Elliptic divisibility sequences and undecidable problems about rational points
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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 Σ_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 Σ_3-theory, and even the Π_2-theory, of Q is undecidable (recall that Hilbert's Tenth Problem for Q is the question whether the Σ_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.
39 pages, uses calrsfs. 3rd version: many small changes, change of title
39 pages, uses calrsfs. 3rd version: many small changes, change of title