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Specialization of Galois groups and integral points on elliptic curves


Author: Siman Wong
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 11G05; Secondary 11J70, 11R09, 11R32, 14G05
DOI: https://doi.org/10.1090/proc/13677
Published electronically: June 22, 2017
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Abstract: Let $ n\not =0, \pm 4$ be an integer. We show that the Galois group of $ x^5-10nx^2-24n $ is $ A_5$ precisely when $ \vert n\vert$ appears in the purely periodic continued fraction expansion $ [\, \vert n\vert, \vert n\vert, \vert n\vert, \ldots \, ] $ of odd positive integer powers of $ (1+\sqrt {5})/2 $; otherwise the Galois group is $ S_5$. This shows that entries A002827 and A135064 of the On-Line Encyclopedia of Integer Sequences agree except for $ n=4$. The proof involves determining all integral points of certain curves of genus 1 and 2. For integral points of an elliptic curve we handle that in two ways: via a computer algebra system and by a method of Tate.


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Additional Information

Siman Wong
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305
Email: siman@math.umass.edu

DOI: https://doi.org/10.1090/proc/13677
Keywords: Elliptic curves, Galois groups, integral points, specialization
Received by editor(s): September 30, 2016
Received by editor(s) in revised form: January 13, 2017
Published electronically: June 22, 2017
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society