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Descent on elliptic curves and Hilbert's tenth problem

Authors: Kirsten Eisenträger and Graham Everest
Journal: Proc. Amer. Math. Soc. 137 (2009), 1951-1959
MSC (2000): Primary 11G05, 11U05
Published electronically: December 18, 2008
MathSciNet review: 2480276
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Abstract: Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.

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Additional Information

Kirsten Eisenträger
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Graham Everest
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Keywords: Elliptic curve, elliptic divisibility sequence, Hilbert's Tenth Problem, isogeny, primitive divisor, $S$-integers, undecidability
Received by editor(s): October 9, 2007
Received by editor(s) in revised form: August 28, 2008
Published electronically: December 18, 2008
Additional Notes: The authors thank the ICMS in Edinburgh for the workshop on Number Theory and Computability in 2007 funded by EPSRC and the LMS
The first author was partially supported by NSF grant DMS-0801123 and a grant from the John Templeton Foundation.
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.