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A threefold violating a local-to-global principle for rationality

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Abstract

In this note we construct an example of a smooth projective threefold that is irrational over \(\mathbb {Q}\) but is rational at all places. Our example is a complete intersection of two quadrics in \(\mathbb P^5\), and we show it has the desired rationality behavior by constructing an explicit element of order \(4\) in the Tate–Shafarevich group of the Jacobian of an associated genus \(2\) curve.

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Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Code availability

The Magma code accompanying this paper is available in our Github repository https://github.com/lena-ji/local-global-2quadrics.

Notes

  1. \(\textrm{Div}^i(C)(k)\) denotes the set of \(k\)-rational divisors of degree \(i\). \({{\,\mathrm{{{\textbf {Pic}}}}\,}}^i_C\) denotes the degree \(i\) component of the Picard scheme, and its \(k\)-points are the \(k\)-rational divisor classes of degree \(i\).

  2. We work with this particular affine patch because, on many other patches, \(F_1(X)\) has no smooth \(\mathbb {F}_2\)-points. In practice, the most difficult condition to verify is the existence of \(\mathbb Q_2\)-points on \(F_1(X)\).

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Acknowledgements

We are grateful to Anthony Várilly-Alvarado for computing advice, to Tom Fisher for suggesting a simplifying change of coordinates, and to Tom Fisher and Jiali Yan for correspondence about [5, 26]. We also thank Eran Assaf, Asher Auel, Nils Bruin, Sebastian Casalaina-Martin, Brendan Creutz, Sam Frengley, Jack Petok, and Bianca Viray for helpful conversations. The computations in Sect. 3 were done in Magma [31].

Funding

L.J. was supported by the National Science Foundation under MSPRF Grant DMS-2202444.

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Author notes

  1. Sarah Frei and Lena Ji have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Dartmouth College, 27 N. Main Street, Hanover, 03755, NH, USA

    Sarah Frei

  2. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, 48109, MI, USA

    Lena Ji

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Frei, S., Ji, L. A threefold violating a local-to-global principle for rationality. Res. number theory 10, 39 (2024). https://doi.org/10.1007/s40993-024-00515-8

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