Algebra & Number Theory

Degree and the Brauer–Manin obstruction

Brendan Creutz and Bianca Viray

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Let Xkn be a smooth projective variety of degree d over a number field k and suppose that X is a counterexample to the Hasse principle explained by the Brauer–Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate–Shafarevich groups) bielliptic surfaces. In the case of Kummer surfaces we show, more specifically, that the obstruction is already given by the 2-primary torsion, and indeed that this holds for higher-dimensional Kummer varieties as well. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.

Article information

Algebra Number Theory, Volume 12, Number 10 (2018), 2445-2470.

Received: 19 December 2017
Revised: 12 July 2018
Accepted: 23 August 2018
First available in Project Euclid: 14 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G05: Rational points
Secondary: 11G35: Varieties over global fields [See also 14G25] 14F22: Brauer groups of schemes [See also 12G05, 16K50]

Brauer–Manin obstruction degree period rational points


Creutz, Brendan; Viray, Bianca. Degree and the Brauer–Manin obstruction. Algebra Number Theory 12 (2018), no. 10, 2445--2470. doi:10.2140/ant.2018.12.2445.

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  • D. Abramovich, K. Karu, K. Matsuki, and J. a. Włodarczyk, “Torification and factorization of birational maps”, J. Amer. Math. Soc. 15:3 (2002), 531–572.
  • C. L. Basile and A. Skorobogatov, “On the Hasse principle for bielliptic surfaces”, pp. 31–40 in Number theory and algebraic geometry, edited by M. Reid and A. Skorobogatov, London Math. Soc. Lecture Note Ser. 303, Cambridge Univ. Press, 2003.
  • A. Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts 34, Cambridge University Press, 1996.
  • V. G. Berkovič, “The Brauer group of abelian varieties”, Funkcional Anal. i Priložen. 6:3 (1972), 10–15. In Russian; translated in Functional Anal. Appl. 6 (1972), 180–184.
  • J. W. S. Cassels, “Arithmetic on curves of genus $1$, IV: Proof of the Hauptvermutung”, J. Reine Angew. Math. 211 (1962), 95–112.
  • P. L. Clark and S. Sharif, “Period, index and potential, III”, Algebra Number Theory 4:2 (2010), 151–174.
  • J.-L. Colliot-Thélène, “L'arithmétique du groupe de Chow des zéro-cycles”, J. Théor. Nombres Bordeaux 7:1 (1995), 51–73.
  • J.-L. Colliot-Thélène, “Conjectures de type local-global sur l'image des groupes de Chow dans la cohomologie étale”, pp. 1–12 in Algebraic $K$-theory (Seattle, 1997), edited by W. Raskind and C. Weibel, Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, “On the Chow groups of certain rational surfaces: a sequel to a paper of S. Bloch”, Duke Math. J. 48:2 (1981), 421–447.
  • J.-L. Colliot-Thélène, A. Pál, and A. N. Skorobogatov, “Pathologies of the Brauer–Manin obstruction”, Math. Z. 282:3-4 (2016), 799–817.
  • P. Corn, “The Brauer–Manin obstruction on del Pezzo surfaces of degree 2”, Proc. Lond. Math. Soc. $(3)$ 95:3 (2007), 735–777.
  • P. Corn and M. Nakahara, “Brauer–Manin obstructions on genus-2 K3 surfaces”, 2017.
  • B. Creutz, B. Viray, and J. F. Voloch, “The $d$-primary Brauer–Manin obstruction for curves”, Res. Number Theory 4:2 (2018), Art. 26, 16.
  • A. Grothendieck, “Le groupe de Brauer, III: Exemples et compléments”, pp. 88–188 in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968.
  • J. van Hamel, “The Brauer–Manin obstruction for zero-cycles on Severi-Brauer fibrations over curves”, J. London Math. Soc. $(2)$ 68:2 (2003), 317–337.
  • D. Harari and T. Szamuely, “Local-global principles for 1-motives”, Duke Math. J. 143:3 (2008), 531–557.
  • Y. Harpaz and A. N. Skorobogatov, “Hasse principle for Kummer varieties”, Algebra Number Theory 10:4 (2016), 813–841.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977.
  • B. Hassett, “Rational surfaces over nonclosed fields”, pp. 155–209 in Arithmetic geometry, edited by H. Darmon et al., Clay Math. Proc. 8, Amer. Math. Soc., Providence, RI, 2009.
  • E. Ieronymou and A. N. Skorobogatov, “Odd order Brauer–Manin obstruction on diagonal quartic surfaces”, Adv. Math. 270 (2015), 181–205.
  • E. Ieronymou and A. N. Skorobogatov, “Corrigendum to “Odd order Brauer–Manin obstruction on diagonal quartic surfaces””, Adv. Math. 307 (2017), 1372–1377.
  • E. Ieronymou, A. N. Skorobogatov, and Y. G. Zarhin, “On the Brauer group of diagonal quartic surfaces”, J. Lond. Math. Soc. $(2)$ 83:3 (2011), 659–672.
  • K. Kato and S. Saito, “Global class field theory of arithmetic schemes”, pp. 255–331 in Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colorado, 1983), edited by S. J. Bloch et al., Contemp. Math. 55, Amer. Math. Soc., Providence, RI, 1986.
  • S. Lang, “Some applications of the local uniformization theorem”, Amer. J. Math. 76 (1954), 362–374.
  • S. Lang and J. Tate, “Principal homogeneous spaces over abelian varieties”, Amer. J. Math. 80 (1958), 659–684.
  • R. Lazarsfeld, Positivity in algebraic geometry, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 49, Springer, 2004.
  • T. LMFDB Collaboration, “The L-functions and modular forms database”, 2013,
  • Y. I. Manin, “Le groupe de Brauer–Grothendieck en géométrie diophantienne”, pp. 401–411 in Actes du Congrès International des Mathématiciens, Tome 1 (Nice, 1970), Gauthier-Villars, Paris, 1971.
  • M. Nakahara, “Index of fibrations and Brauer classes that never obstruct the Hasse principle”, 2017.
  • H. Nishimura, “Some remarks on rational points”, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 29 (1955), 189–192.
  • B. Poonen, “Existence of rational points on smooth projective varieties”, J. Eur. Math. Soc. $($JEMS$)$ 11:3 (2009), 529–543.
  • J.-J. Sansuc, “Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres”, J. Reine Angew. Math. 327 (1981), 12–80.
  • F. Serrano, “Divisors of bielliptic surfaces and embeddings in ${\bf P}^4$”, Math. Z. 203:3 (1990), 527–533.
  • F. Serrano, “Isotrivial fibred surfaces”, Ann. Mat. Pura Appl. $(4)$ 171 (1996), 63–81.
  • A. N. Skorobogatov, “Beyond the Manin obstruction”, Invent. Math. 135:2 (1999), 399–424.
  • A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics 144, Cambridge University Press, 2001.
  • A. Skorobogatov and P. Swinnerton-Dyer, “2-descent on elliptic curves and rational points on certain Kummer surfaces”, Adv. Math. 198:2 (2005), 448–483.
  • A. N. Skorobogatov and Y. G. Zarhin, “A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces”, J. Algebraic Geom. 17:3 (2008), 481–502.
  • A. N. Skorobogatov and Y. G. Zarhin, “Kummer varieties and their Brauer groups”, Pure Appl. Math. Q. 13:2 (2017), 337–368.
  • P. Swinnerton-Dyer, “The Brauer group of cubic surfaces”, Math. Proc. Cambridge Philos. Soc. 113:3 (1993), 449–460.
  • A. Várilly-Alvarado, “Arithmetic of del Pezzo surfaces”, pp. 293–319 in Birational geometry, rational curves, and arithmetic, edited by F. Bogomolov et al., Springer, 2013.
  • O. Wittenberg, “Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque”, Duke Math. J. 161:11 (2012), 2113–2166.