Specialization of Galois groups and integral points on elliptic curves
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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 $|n|$ appears in the purely periodic continued fraction expansion $[ |n|, |n|, |n|, \ldots ]$ 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.References
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Additional Information
- Siman Wong
- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305
- MR Author ID: 643528
- Email: siman@math.umass.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5179-5190
- MSC (2010): Primary 11G05; Secondary 11J70, 11R09, 11R32, 14G05
- DOI: https://doi.org/10.1090/proc/13677
- MathSciNet review: 3717947