On Diophantine definability and decidability in some infinite totally real extensions of ℚ

Abstract

Abstract. Let M be a number field, and W M a set of its non-Archimedean. . , pr} be a finite set of prime numbers. Let F inf be the field generated by all the p j i -th roots of unity as j → ∞ and i = 1, . . . , r. Let K inf be the largest totally real subfield of F inf . Then for any ε > 0, there exist a number field M ⊂ K inf , and a set W M of non-Archimedean primes of M such that W M has density greater than 1 − ε, and Z has a Diophantine definition over the integral closure of O M,W M in K inf .

Rings of algebraic numbers in infinite extensions of $${\mathbb {Q}}$$ and elliptic curves retaining their rank

“…In this paper we refine our results on bounds as used in [25], [31], and [33]. We also generalize Denef's results to any totally real infinite extension K ∞ of Q and any of its extensions of degree 2 assuming some finite extension of K ∞ has an elliptic curve of positive rank with a finitely generated Mordel-Weil group.…”

“…Note that in this paper we will have no assumptions on the nature of the totally real field besides the assumption on the existence of the elliptic curve satisfying the conditions above. Thus we will be able to consider a larger class of fields than in [25], [31], and [33].…”

“…The bounds also come in the following lemma which we will use below. It is a slight modification of Lemma 5.1 of [31].…”

Characterizing diophantine henselian valuation rings and valuation ideals

“…(From a model theoretic point of view, a subset of K is diophantine if it is definable by an existential parameter-free formula in the language L ring of rings.) See for example [Den78,Shl03,Kol08,Koe16] and the references therein for problems and results on diophantine subsets of number fields, function fields, and certain infinite algebraic extensions of Q.…”

Hilbert’s Tenth Problem and Mazur’s Conjectures in Complementary Subrings of Number Fields

“…To measure the "size" of a set of primes one can use natural density defined below. The study of Hilbert's Tenth Problem and of the archimedean version of Mazur's conjecture over rings of S-integers has produced Diophantine definitions of Z and discrete archimedean sets over large subrings of some number fields ( [23], [24], [25], [26], [27], and [29]). In 2003 Poonen proved that there exists a recursive set S of primes of natural density one such that Hilbert's Tenth Problem is undecidable for Z[S −1 ].…”