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Two torsion in the Brauer group of a hyperelliptic curve

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Abstract

We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C 1. In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.

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References

  1. Bruin N., Stoll M.: Two-cover descent on hyperelliptic curves. Math. Comput. 78(268), 2347–2370 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cassels J.W.S.: Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math. 211, 95–112 (1962)

    MATH  MathSciNet  Google Scholar 

  3. Chernousov, V., Guletskiĭ, V.: 2-torsion of the Brauer group of an elliptic curve: generators and relations. In: Proceedings of the Conference on Quadratic Forms and Related Topics (Baton Rouge, LA), pp. 85–120 (2001)

  4. Colliot-Thélène, J.-L.: Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale, Algebraic K-theory (Seattle, WA). In: Proceedings of the Symposium on Pure Mathematics, vol. 67, pp. 1–12. American Mathematical Society, Providence, RI (French). MR1743234 (2001d:11067) (1999)

  5. Colliot-Thélène, J.-L., Sansuc, J.-J.: La R-équivalence sur les tores. Ann. Sci. École Norm. Sup. (4) 10(2), 175–229 (1977) (French)

  6. Creutz B.: Locally trivial torsors that are not Weil–Châtelet divisible. Bull. Lond. Math. Soc. 45(5), 935–942 (2013). doi:10.1112/blms/bdt019

    Article  MATH  MathSciNet  Google Scholar 

  7. Creutz, B.: Explicit descent in the Picard group of a cyclic cover of the projective line. In: Howe, E.W., Kedlaya K.S. (eds.) ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, San Diego 2012. Open Book Series, vol. 1, pp. 295–315. Mathematical Science, Berkeley, California (2013)

  8. Creutz, B., Viray, B.: On Brauer groups of double covers of ruled surfaces. Math. Ann. (2014). doi:10.1007/s00208-014-1153-0

  9. Eriksson D., Scharaschkin V.: On the Brauer–Manin obstruction for zero-cycles on curves. Acta Arith. 135(2), 99–110 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber). Algebraic geometry 2000, Azumino (Hotaka). In: Advanced Studies in Pure Mathematics, vol. 36, pp. 153–183. Mathematical Society of Japan, Tokyo (2002)

  11. Gille, P., Szamuely, T.: Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics, vol. 101. Cambridge University Press, Cambridge (2006)

  12. Lind, C.-E.: Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, vol 1940, p. 97. Thesis, University of Uppsala (1940) (German)

  13. Manin, Y.I.: Le groupe de Brauer-Grothendieck en géométrie diophantienne. Actes du Congrès International des Mathématiciens (Nice, 1970), pp. 401–411. Gauthier-Villars, Paris (1971)

  14. Merkurjev, A.S.: On the norm residue symbol of degree 2.Dokl. Akad. Nauk SSSR, 261(3), 542–547 (1981) (Russian)

  15. Poonen B., Schaefer E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488, 141–188 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Poonen B., Stoll M.: The Cassels–Tate pairing on polarized abelian varieties. Ann. Math. 150(3), 1109–1149 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rehmann, U., Tikhonov, S.V.,Yanchevskii, V.I.: Two-torsion of the Brauer groups of hyperelliptic curves and unramified algebras over their function fields. Commun. Algebr. 29(9), 3971–3987 (2001). doi:10.1081/AGB-100105985 (special issue dedicated to Alexei Ivanovich Kostrikin)

  18. Reichardt, H.: Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184, 12–18 (1942)(German)

  19. Saito, S.: Some observations on motivic cohomology of arithmetic schemes. Invent. Math. 98(2), 371–404. MR1016270 (90k:11077) (1989)

  20. Schaefer Edward F.: 2-descent on the Jacobians of hyperelliptic curves. J. Number Theory 51(2), 219–232 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Skorobogatov, A.N.: Torsors and rational points. In: Cambridge Tracts in Mathematics, vol. 144. Cambridge University Press, Cambridge (2001)

  22. Stoll M.: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98(3), 245–277 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stoll M.: Finite descent obstructions and rational points on curves. Algebr. Number Theory 1(4), 349–391 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tate, J.: Duality theorems in Galois cohomology over number fields. In: Proceedings of the International Congr. Mathematicians (Stockholm, 1962), pp. 288–295. Inst. Mittag-Leffler, Djursholm (1963)

  25. Wittenberg, O.: Transcendental Brauer–Manin obstruction on a pencil of elliptic curves. In: Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, CA, 2002). Progr. Math., vol. 226, pp. 259–267. Birkhäuser Boston, Boston (2004)

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Correspondence to Bianca Viray.

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Bianca Viray was partially supported by NSF Grant DMS-1002933.

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Creutz, B., Viray, B. Two torsion in the Brauer group of a hyperelliptic curve. manuscripta math. 147, 139–167 (2015). https://doi.org/10.1007/s00229-014-0721-7

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  • DOI: https://doi.org/10.1007/s00229-014-0721-7

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