Pasten, Hector; Sun, Chia-Liang Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle. (English) Zbl 1331.12007 Proc. Am. Math. Soc. 144, No. 6, 2361-2373 (2016). Summary: We study a local-global principle for polynomial equations with coefficients in a finite field and solutions restricted in a rank-one multiplicative subgroup in a function field over this finite field. We prove such a local-global principle for all sufficiently large characteristics, and we show that the result should hold in full generality under a certain reasonable hypothesis related to the existence of large multiplicative subgroups of finite fields avoiding linear relations. We give a method for verifying the latter hypothesis in specific cases, and we show that it is a consequence of the classical Artin primitive root conjecture. In particular, this function field local-global principle is a consequence of GRH. We also discuss the relation of these problems with a finite field version of the Manin-Mumford conjecture. Cited in 1 Document MSC: 12E20 Finite fields (field-theoretic aspects) 14G05 Rational points × Cite Format Result Cite Review PDF Full Text: DOI Link References: [1] Alon, Noga; Bourgain, Jean, Additive patterns in multiplicative subgroups, Geom. Funct. Anal., 24, 3, 721-739 (2014) · Zbl 1377.11013 · doi:10.1007/s00039-014-0270-y [2] Bartolome, Boris; Bilu, Yuri; Luca, Florian, On the exponential local-global principle, Acta Arith., 159, 2, 101-111 (2013) · Zbl 1330.11019 · doi:10.4064/aa159-2-1 [3] Erd{\H{o}}s, P{\'a}l; Murty, M. Ram, On the order of \(a\pmod p\). Number theory, Ottawa, ON, 1996, CRM Proc. Lecture Notes 19, 87-97 (1999), Amer. Math. Soc., Providence, RI · Zbl 0931.11034 [4] Gupta, Rajiv; Murty, M. Ram, A remark on Artin’s conjecture, Invent. Math., 78, 1, 127-130 (1984) · Zbl 0549.10037 · doi:10.1007/BF01388719 [5] Harari, David; Voloch, Jos{\'e} Felipe, The Brauer-Manin obstruction for integral points on curves, Math. Proc. Cambridge Philos. Soc., 149, 3, 413-421 (2010) · Zbl 1280.11038 · doi:10.1017/S0305004110000381 [6] Heath-Brown, D. R., Artin’s conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2), 37, 145, 27-38 (1986) · Zbl 0586.10025 · doi:10.1093/qmath/37.1.27 [7] Hindry, Marc, Autour d’une conjecture de Serge Lang, Invent. Math., 94, 3, 575-603 (1988) · Zbl 0638.14026 · doi:10.1007/BF01394276 [8] Hooley, Christopher, On Artin’s conjecture, J. Reine Angew. Math., 225, 209-220 (1967) · Zbl 0221.10048 [9] Hrushovski, Ehud, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc., 9, 3, 667-690 (1996) · Zbl 0864.03026 · doi:10.1090/S0894-0347-96-00202-0 [10] Katz, Nicholas M., Wieferich past and future. Topics in finite fields, Contemp. Math. 632, 253-270 (2015), Amer. Math. Soc., Providence, RI · Zbl 1416.11082 · doi:10.1090/conm/632/12632 [11] Murty, M. Ram; Srinivasan, S., Some remarks on Artin’s conjecture, Canad. Math. Bull., 30, 1, 80-85 (1987) · Zbl 0574.10005 · doi:10.4153/CMB-1987-012-5 [12] Raynaud, M., Courbes sur une vari\'et\'e ab\'elienne et points de torsion, Invent. Math., 71, 1, 207-233 (1983) · Zbl 0564.14020 · doi:10.1007/BF01393342 [13] [Sko37] T. Skolem, Anwendung exponentieller Kongruenzen zum Beweis der Unlosbarkeit gewisser diophantischer Gleichungen. Avhandlinger Utgitt av det Norske Videnskaps-Akademi i Oslo I. Mat.-Naturv. Klasse. Ny Serie, 12: 1-16, 1937. · Zbl 0017.24606 [14] Sun, Chia-Liang, Product of local points of subvarieties of almost isotrivial semi-abelian varieties over a global function field, Int. Math. Res. Not. IMRN, 19, 4477-4498 (2013) · Zbl 1312.14105 [15] Sun, Chia-Liang, Local-global principle of affine varieties over a subgroup of units in a function field, Int. Math. Res. Not. IMRN, 11, 3075-3095 (2014) · Zbl 1326.14056 [16] Sun, Chia-Liang, The Brauer-Manin-Scharaschkin obstruction for subvarieties of a semi-abelian variety and its dynamical analog, J. Number Theory, 147, 533-548 (2015) · Zbl 1394.11078 · doi:10.1016/j.jnt.2014.07.015 [17] Tzermias, Pavlos, The Manin-Mumford conjecture: a brief survey, Bull. London Math. Soc., 32, 6, 641-652 (2000) · Zbl 1073.14525 · doi:10.1112/S0024609300007578 [18] Voloch, Jos{\'e} Felipe, On the order of points on curves over finite fields, Integers, 7, A49, 4 pp. (2007) · Zbl 1136.11042 [19] Yekhanin, Sergey, A note on plane pointless curves, Finite Fields Appl., 13, 2, 418-422 (2007) · Zbl 1146.14014 · doi:10.1016/j.ffa.2006.11.001 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.