result of Bogomolov and Tschinkel. In particular, we show that any K3 surface defined over an algebraically closed field of arbitrary characteristic and admitting a non-isotrivial elliptic fibration contains infinitely many rational curves.
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Submitted on : Wednesday, July 3, 2019 - 3:58:08 PM
Last modification on : Wednesday, September 16, 2020 - 5:32:35 PM


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Salim Tayou. Sur certains aspects géométriques et arithmétiques des variétés de Shimura orthogonales. Géométrie algébrique [math.AG]. Université Paris-Saclay, 2019. Français. ⟨NNT : 2019SACLS144⟩. ⟨tel-02172234⟩



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