Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields. (English) Zbl 1440.14035
In the paper under review, the authors discuss local-global principle for zero-cycles on certain homogeneous spaces defined over certain function fields.
The concerned varieties are defined over function fields \(K\) of curves over complete discretely valued fields (so-called semiglobal fields), these fields are two-dimensional. Over such fields, patching methods have been used to prove local-global principle for rational points on several homogeneous spaces by the authors of this paper [Comment. Math. Helv. 87, No. 4, 1011–1033 (2012; Zbl 1332.11065); Trans. Am. Math. Soc. 368, No. 6, 4219–4255 (2016; Zbl 1360.11068); Am. J. Math. 137, No. 6, 1559–1612 (2015; Zbl 1348.11036)]. Here local-global principle means that such varieties possess \(K\)-rational points if and only if they possess rational points over certain collections of overfields of \(K\). One naturally asks whether local-global principle holds for zero-cycles of degree one.
Roughly speaking, the authors prove several variants (with different fields \(K\) and different collections of overfields of \(K\)) of the following statement:
This is an analogue of a result proved by the reviewer [Y. Liang, Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 35–56 (2013; Zbl 1264.14033)] concerning local-global principle with Brauer-Manin obstruction for rationally connected varieties defined over number fields.
The concerned varieties are defined over function fields \(K\) of curves over complete discretely valued fields (so-called semiglobal fields), these fields are two-dimensional. Over such fields, patching methods have been used to prove local-global principle for rational points on several homogeneous spaces by the authors of this paper [Comment. Math. Helv. 87, No. 4, 1011–1033 (2012; Zbl 1332.11065); Trans. Am. Math. Soc. 368, No. 6, 4219–4255 (2016; Zbl 1360.11068); Am. J. Math. 137, No. 6, 1559–1612 (2015; Zbl 1348.11036)]. Here local-global principle means that such varieties possess \(K\)-rational points if and only if they possess rational points over certain collections of overfields of \(K\). One naturally asks whether local-global principle holds for zero-cycles of degree one.
Roughly speaking, the authors prove several variants (with different fields \(K\) and different collections of overfields of \(K\)) of the following statement:
- For a \(K\)-scheme of finite type \(X\), if local-global principle holds for rational points on the base change \(X_L\) for all finite separable field extensions \(L/K\), then local-global principle holds for zero-cycles of degree one on \(X\).
This is an analogue of a result proved by the reviewer [Y. Liang, Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 35–56 (2013; Zbl 1264.14033)] concerning local-global principle with Brauer-Manin obstruction for rationally connected varieties defined over number fields.
Reviewer: Yongqi Liang (Hefei)
MSC:
| 14C25 | Algebraic cycles |
| 14G05 | Rational points |
| 14H25 | Arithmetic ground fields for curves |
| 11E72 | Galois cohomology of linear algebraic groups |
| 12G05 | Galois cohomology |
| 12F10 | Separable extensions, Galois theory |
Keywords:
linear algebraic groups and torsors; zero-cycles; local-global principles; semiglobal fields; discrete valuation ringsReferences:
| [1] | Artin, M., Algebraic approximation of structures over complete local rings, Inst. Hautes \'{E}tudes Sci. Publ. Math., 36, 23-58 (1969) · Zbl 0181.48802 |
| [2] | Bourbaki, Nicolas, Elements of mathematics. Commutative algebra, xxiv+625 pp. (1972), Hermann, Paris; Addison-Wesley Publishing Co., Reading, MA · Zbl 0902.13001 |
| [3] | Borovoi, Mikhail V., Abelianization of the second nonabelian Galois cohomology, Duke Math. J., 72, 1, 217-239 (1993) · Zbl 0849.12011 · doi:10.1215/S0012-7094-93-07209-2 |
| [4] | Borovoi, Mikhail; Kunyavski\u{\i}, Boris, Formulas for the unramified Brauer group of a principal homogeneous space of a linear algebraic group, J. Algebra, 225, 2, 804-821 (2000) · Zbl 1054.14510 · doi:10.1006/jabr.1999.8153 |
| [5] | Borovoi, Mikhail; Kunyavski\u{\i}, Boris; Gille, Philippe, Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields, J. Algebra, 276, 1, 292-339 (2004) · Zbl 1057.11023 · doi:10.1016/j.jalgebra.2003.10.024 |
| [6] | Colliot-Th\'{e}l\`“ene, Jean-Louis, R\'”{e}solutions flasques des groupes lin\'{e}aires connexes, J. Reine Angew. Math., 618, 77-133 (2008) · Zbl 1158.14021 · doi:10.1515/CRELLE.2008.034 |
| [7] | J.-L. Colliot-Th\'el\`ene, Local-global principle for rational points and zero-cycles, Notes from the Arizona Winter School 2015, http://math.arizona.edu/ swc/aws/2015/2015Colliot-TheleneNotes.pdf · Zbl 0774.12002 |
| [8] | Colliot-Th\'{e}l\`ene, 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 |
| [9] | Colliot-Th\'{e}l\`ene, J.-L.; Ojanguren, M.; Parimala, R., Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes. Algebra, arithmetic and geometry, Part I, II, Mumbai, 2000, Tata Inst. Fund. Res. Stud. Math. 16, 185-217 (2002), Tata Inst. Fund. Res., Bombay · Zbl 1055.14019 |
| [10] | Colliot-Th\'{e}l\`ene, 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 |
| [11] | Colliot-Th\'{e}l\`“ene, J.-L.; Parimala, R.; Suresh, V., Lois de r\'”{e}ciprocit\'{e} sup\'{e}rieures et points rationnels, Trans. Amer. Math. Soc., 368, 6, 4219-4255 (2016) · Zbl 1360.11068 · doi:10.1090/tran/6519 |
| [12] | Colliot-Th\'{e}l\`“ene, J.-L.; Sansuc, J.-J., Fibr\'”{e}s quadratiques et composantes connexes r\'{e}elles, Math. Ann., 244, 2, 105-134 (1979) · Zbl 0418.14016 · doi:10.1007/BF01420486 |
| [13] | Gabber, Ofer; Liu, Qing; Lorenzini, Dino, The index of an algebraic variety, Invent. Math., 192, 3, 567-626 (2013) · Zbl 1268.13009 · doi:10.1007/s00222-012-0418-z |
| [14] | Harbater, David; Hartmann, Julia, Patching over fields, Israel J. Math., 176, 61-107 (2010) · Zbl 1213.14052 · doi:10.1007/s11856-010-0021-1 |
| [15] | Harbater, David; Hartmann, Julia; Krashen, Daniel, Applications of patching to quadratic forms and central simple algebras, Invent. Math., 178, 2, 231-263 (2009) · Zbl 1259.12003 · doi:10.1007/s00222-009-0195-5 |
| [16] | Harbater, David; Hartmann, Julia; Krashen, Daniel, Local-global principles for Galois cohomology, Comment. Math. Helv., 89, 1, 215-253 (2014) · Zbl 1332.11046 · doi:10.4171/CMH/317 |
| [17] | 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 |
| [18] | Harbater, David; Hartmann, Julia; Krashen, Daniel, Refinements to patching and applications to field invariants, Int. Math. Res. Not. IMRN, 20, 10399-10450 (2015) · Zbl 1386.14101 · doi:10.1093/imrn/rnu278 |
| [19] | D. Harbater, J. Hartmann, D. Krashen, R. Parimala, and V. Suresh, Local-global Galois theory of arithmetic function fields, arXiv:1710.03635 (2017). Israel J. Math. (to appear). · Zbl 1451.11126 |
| [20] | Krashen, Daniel, Field patching, factorization, and local-global principles. Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, 57-82 (2010), Springer, New York · Zbl 1251.12007 · doi:10.1007/978-1-4419-6211-9\_4 |
| [21] | Lang, Serge, Algebraic number theory, xi+354 pp. (1970), Addison-Wesley Publishing Co., Inc., Reading, MA\textendash London\textendash Don Mills, ON · Zbl 0211.38404 |
| [22] | Liang, Yongqi, Arithmetic of \(0\)-cycles on varieties defined over number fields, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4), 46, 1, 35-56 (2013) (2013) · Zbl 1264.14033 · doi:10.24033/asens.2184 |
| [23] | Lipman, Joseph, Introduction to resolution of singularities. Algebraic geometry, Proc. Sympos. Pure Math., vol. 29, Humboldt State Univ., Arcata, Calif., 1974, 187-230 (1975), Amer. Math. Soc., Providence, RI · Zbl 0306.14007 |
| [24] | Nisnevich, Yevsey A., Espaces homog\`“enes principaux rationnellement triviaux et arithm\'”{e}tique des sch\'{e}mas en groupes r\'{e}ductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris S\'{e}r. I Math., 299, 1, 5-8 (1984) · Zbl 0587.14033 |
| [25] | Parimala, R., Homogeneous varieties-Zero-cycles of degree one versus rational points, Asian J. Math., 9, 2, 251-256 (2005) · Zbl 1095.14018 · doi:10.4310/AJM.2005.v9.n2.a9 |
| [26] | Parimala, R.; Preeti, R.; Suresh, V., Local-global principle for reduced norms over function fields of \(p\)-adic curves, Compos. Math., 154, 2, 410-458 (2018) · Zbl 1469.11061 · doi:10.1112/S0010437X17007618 |
| [27] | Parimala, R.; Suresh, V., Period-index and \(u\)-invariant questions for function fields over complete discretely valued fields, Invent. Math., 197, 1, 215-235 (2014) · Zbl 1356.11018 · doi:10.1007/s00222-013-0483-y |
| [28] | Parimala, R., Arithmetic of linear algebraic groups over two-dimensional fields. Proceedings of the International Congress of Mathematicians. Volume I, 339-361 (2010), Hindustan Book Agency, New Delhi · Zbl 1251.20044 |
| [29] | Raynaud, Michel, G\'{e}om\'{e}trie analytique rigide d’apr\`“es Tate, Kiehl \(, \ldots \). Table Ronde d”Analyse non archim\'{e}dienne, Paris, 1972, 319-327. Bull. Soc. Math. France, M\'”{e}m. No. 39-40 (1974), Soc. Math. France, Paris · Zbl 0299.14003 · doi:10.24033/msmf.170 |
| [30] | Serre, Jean-Pierre, Galois cohomology, Springer Monographs in Mathematics, x+210 pp. (2002), Springer-Verlag, Berlin · Zbl 1004.12003 |
| [31] | J. Starr, Rationally simply connected varieties and pseudo algebraically closed fields, arXiv:1704.02932 (2017). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
