Elliptic divisibility sequences and undecidable problems about rational points
| dc.creator | Cornelissen, Gunther | |
| dc.creator | Zahidi, Karim | |
| dc.date | 2004-12-23 | |
| dc.date | 2006-06-23 | |
| dc.date.accessioned | 2026-07-07T06:39:12Z | |
| dc.date.available | 2026-07-07T06:39:12Z | |
| dc.description | 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. | |
| dc.description | 39 pages, uses calrsfs. 3rd version: many small changes, change of title | |
| dc.identifier | https://arxiv.org/abs/math/0412473 | |
| dc.identifier | http://arxiv.org/abs/math/0412473 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100993 | |
| dc.subject | Number Theory | |
| dc.subject | Logic | |
| dc.subject | 03B25, 11U05 | |
| dc.title | Elliptic divisibility sequences and undecidable problems about rational points | |
| dc.type | text |