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A positive proportion of Hasse principle failures in a family of Châtelet surfaces. (English) Zbl 1435.14023

Summary: We investigate a family of Châtelet surfaces over \(\mathbb{Q}\) and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fails the Hasse principle, by building on previous work of R. de la Bretèche and T. D. Browning [Proc. Lond. Math. Soc. (3) 108, No. 4, 1030–1078 (2014; Zbl 1291.14041)].

MSC:

14G12 Hasse principle, weak and strong approximation, Brauer-Manin obstruction
14G05 Rational points
11G35 Varieties over global fields
11D25 Cubic and quartic Diophantine equations

Citations:

Zbl 1291.14041
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References:

[1] Browning, T. D. and Gorodnik, A., Power-free values of polynomials on symmetric varieties, Proc. Lond. Math. Soc.114 (2017) 1044-1080. · Zbl 1429.11181
[2] De La Bretèche, R. and Browning, T. D., Density of Châtelet surfaces failing the Hasse principle, Proc. Lond. Math. Soc.108 (2014) 1036-1078. · Zbl 1291.14041
[3] Colliot-Thélène, J.-L., Coray, D. and Sansuc, J.-J., Descent and the Hasse principle for certain rational varieties, J. Reine Angew. Math.320 (1980) 150-191. · Zbl 0434.14019
[4] J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces, I, II, J. Reine Angew. Math.373 (1987) 37-107; 374 (1987) 72-168. · Zbl 0622.14030
[5] Iskovskikh, V. A., A counterexample to the Hasse principle for systems of two quadratic forms in five variables, Mat. Z.10 (1971) 253-257. · Zbl 0221.10028
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