The number of varieties in a family which contain a rational point

Daniel Loughran

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    Abstract

    We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer-Severi varieties and closely related counting functions associated to Brauer group elements. Using harmonic analysis on toric varieties, we provide a positive answer to a question of Serre on such counting functions in some cases. We also formulate some conjectures on generalisations of Serre's problem.
    Original languageEnglish
    Pages (from-to)2539-2588
    Number of pages50
    JournalJournal of the European Mathematical Society
    Volume20
    Issue number10
    Early online date20 Jul 2018
    DOIs
    Publication statusPublished - 2018

    Keywords

    • Brauer groups
    • Families of varieties
    • Rational points
    • Toric varieties

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