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Journal für die reine und angewandte Mathematik

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Elliptic divisibility sequences and undecidable problems about rational points

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2007, Issue 613, Pages 1–33, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2007.089, February 2008

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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 (∀∃∀∃) (F = 0) where the ∀-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the Σ5-theory of ℚ is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of ℤ in ℚ with quantifier complexity ∀∃, involving only one universally quantified variable. This improves the complexity of defining ℤ in ℚ in two ways, and implies that the Σ3-theory, and even the Π2-theory, of ℚ is undecidable (recall that Hilbert's Tenth Problem for ℚ is the question whether the Σ1-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.

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