Abstract
By using symbolic generators of the Brauer group of diagonal cubic surfaces, we compute the degree-zero part of the Chow group of zero-cycles on some diagonal cubic surfaces over \(p\)-adic fields. We also find that the Chow group of zero-cycles on them are generated by the classes of rational points.
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References
Dalawat, C.S.: Le groupe de chow d’une surface de chátelet sur un corps local. Indag. Math. (N.S) 11(2) (2000), 173–185
Pisolkar, S.: The Chow group of zero-cycles on certain Châtelet surfaces over local fields. Indag. Math. (N.S.) 19(3), 427–439 (2008)
Colliot-Thélène, J.L., Saito, S.: Zéro-cycles sur les variétés \(p\)-adiques groupe de Brauer. Int. Math. Res. Not. 4, 151–160 (1996)
Saito, S., Sato, K.: Zero-cycles on varieties over \(p\)-adic fields and Brauer groups. Ann. Sci. Éc. Norm. Supér. (4) 47(3), 505–537 (2014)
Manin, Y.I.: Cubic Forms: Algebra, Geometry, Arithmetic, vol. 4. North-Holland Mathematical Library. North-Holland, Amsterdam. Translated from Russian by Hazewinkel, M. (1986)
Uematsu, T.: On the Brauer group of diagonal cubic surfaces. Q. J. Math. 65(2), 677–701 (2014)
Uematsu, T.: Symbolic generators of the Brauer group of diagonal cubic surfaces and their applications to zero-cycles. RIMS Kokyuroku Bessatsu, Algebraic Number Theory and Related Topics (2012, to appear)
Colliot-Thélène, J.L., Kanevsky, D., Sansuc, J.J.: Arithmétique des surfaces cubiques diagonales. In: Diophantine approximation and transcendence theory (Bonn, 1985). Lecture Notes in Mathematics, vol. 1290, pp. 1–108. Springer, Berlin (1987)
Manin, Y.I.: Le groupe de Brauer–Grothendieck en géométrie diophantienne. In: Actes du Congrés International des Mathématiciens (Nice, 1970), Tome 1, pp. 401–411. Gauthier-Villars, Paris (1971)
Colliot-Thélène, J.L.: Hilbert’s theorem 90 for \(k_2\), with application to the Chow groups of rational surfaces. Invent. Math. 71, 1–20 (1983)
Bloch, S., Kato, K.: \(p\)-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. 63, 107–152 (1986)
Madore, D.A.: Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction. J. Number Theory 128(4), 926–944 (2008)
Acknowledgments
The author is deeply grateful to Professor Kanetomo Sato for suggesting to him problems of constructing zero-cycles and for discussing about the proof of Proposition 13. He would like to express his gratitude to the referee for pointing out many mistakes and giving valuable comments.
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This work was supported by MEXT Grant-in-Aid for the Global COE Program ‘The research and training center for new development in mathematics’ at Graduate School of Mathematical Sciences, the University of Tokyo, and by JSPS Grant-in-Aid for Scientific Research (B) Grant Number 23340003 (Kanetomo Sato).
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Uematsu, T. Zero-cycles on diagonal cubic surfaces over \(p\)-adic fields. Math. Z. 279, 1047–1066 (2015). https://doi.org/10.1007/s00209-014-1402-7
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DOI: https://doi.org/10.1007/s00209-014-1402-7