Abstract
We study rational points on conic bundles over elliptic curves with positive rank over a number field. We show that the étale Brauer–Manin obstruction is insufficient to explain failures of the Hasse principle for such varieties. We then further consider properties of the distribution of the set of rational points with respect to its image in the rational points of the elliptic curve. In the process, we prove results on a local-to-global principle for torsion points on elliptic curves over \({{{\mathbb {Q}}}}\).
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Acknowledgements
We thank J.-L. Colliot-Thélène, Dan Loughran and Tony Várilly-Alvarado for fruitful discussions and their comments on a preliminary draft of this paper. We thank the anonymous reviewer for their suggestions on both mathematical content and exposition. Part of the work was done at the Institut Henri Poincaré, we thank the IHP for their hospitality.
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Berg, J., Nakahara, M. Rational points on conic bundles over elliptic curves. Math. Z. 300, 2429–2449 (2022). https://doi.org/10.1007/s00209-021-02870-z
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DOI: https://doi.org/10.1007/s00209-021-02870-z