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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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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