Home Arithmetic of Châtelet surface bundles revisited
Article
Licensed
Unlicensed Requires Authentication

Arithmetic of Châtelet surface bundles revisited

  • Guang Hu and Yongqi Liang ORCID logo EMAIL logo
Published/Copyright: June 1, 2023

Abstract

We study the arithmetic of algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate–Shafarevich group of a certain elliptic curve, we show, for Châtelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer–Manin obstruction is not invariant under an arbitrary finite extension of the ground field.


Communicated by Jan Bruinier


Award Identifier / Grant number: 12071448

Funding statement: Both authors are partially supported by the National Natural Science Foundation of China (Grant No. 12071448) and Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302902).

Acknowledgements

The authors would like to thank the referee for his suggestions on the organization of the paper.

References

[1] F. Balestrieri and R. Newton, Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces, Int. Math. Res. Not. IMRN 2021 (2021), no. 6, 4255–4279. 10.1093/imrn/rny303Search in Google Scholar

[2] Y. Cao and Y. Liang, Étale Brauer–Manin obstruction for Weil restrictions, Adv. Math. 410 (2022), Article ID 108718. 10.1016/j.aim.2022.108718Search in Google Scholar

[3] J.-L. Colliot-Thélène, Zéro-cycles de degré 1 sur les solides de Poonen, Bull. Soc. Math. France 138 (2010), no. 2, 249–257. 10.24033/bsmf.2590Search in Google Scholar

[4] J.-L. Colliot-Thélène, A. Pál and A. N. Skorobogatov, Pathologies of the Brauer–Manin obstruction, Math. Z. 282 (2016), no. 3–4, 799–817. 10.1007/s00209-015-1565-xSearch in Google Scholar

[5] J.-L. Colliot-Thélène and B. Poonen, Algebraic families of nonzero elements of Shafarevich–Tate groups, J. Amer. Math. Soc. 13 (2000), no. 1, 83–99. 10.1090/S0894-0347-99-00315-XSearch in Google Scholar

[6] J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492. 10.1215/S0012-7094-87-05420-2Search in Google Scholar

[7] J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces. I, J. Reine Angew. Math. 373 (1987), 37–107. 10.1515/crll.1987.373.37Search in Google Scholar

[8] J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces. II, J. Reine Angew. Math. 374 (1987), 72–168. 10.1515/crll.1987.374.72Search in Google Scholar

[9] J.-L. Colliot-Thélène and A. N. Skorobogatov, The Brauer–Grothendieck Group, Ergeb. Math. Grenzgeb. (3) 71, Springer, Cham, 2021. 10.1007/978-3-030-74248-5Search in Google Scholar

[10] B. Creutz and B. Viray, Quadratic points on intersections of two quadrics, preprint (2021), https://arxiv.org/abs/2106.08560. Search in Google Scholar

[11] Y. Liang, Non-invariance of weak approximation properties under extension of the ground field, preprint (2018), https://arxiv.org/abs/1805.08851. Search in Google Scholar

[12] B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575. 10.1007/s00222-010-0252-0Search in Google Scholar

[13] B. Poonen, Existence of rational points on smooth projective varieties, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 529–543. 10.4171/JEMS/159Search in Google Scholar

[14] B. Poonen, Insufficiency of the Brauer–Manin obstruction applied to étale covers, Ann. of Math. (2) 171 (2010), no. 3, 2157–2169. 10.4007/annals.2010.171.2157Search in Google Scholar

[15] B. Poonen, Rational Points on Varieties, Grad. Stud. Math. 186, American Mathematical Society, Providence, 2017. 10.1090/gsm/186Search in Google Scholar

[16] M. Riman, The Brauer–Manin obstruction of del Pezzo surfaces of degree 4, preprint. Search in Google Scholar

[17] C. Rivera and B. Viray, Persistence of the Brauer–Manin obstruction on cubic surfaces, preprint (2021), https://arxiv.org/abs/2111.03546. Search in Google Scholar

[18] J.-P. Serre, Corps locaux, 4th ed., Hermann, Paris, 1968. Search in Google Scholar

[19] A. Skorobogatov, Torsors and Rational Points, Cambridge Tracts in Math. 144, Cambridge University, Cambridge, 2001. 10.1017/CBO9780511549588Search in Google Scholar

[20] M. Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), 349–391. 10.2140/ant.2007.1.349Search in Google Scholar

[21] H. Wu, Châtelet surfaces and non-invariance of the Brauer–Manin obstruction for 3-folds, preprint (2020), https://arxiv.org/abs/2010.04919. Search in Google Scholar

[22] H. Wu, Non-invariance of the Brauer–Manin obstruction for surfaces, preprint (2021), https://arxiv.org/abs/2103.01784. Search in Google Scholar

[23] LMFDB, The L-functions and modular forms database, http://www.lmfdb.org, accessed: 2021-09-29. Search in Google Scholar

Received: 2022-02-08
Revised: 2023-04-04
Published Online: 2023-06-01
Published in Print: 2023-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 15.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0049/html
Scroll to top button