Elliptic curves retaining their rank in finite extensions and Hilbert’s Tenth Problem for rings of algebraic numbers
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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.References
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Additional Information
- Alexandra Shlapentokh
- Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
- MR Author ID: 288363
- ORCID: 0000-0003-1990-909X
- Email: shlapentokha@ecu.edu
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3541-3555
- MSC (2000): Primary 11U05; Secondary 11G05, 03C07, 03B25
- DOI: https://doi.org/10.1090/S0002-9947-08-04302-X
- MathSciNet review: 2386235