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Étale Steenrod operations and the Artin–Tate pairing

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  13 July 2020

Tony Feng
Tony Feng*
Affiliation:
Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA02139, USA email fengt@mit.edu
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Abstract

We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.

Keywords

arithmetic geometryBrauer groupsSteenrod operationsArtin–Tate conjecture

MSC classification

Primary: 11G25: Varieties over finite and local fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 7 , July 2020 , pp. 1476 - 1515
Copyright
© The Author 2020

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Footnotes

The author was supported by an NSF Graduate Fellowship, a Stanford ARCS Fellowship, and an NSF Postdoctoral Fellowship during the completion of this paper.

References

Barnea, I. and Schlank, T. M., A projective model structure on pro-simplicial sheaves, and the relative étale homotopy type, Adv. Math. 291 (2016), 784–858; MR 3459031.10.1016/j.aim.2015.11.014CrossRefGoogle Scholar
Browder, W., Torsion in H-spaces, Ann. of Math. (2) 74 (1961), 24–51; MR 0124891.10.2307/1970305CrossRefGoogle Scholar
Browder, W., Remark on the Poincaré duality theorem, Proc. Amer. Math. Soc. 13 (1962), 927–930; MR 0143205.Google Scholar
Bhatt, B. and Scholze, P., The pro-étale topology for schemes, Astérisque 369 (2015), 99–201; MR 3379634.Google Scholar
Cassels, J. W. S., Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton–Dyer, J. Reine Angew. Math. 217 (1965), 180–199; MR 0179169.Google Scholar
Epstein, D. B. A., Steenrod operations in homological algebra, Invent. Math. 1 (1966), 152–208; MR 0199240.10.1007/BF01389726CrossRefGoogle Scholar
Freitag, E. and Kiehl, R., Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13 (Springer, Berlin, 1988); translated from the German by Betty S. Waterhouse and William C. Waterhouse, with an historical introduction by J. A. Dieudonné; MR 926276.10.1007/978-3-662-02541-3CrossRefGoogle Scholar
Friedlander, E. M., Étale homotopy of simplicial schemes, Annals of Mathematics Studies, vol. 104 (Princeton University Press and University of Tokyo Press, 1982); MR 676809.Google Scholar
Fulton, W., Intersection theory, 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], vol. 2, second edition (Springer, Berlin, 1998); MR 1644323.10.1007/978-1-4612-1700-8CrossRefGoogle Scholar
Geisser, T., Comparing the Brauer group to the Tate–Shafarevich group, J. Inst. Math. Jussieu 19 (2020), 965–970.10.1017/S1474748018000294CrossRefGoogle Scholar
Gordon, W. J., Linking the conjectures of Artin–Tate and Birch–Swinnerton–Dyer, Compos. Math. 38 (1979), 163–199; MR 528839.Google Scholar
Grothendieck, A., La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154; MR 0116023.CrossRefGoogle Scholar
Harpaz, Y. and Schlank, T. M., Homotopy obstructions to rational points, in Torsors, étale homotopy and applications to rational points, London Mathematical Society Lecture Note Series, vol. 405 (Cambridge University Press, Cambridge, 2013), 280–413; MR 3077173.CrossRefGoogle Scholar
Jahn, T., The order of higher Brauer groups, Math. Ann. 362 (2015), 43–54; MR 3343869.CrossRefGoogle Scholar
Jardine, J. F., Steenrod operations in the cohomology of simplicial presheaves, in Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 279 (Kluwer Academic, Dordrecht, 1989), 103–116; MR 1045847.10.1007/978-94-009-2399-7_5CrossRefGoogle Scholar
Lawson, H. B. Jr. and Michelsohn, M.-L., Spin geometry, Princeton Mathematical Series, vol. 38 (Princeton University Press, Princeton, NJ, 1989); MR 1031992.Google Scholar
Liu, Q., Lorenzini, D. and Raynaud, M., On the Brauer group of a surface, Invent. Math. 159 (2005), 673–676; MR 2125738.10.1007/s00222-004-0403-2CrossRefGoogle Scholar
Liu, Q., Lorenzini, D. and Raynaud, M., Corrigendum to Néron models, Lie algebras, and reduction of curves of genus one and the Brauer group of a surface, Invent. Math. 214 (2018), 593–604; MR 3858404.10.1007/s00222-018-0809-xCrossRefGoogle Scholar
Manin, Ju. I., Rational surfaces over perfect fields. II, Mat. Sb. (N.S.) 72 (1967), 161–192; MR 0225781.Google Scholar
Manin, Yu. I., Cubic forms: Algebra, geometry, arithmetic, North-Holland Mathematical Library, vol. 4, second edition (North-Holland, Amsterdam, 1986); translated from the Russian by M. Hazewinkel; MR 833513.Google Scholar
Milnor, J. W. and Stasheff, J. D., Characteristic classes, Annals of Mathematics Studies, vol. 76 (Princeton University Press and University of Tokyo Press, 1974); MR 0440554.10.1515/9781400881826CrossRefGoogle Scholar
Mosher, R. E. and Tangora, M. C., Cohomology operations and applications in homotopy theory (Harper & Row, New York–London, 1968); MR 0226634.Google Scholar
Poonen, B. and Stoll, M., The Cassels–Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), 1109–1149; MR 1740984.10.2307/121064CrossRefGoogle Scholar
Deligne, P., Séminaire de géométrie algébrique du Bois Marie – Cohomologie étale (SGA 4[[()[]mml:mfrac[]()]][[()[]mml:mrow []()]]1[[()[]/mml:mrow[]()]] [[()[]mml:mrow []()]]2[[()[]/mml:mrow[]()]][[()[]/mml:mfrac[]()]]), Lecture Notes in Mathematics, vol. 569 (Springer, 1970).Google Scholar
Sherman, D., The Cassels–Tate pairing and work of Poonen–Stoll, (2015), available athttp://math.stanford.edu/∼conrad/BSDseminar/Notes/L7.pdf.Google Scholar
Smith, J. R., Steenrod coalgebras, Topol. Appl. 185/186 (2015), 93–123; MR 3319467.CrossRefGoogle Scholar
Tate, J., Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) (Inst. Mittag-Leffler, Djursholm, 1963), 288–295; MR 0175892.Google Scholar
Tate, J., On the conjectures of Birch and Swinnerton–Dyer and a geometric analog, Séminaire Bourbaki, vol. 9 (Société mathématique de France, Paris, 1995), Exp. No. 306, 415–440; MR 1610977.Google Scholar
Thom, R., Espaces fibrés en sphères et carrés de Steenrod, Ann. Sci. Éc. Norm. Supér. (3) 69 (1952), 109–182; MR 0054960.CrossRefGoogle Scholar
Urabe, T., The bilinear form of the Brauer group of a surface, Invent. Math. 125 (1996), 557–585; MR 1400317.10.1007/s002220050086CrossRefGoogle Scholar
Zarhin, Y. G., The Brauer group of a surface over a finite field (Russian), Arith. Geom. Varieties (1989), 57–67.Google Scholar