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

Shlapentokh, Alexandra

doi:10.1090/S0002-9947-08-04302-X

Elliptic curves retaining their rank in finite extensions and Hilbert’s Tenth Problem for rings of algebraic numbers
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Trans. Amer. Math. Soc. 360 (2008), 3541-3555 Request permission

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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