88 research outputs found
Elliptic divisibility sequences and undecidable problems about rational points
Julia Robinson has given a first-order definition of the rational integers Z
in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0)
where the \forall-quantifiers run over a total of 8 variables, and where F is a
polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove
that a conjecture about elliptic curves provides an interpretation of Z in Q
with quantifier complexity \forall \exists, involving only one universally
quantified variable. This improves the complexity of defining Z in Q in two
ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is
undecidable (recall that Hilbert's Tenth Problem for Q is the question whether
the \Sigma_1-theory of Q is undecidable).
In short, granting the conjecture, there is a one-parameter family of
hypersurfaces over Q for which one cannot decide whether or not they all have a
rational point.
The conjecture is related to properties of elliptic divisibility sequences on
an elliptic curve and its image under rational 2-descent, namely existence of
primitive divisors in suitable residue classes, and we discuss how to prove
weaker-in-density versions of the conjecture and present some heuristics.Comment: 39 pages, uses calrsfs. 3rd version: many small changes, change of
titl
Division-ample sets and the Diophantine problem for rings of integers
We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties
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