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Non-existence of points rational over number fields on Shimura curves

Volume 172 / 2016

Keisuke Arai Acta Arithmetica 172 (2016), 243-250 MSC: Primary 11G18, 14G05; Secondary 11G10, 11G15. DOI: 10.4064/aa8071-10-2015 Published online: 20 January 2016

Abstract

Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we extend their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.

Authors

  • Keisuke AraiDepartment of Mathematics
    School of Science and Technology for Future Life
    Tokyo Denki University
    5 Senju Asahi-cho, Adachi-ku
    Tokyo 120-8551, Japan
    e-mail

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