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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers


Author: Alexandra Shlapentokh
Journal: Trans. Amer. Math. Soc. 360 (2008), 3541-3555
MSC (2000): Primary 11U05; Secondary 11G05, 03C07, 03B25
Published electronically: January 25, 2008
MathSciNet review: 2386235
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Abstract: Using Poonen's version of the ``weak vertical method'' we produce new examples of ``large'' and ``small'' rings of algebraic numbers (including rings of integers) where $ \mathbb{Z}$ and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.


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Additional Information

Alexandra Shlapentokh
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: shlapentokha@ecu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04302-X
PII: S 0002-9947(08)04302-X
Keywords: Hilbert's Tenth Problem, elliptic curves, Diophantine definitions
Received by editor(s): October 4, 2004
Received by editor(s) in revised form: April 19, 2006
Published electronically: January 25, 2008
Additional Notes: The research for this paper was partially supported by NSF grants DMS-9988620 and DMS-0354907.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.