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Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
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
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