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

More Than 1700 Years of Word Equations

Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of word equations over a unary alphabet, and, more importantly, a word equation can be viewed as a special case of a Diophantine equation. Hence, the problem WordEquations: "Is a given word equation solvable?" is intimately related to Hilbert's 10th problem on the solvability of Diophantine equations. This became clear to the Russian school of mathematics at the latest in the mid 1960s, after which a systematic study of that relation began. Here, we review some recent developments which led to an amazingly simple decision procedure for WordEquations, and to the description of the set of all solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings of CAI 2015, Stuttgart, Germany, September 1 - 4, 201

Diophantine equations in two variables

This is an expository lecture on the subject of the title delivered at the Park-IAS mathematical institute in Princeton (July, 2000).Comment: Not for separate publicatio