Prime torsion in the Brauer group of an elliptic curve

Ure, Charlotte

doi:10.1090/tran/9043

Prime torsion in the Brauer group of an elliptic curve
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by Charlotte Ure;

Trans. Amer. Math. Soc. 377 (2024), 2413-2437

DOI: https://doi.org/10.1090/tran/9043

Published electronically: February 14, 2024

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Abstract:

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of unity. These elements of the Brauer group are given as tensor products of symbol algebras over the function field of the elliptic curve. We give sufficient conditions to determine if the Brauer classes that arise are trivial. Using our algorithm, we derive an upper bound on the symbol length of the prime torsion of $Br(E)/Br(k)$.

References

Ettore Aldrovandi and Niranjan Ramachandran, Cup products, the Heisenberg group, and codimension two algebraic cycles, Doc. Math. 21 (2016), 1313–1344. MR 3578206, DOI 10.4171/dm/x3
M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95. MR 321934, DOI 10.1112/plms/s3-25.1.75
Andrea Bandini and Laura Paladino, Number fields generated by the 3-torsion points of an elliptic curve, Monatsh. Math. 168 (2012), no. 2, 157–181. MR 2984145, DOI 10.1007/s00605-012-0377-x
Jennifer Berg and Anthony Várilly-Alvarado, Odd order obstructions to the Hasse principle on general K3 surfaces, Math. Comp. 89 (2020), no. 323, 1395–1416. MR 4063322, DOI 10.1090/mcom/3485
Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
V. Chernousov and V. Guletskiĭ, 2-torsion of the Brauer group of an elliptic curve: generators and relations, Proceedings of the Conference on Quadratic Forms and Related Topics (Baton Rouge, LA, 2001), 2001, pp. 85–120. MR 1869390
Vladimir I. Chernousov, Andrei S. Rapinchuk, and Igor A. Rapinchuk, On the size of the genus of a division algebra, Tr. Mat. Inst. Steklova 292 (2016), no. Algebra, Geometriya i Teoriya Chisel, 69–99; English transl., Proc. Steklov Inst. Math. 292 (2016), no. 1, 63–93. MR 3628454, DOI 10.1134/S0371968516010052
Mirela Ciperiani and Daniel Krashen, Relative Brauer groups of genus 1 curves, Israel J. Math. 192 (2012), no. 2, 921–949. MR 3009747, DOI 10.1007/s11856-012-0057-5
Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 113–186. MR 2348904
Brendan Creutz and Bianca Viray, Two torsion in the Brauer group of a hyperelliptic curve, Manuscripta Math. 147 (2015), no. 1-2, 139–167. MR 3336942, DOI 10.1007/s00229-014-0721-7
Brendan Creutz, Bianca Viray, and José Felipe Voloch, The $d$-primary Brauer-Manin obstruction for curves, Res. Number Theory 4 (2018), no. 2, Paper No. 26, 16. MR 3807414, DOI 10.1007/s40993-018-0120-3
D. K. Faddeev, Simple algebras over a field of algebraic functions of one variable, Amer. Math. Soc. Transl. (2) 3 (1956), 15–38. MR 77505, DOI 10.1090/trans2/003/02
Ofer Gabber, A note on the unramified Brauer group and purity, Manuscripta Math. 95 (1998), no. 1, 107–115. MR 1492372, DOI 10.1007/BF02678018
Alexander Grothendieck, Le groupe de Brauer. III. Exemples et compléments, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 88–188 (French). MR 244271
V. I. Guletskiĭ, G. L. Margolin, and V. I. Yanchevskiĭ, Representation of $2$-torsion in Brauer groups of curves by quaternion algebras, Dokl. Akad. Nauk Belarusi 41 (1997), no. 6, 8–12, 122 (Russian, with English and Russian summaries). MR 1668943
V. I. Guletskiĭ and V. I. Yanchevskiĭ, Representation of torsion in Brauer groups of curves by cyclic algebras, Dokl. Nats. Akad. Nauk Belarusi 42 (1998), no. 2, 52–55, 124 (Russian, with English and Russian summaries). MR 1699137
Ilseop Han, Relative Brauer groups of function fields of curves of genus one, Comm. Algebra 31 (2003), no. 9, 4301–4328. MR 1995537, DOI 10.1081/AGB-120022794
Stephen Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math. 7 (1969), 120–136. MR 242831, DOI 10.1007/BF01389795
Y. I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars Éditeur, Paris, 1971, pp. 401–411. MR 427322
A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
Laura Paladino, Elliptic curves with $\Bbb Q({\scr E}[3])=\Bbb Q(\zeta _3)$ and counterexamples to local-global divisibility by 9, J. Théor. Nombres Bordeaux 22 (2010), no. 1, 139–160 (English, with English and French summaries). MR 2675877, DOI 10.5802/jtnb.708
Bjorn Poonen and Eric Rains, Self cup products and the theta characteristic torsor, Math. Res. Lett. 18 (2011), no. 6, 1305–1318. MR 2915483, DOI 10.4310/MRL.2011.v18.n6.a18
S. Pumplün, Quaternion algebras over elliptic curves, Comm. Algebra 26 (1998), no. 12, 4357–4373. MR 1661277, DOI 10.1080/00927879808826415
I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204, DOI 10.1093/oso/9780198526735.001.0001
Shmuel Rosset and John Tate, A reciprocity law for $K_{2}$-traces, Comment. Math. Helv. 58 (1983), no. 1, 38–47. MR 699005, DOI 10.1007/BF02564623
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
Alexei N. Skorobogatov, Beyond the Manin obstruction, Invent. Math. 135 (1999), no. 2, 399–424. MR 1666779, DOI 10.1007/s002220050291
Alexei Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. MR 1845760, DOI 10.1017/CBO9780511549588
Ure, C 2019. Prime torsion in the Brauer group of an elliptic curve, Ph.D. Thesis, Michigan State University.