320 research outputs found

    Modularity of Calabi-Yau varieties

    Full text link
    In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror symmetry. Finally, we address some questions and open problems.Comment: Further references adde

    Model theory of special subvarieties and Schanuel-type conjectures

    Full text link
    We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure)

    On the distribution of rational points on ramified covers of abelian varieties

    Get PDF
    We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields kk of characteristic zero. For example, given a ramified cover π:X→A\pi : X \to A, where AA is an abelian variety over kk with a dense set of kk-rational points, we prove that there is a finite-index coset C⊂A(k)C \subset A(k) such that π(X(k))\pi(X(k)) is disjoint from CC. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the Inverse Galois Problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.Comment: 38 pages. Title changed. Introduction and abstract improved. No other changes. Comments more than welcome
    • …
    corecore