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

DOI10.1090/S0002-9947-08-04302-XzbMath1222.11147OpenAlexW2076040392MaRDI QIDQ3508028

Alexandra Shlapentokh

Publication date: 27 June 2008

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0002-9947-08-04302-x

zbMATH Keywords

elliptic curvesHilbert's tenth problemDiophantine definitions

Mathematics Subject Classification ID

Decidability (number-theoretic aspects) (11U05) Elliptic curves over global fields (11G05) Decidability of theories and sets of sentences (03B25) Basic properties of first-order languages and structures (03C07)

Related Items

Diophantine definability and decidability in extensions of degree 2 of totally real fields ⋮ Defining the integers in large rings of a number field using one universal quantifier ⋮ ELLIPTIC CURVE POINTS AND DIOPHANTINE MODELS OF ℤ IN LARGE SUBRINGS OF NUMBER FIELDS ⋮ NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES ⋮ First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers ⋮ Existential decidability for addition and divisibility in holomorphy subrings of global fields ⋮ Elliptic curves, \(L\)-functions, and Hilbert's tenth problem ⋮ Rank growth of elliptic curves over 𝑁-th root extensions ⋮ Existential definability and diophantine stability ⋮ REMARKS ON HILBERT’S TENTH PROBLEM AND THE IWASAWA THEORY OF ELLIPTIC CURVES ⋮ Superficies elípticas y el décimo problema de Hilbert ⋮ As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields ⋮ Extensions of Hilbert’s Tenth Problem: Definability and Decidability in Number Theory ⋮ Defining Integers ⋮ Rings of algebraic numbers in infinite extensions of \(\mathbb Q\) and elliptic curves retaining their rank ⋮ Towards Hilbert's tenth problem for rings of integers through Iwasawa theory and Heegner points

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