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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.
Cite this article
Daniel Loughran, The number of varieties in a family which contain a rational point. J. Eur. Math. Soc. 20 (2018), no. 10, pp. 2539–2588