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COMPARING THE BRAUER GROUP TO THE TATE–SHAFAREVICH GROUP

  • Thomas H. Geisser (a1)
Abstract

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate–Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.

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1. Demarche, C. and Harari, D., Artin–Mazur–Milne duality Theorem for fppf cohomology, Preprint, 2018, https://arxiv.org/abs/1804.03941.
2. González-Avilés, C. D., Brauer groups and Tate–Shafarevich groups, J. Math. Sci. Univ. Tokyo 10(2) (2003), 391419.
3. González-Avilés, C. D. and Tan, K.-S., A generalization of the Cassels–Tate dual exact sequence, Math. Res. Lett. 14(2) (2007), 295302.
4. Gordon, W., Linking the conjectures of Artin–Tate and Birch–Swinnerton-Dyer, Compos. Math. 38 (1979), 163199.
5. Grothendieck, A., Le groupe de Brauer. III. Exemples et compléments, in Dix exposés sur la cohomologie des schémas, Advanced Studies in Pure Mathematics, 3, pp. 88188 (North-Holland, Amsterdam, 1968).
6. Lichtenbaum, S., Duality theorems for curves over p-adic fields, Invent. Math. 7 (1969), 120136.
7. Liu, Q., Lorenzini, D. and Raynaud, M., Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math. 157(3) (2004), 455518.
8. Liu, Q., Lorenzini, D. and Raynaud, M., On the Brauer group of a surface, Invent. Math. 159(3) (2005), 673676.
9. Liu, Q., Lorenzini, D. and Raynaud, M., Corrigendum to [8], Invent. Math. to appear. https://arxiv.org/abs/1804.11158.
10. Milne, J. S., Comparison of the Brauer group with the Tate–Safarevic group, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3) (1981), 735743.
11. Milne, J. S., Arithmetic Duality Theorems, 2nd edn (BookSurge, LLC, Charleston, SC, 2006). viii+339 pp.
12. Saito, S., Arithmetic on two-dimensional local rings, Invent. Math. 85(2) (1986), 379414.
13. Saito, S., Arithmetic theory of arithmetic surfaces, Ann. of Math. (2) 129(3) (1989), 547589.
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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