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Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle


Authors: Hector Pasten and Chia-Liang Sun
Journal: Proc. Amer. Math. Soc. 144 (2016), 2361-2373
MSC (2010): Primary 12E20, 14G05
DOI: https://doi.org/10.1090/proc/12941
Published electronically: October 21, 2015
MathSciNet review: 3477053
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Abstract: 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.


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  • [1] Noga Alon and Jean Bourgain, Additive patterns in multiplicative subgroups, Geom. Funct. Anal. 24 (2014), no. 3, 721-739. MR 3213827, https://doi.org/10.1007/s00039-014-0270-y
  • [2] Boris Bartolome, Yuri Bilu, and Florian Luca, On the exponential local-global principle, Acta Arith. 159 (2013), no. 2, 101-111. MR 3062909, https://doi.org/10.4064/aa159-2-1
  • [3] Pál Erdős and M. Ram Murty, On the order of $ a\pmod p$, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 87-97. MR 1684594 (2000c:11152)
  • [4] Rajiv Gupta and M. Ram Murty, A remark on Artin's conjecture, Invent. Math. 78 (1984), no. 1, 127-130. MR 762358 (86d:11003), https://doi.org/10.1007/BF01388719
  • [5] David Harari and José Felipe Voloch, The Brauer-Manin obstruction for integral points on curves, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 3, 413-421. MR 2726726 (2012c:14047), https://doi.org/10.1017/S0305004110000381
  • [6] D. R. Heath-Brown, Artin's conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27-38. MR 830627 (88a:11004), https://doi.org/10.1093/qmath/37.1.27
  • [7] Marc Hindry, Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575-603 (French). MR 969244 (89k:11046), https://doi.org/10.1007/BF01394276
  • [8] Christopher Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209-220. MR 0207630 (34 #7445)
  • [9] Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667-690. MR 1333294 (97h:11154), https://doi.org/10.1090/S0894-0347-96-00202-0
  • [10] Nicholas M. Katz, Wieferich past and future, Topics in finite fields, Contemp. Math., vol. 632, Amer. Math. Soc., Providence, RI, 2015, pp. 253-270. MR 3329985, https://doi.org/10.1090/conm/632/12632
  • [11] M. Ram Murty and S. Srinivasan, Some remarks on Artin's conjecture, Canad. Math. Bull. 30 (1987), no. 1, 80-85. MR 879875 (88e:11094), https://doi.org/10.4153/CMB-1987-012-5
  • [12] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), no. 1, 207-233 (French). MR 688265 (84c:14021), https://doi.org/10.1007/BF01393342
  • [13] 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.
  • [14] Chia-Liang Sun, Product of local points of subvarieties of almost isotrivial semi-abelian varieties over a global function field, Int. Math. Res. Not. IMRN 19 (2013), 4477-4498. MR 3116170
  • [15] Chia-Liang Sun, Local-global principle of affine varieties over a subgroup of units in a function field, Int. Math. Res. Not. IMRN 11 (2014), 3075-3095. MR 3214315
  • [16] Chia-Liang Sun, The Brauer-Manin-Scharaschkin obstruction for subvarieties of a semi-abelian variety and its dynamical analog, J. Number Theory 147 (2015), 533-548. MR 3276339, https://doi.org/10.1016/j.jnt.2014.07.015
  • [17] Pavlos Tzermias, The Manin-Mumford conjecture: a brief survey, Bull. London Math. Soc. 32 (2000), no. 6, 641-652. MR 1781574 (2001g:11091), https://doi.org/10.1112/S0024609300007578
  • [18] José Felipe Voloch, On the order of points on curves over finite fields, Integers 7 (2007), A49, 4. MR 2373111 (2009j:14028)
  • [19] Sergey Yekhanin, A note on plane pointless curves, Finite Fields Appl. 13 (2007), no. 2, 418-422. MR 2307138 (2008b:14032), https://doi.org/10.1016/j.ffa.2006.11.001

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Additional Information

Hector Pasten
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
Email: hpasten@math.harvard.edu

Chia-Liang Sun
Affiliation: Institute of Mathematics, Academia Sinica, Room 626, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
Email: csun@math.sinica.edu.tw

DOI: https://doi.org/10.1090/proc/12941
Received by editor(s): February 24, 2015
Received by editor(s) in revised form: July 19, 2015
Published electronically: October 21, 2015
Additional Notes: The first author was supported by a Benjamin Peirce Fellowship
The second author was supported by an Academia Sinica Postdoctoral Fellowship.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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