Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle
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- by Hector Pasten and Chia-Liang Sun
- Proc. Amer. Math. Soc. 144 (2016), 2361-2373
- DOI: https://doi.org/10.1090/proc/12941
- Published electronically: October 21, 2015
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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.References
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Bibliographic Information
- Hector Pasten
- Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
- MR Author ID: 891758
- 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
- MR Author ID: 1040889
- Email: csun@math.sinica.edu.tw
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2361-2373
- MSC (2010): Primary 12E20, 14G05
- DOI: https://doi.org/10.1090/proc/12941
- MathSciNet review: 3477053