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Local-global questions for tori over \(p\)-adic function fields. (English) Zbl 1355.14018

Let \(K\) be the function field of a curve over a \(p\)-adic field. The paper under review mainly addresses local-global principles for torsors under tori defined over \(K\). Although the cohomological dimension of \(K\) is strictly greater than 2, it has been shown in recent works, especially [J.-L. Colliot-Thélène et al., Comment. Math. Helv. 87, No. 4, 1011–1033 (2012; Zbl 1332.11065)] and [J. L. Colliot-Thélène et al., Trans. Am. Math. Soc. 368, No. 6, 4219–4255 (2016; Zbl 1360.11068)], that when all the discrete valuations of \(K\) are considered, local-global results closely analogous to the classical ones (over number fields) may continue to hold. In the present work, the authors restrict themselves to discrete valuations that are trivial on the constant field of \(K\) and study the associated Tate-Shafarevich groups \font\fontWCA=wncyr10 {\fontWCA SH}\({}^i(T)\), \(i=1,\,2\), for algebraic tori \(T\) over \(K\).
The local completions \(K_v\), which some people call 2-local fields, have very nice cohomological properties. From local duality results the authors utilize all necessary cohomological machineries to obtain the following duality theorem:
There is a perfect pairing of finite groups \[ \text{\font\fontWCA=wncyr10 {\fontWCA SH}}^1(T)\times \text{\font\fontWCA=wncyr10 {\fontWCA SH}}^2(T')\longrightarrow \mathbb{Q}/\mathbb{Z} \] where \(T'\) denotes the dual torus of \(T\).
This duality theorem then leads to proofs that Sansuc’s (by now classical) results over number fields have analogues over \(K\). Given a torsor \(Y\) under the torus \(T\), a description of the obstruction to the local-global principle for rational points on \(Y\) is given in terms of a map defined on a subquotient of \(H^3(Y,\,\mathbb{Q}/\mathbb{Z}(2))\), and this obstruction is shown to be the only one.
In the last section of the paper is proved a generalization to torsors under certain reductive groups \(G\). There the simply connected cover \(G^{sc}\) of the derived group of \(G\) is assumed to be quasi-split without \(E_8\) factor. The proof relies on facts about the Rost invariant of \(G^{sc}\).
The aforementioned duality theorem also has a variant for finite commutative groups (in places of tori), as stated in Theorem4.4.
The interested readers may be referred to more recent works of D. Izquierdo [Math. Z. 284, No. 1–2, 615–642 (2016; Zbl 1407.11130)] for variants and generalizations over similar fields, which many have even higher cohomological dimensions.

MSC:

14G05 Rational points
11G20 Curves over finite and local fields
14G20 Local ground fields in algebraic geometry
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[1] Bloch, Spencer, Algebraic cycles and higher \(K\)-theory, Adv. in Math., 61, 3, 267-304 (1986) · Zbl 0608.14004 · doi:10.1016/0001-8708(86)90081-2
[2] Borovoi, Mikhail, Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc., 132, 626, viii+50 pp. (1998) · Zbl 0918.20037 · doi:10.1090/memo/0626
[3] Borovoi, Mikhail; van Hamel, Joost, Extended Picard complexes and linear algebraic groups, J. Reine Angew. Math., 627, 53-82 (2009) · Zbl 1170.14015 · doi:10.1515/CRELLE.2009.011
[4] Chernousov, V. I., A remark on the \(({\rm mod}\, 5)\)-invariant of Serre for groups of type \(E_8\), Mat. Zametki. Math. Notes, 56 56, 1-2, 730-733 (1995) (1994) · Zbl 0835.20059 · doi:10.1007/BF02110564
[5] Chernousov, Vladimir, On the kernel of the Rost invariant for \(E_8\) modulo \(3\). Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, 199-214 (2010), Springer, New York · Zbl 1231.20046 · doi:10.1007/978-1-4419-6211-9\_11
[6] Colliot-Th{\'e}l{\`“e}ne, Jean-Louis, R\'”esolutions flasques des groupes lin\'eaires connexes, J. Reine Angew. Math., 618, 77-133 (2008) · Zbl 1158.14021 · doi:10.1515/CRELLE.2008.034
[7] Colliot-Th{\'e}l{\`e}ne, J.-L.; Gille, P.; Parimala, R., Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J., 121, 2, 285-341 (2004) · Zbl 1129.11014 · doi:10.1215/S0012-7094-04-12124-4
[8] Colliot-Th{\'e}l{\`e}ne, Jean-Louis; Parimala, Raman; Suresh, Venapally, Patching and local-global principles for homogeneous spaces over function fields of \(p\)-adic curves, Comment. Math. Helv., 87, 4, 1011-1033 (2012) · Zbl 1332.11065 · doi:10.4171/CMH/276
[9] Colliot-Th{\'e}l{\`“e}ne, J.-L.; Parimala, R.; Suresh, V., Lois de r\'”eciprocit\'e sup\'erieures et points rationnels, Trans. Amer. Math. Soc., 368, 6, 4219-4255 (2016) · Zbl 1360.11068 · doi:10.1090/tran/6519
[10] Colliot-Th{\'e}l{\`“e}ne, Jean-Louis; Sansuc, Jean-Jacques, \(La R\)-\'”equivalence sur les tores, Ann. Sci. \'Ecole Norm. Sup. (4), 10, 2, 175-229 (1977) · Zbl 0356.14007
[11] Colliot-Th{\'e}l{\`“e}ne, Jean-Louis; Sansuc, Jean-Jacques, La descente sur les vari\'”et\'es rationnelles. II, Duke Math. J., 54, 2, 375-492 (1987) · Zbl 0659.14028 · doi:10.1215/S0012-7094-87-05420-2
[12] Geisser, Thomas, Duality via cycle complexes, Ann. of Math. (2), 172, 2, 1095-1126 (2010) · Zbl 1215.19001 · doi:10.4007/annals.2010.172.1095
[13] Geisser, Thomas; Levine, Marc, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, J. Reine Angew. Math., 530, 55-103 (2001) · Zbl 1023.14003 · doi:10.1515/crll.2001.006
[14] Greenberg, Marvin J., Rational points in Henselian discrete valuation rings, Inst. Hautes \'Etudes Sci. Publ. Math., 31, 59-64 (1966)
[15] Harari, David; Scheiderer, Claus; Szamuely, Tam{\'a}s, Weak approximation for tori over \(p\)-adic function fields, Int. Math. Res. Not. IMRN, 10, 2751-2783 (2015) · Zbl 1349.11098 · doi:10.1093/imrn/rnu019
[16] Harari, David; Szamuely, Tam{\'a}s, Arithmetic duality theorems for 1-motives, J. Reine Angew. Math., 578, 93-128 (2005) · Zbl 1088.14012 · doi:10.1515/crll.2005.2005.578.93
[17] Harari, David; Szamuely, Tam{\'a}s, Local-global principles for 1-motives, Duke Math. J., 143, 3, 531-557 (2008) · Zbl 1155.14020 · doi:10.1215/00127094-2008-028
[18] Harbater, David; Hartmann, Julia; Krashen, Daniel, Local-global principles for torsors over arithmetic curves, Amer. J. Math., 137, 6, 1559-1612 (2015) · Zbl 1348.11036 · doi:10.1353/ajm.2015.0039
[19] Harder, G{\`“u}nter, Halbeinfache Gruppenschemata \'”uber vollst\"andigen Kurven, Invent. Math., 6, 107-149 (1968) · Zbl 0186.25902
[20] Hu, Yong, Hasse principle for simply connected groups over function fields of surfaces, J. Ramanujan Math. Soc., 29, 2, 155-199 (2014) · Zbl 1320.11033
[21] Kahn, Bruno, Classes de cycles motiviques \'etales, Algebra Number Theory, 6, 7, 1369-1407 (2012) · Zbl 1263.14011 · doi:10.2140/ant.2012.6.1369
[22] Kato, Kazuya, A generalization of local class field theory by using \(K\)-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27, 3, 603-683 (1980) · Zbl 0463.12006
[23] Kato, Kazuya, A Hasse principle for two-dimensional global fields, with an appendix by Jean-Louis Colliot-Th\'el\`ene, J. Reine Angew. Math., 366, 142-183 (1986) · Zbl 0576.12012 · doi:10.1515/crll.1986.366.142
[24] Kneser, Martin, Starke Approximation in algebraischen Gruppen. I, J. Reine Angew. Math., 218, 190-203 (1965) · Zbl 0143.04701
[25] Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre, The book of involutions, with a preface in French by J.Tits, American Mathematical Society Colloquium Publications 44, xxii+593 pp. (1998), American Mathematical Society, Providence, RI · Zbl 0955.16001
[26] Lichtenbaum, Stephen, Duality theorems for curves over \(p\)-adic fields, Invent. Math., 7, 120-136 (1969) · Zbl 0186.26402
[27] Milne, James S., \'Etale cohomology, Princeton Mathematical Series 33, xiii+323 pp. (1980), Princeton University Press, Princeton, N.J.
[28] Milne, J. S., Arithmetic duality theorems, Perspectives in Mathematics 1, x+421 pp. (1986), Academic Press, Inc., Boston, MA · Zbl 0613.14019
[29] [mishih] J. S. Milne and K.-y. Shih, Conjugates of Shimura varieties, in P. Deligne at al. Hodge Cycles, Motives and Shimura Varieties, Lecture Notes in Math. vol. 900, Springer-Verlag, 1982, pp. 280-356.
[30] Preeti, R., Classification theorems for Hermitian forms, the Rost kernel and Hasse principle over fields with \(cd_2(k)\leq 3\), J. Algebra, 385, 294-313 (2013) · Zbl 1292.11056 · doi:10.1016/j.jalgebra.2013.02.038
[31] Sansuc, J.-J., Groupe de Brauer et arithm\'etique des groupes alg\'ebriques lin\'eaires sur un corps de nombres, J. Reine Angew. Math., 327, 12-80 (1981) · Zbl 0468.14007 · doi:10.1515/crll.1981.327.12
[32] Scheiderer, Claus; van Hamel, Joost, Cohomology of tori over \(p\)-adic curves, Math. Ann., 326, 1, 155-183 (2003) · Zbl 1050.14016 · doi:10.1007/s00208-003-0416-y
[33] Serre, Jean-Pierre, Cohomologie galoisienne, Lecture Notes in Mathematics 5, x+181 pp. (1994), Springer-Verlag, Berlin · Zbl 0812.12002
[34] Semenov, Nikita, Motivic construction of cohomological invariants, Comment. Math. Helv., 91, 1, 163-202 (2016) · Zbl 1344.14005 · doi:10.4171/CMH/382
[35] Skorobogatov, Alexei, Torsors and rational points, Cambridge Tracts in Mathematics 144, viii+187 pp. (2001), Cambridge University Press, Cambridge · Zbl 0972.14015 · doi:10.1017/CBO9780511549588
[36] Springer, T. A., Linear algebraic groups, Progress in Mathematics 9, xiv+334 pp. (1998), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0927.20024 · doi:10.1007/978-0-8176-4840-4
[37] Suslin, Andrei; Voevodsky, Vladimir, Bloch-Kato conjecture and motivic cohomology with finite coefficients. The arithmetic and geometry of algebraic cycles, Banff, AB, 1998, NATO Sci. Ser. C Math. Phys. Sci. 548, 117-189 (2000), Kluwer Acad. Publ., Dordrecht · Zbl 1005.19001
[38] Totaro, Burt, Milnor \(K\)-theory is the simplest part of algebraic \(K\)-theory, \(K\)-Theory, 6, 2, 177-189 (1992) · Zbl 0776.19003 · doi:10.1007/BF01771011
[39] Voskresenski{\u \i }, V. E., Birational properties of linear algebraic groups, Izv. Akad. Nauk SSSR Ser. Mat., 34, 3-19 (1970)
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