57 research outputs found
Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations
We prove the following special case of Mazur's conjecture on the topology of
rational points. Let be an elliptic curve over with
-invariant . For a class of elliptic pencils which are quadratic
twists of by quartic polynomials, the rational points on the projective
line with positive rank fibres are dense in the real topology. This extends
results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic
polynomials.
For the proof, we investigate a highly singular rational curve on the Kummer
surface associated to a product of two elliptic curves over ,
which previously appeared in publications by Mestre, Kuwata-Wang and Satg\'e.
We produce this curve in a simpler manner by finding algebraic equations which
give a direct proof of rationality. We find that the same equations give rise
to rational curves on a class of more general surfaces extending the Kummer
construction. This leads to further applications apart from Mazur's conjecture,
for example the existence of rational points on simultaneous twists of
superelliptic curves.
Finally, we give a proof of Mazur's conjecture for the Kummer surface
without any restrictions on the -invariants of the two elliptic curves.Comment: 14 pages, same content as published version except for added remark
acknowledging overlap with prior work by Ula
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of
characteristic 0. Let be a holomorphy subring of , not equal to . We
prove the following undecidability results for : If is recursive, then
Hilbert's Tenth Problem is undecidable in . In general, there exist
such that there is no algorithm to tell whether a
polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in
.Comment: This version contains minor revisions and will appear in Annales de l
Institut Fourie
Density of rational points on isotrivial rational elliptic surfaces
For a large class of isotrivial rational elliptic surfaces (with section), we
show that the set of rational points is dense for the Zariski topology, by
carefully studying variations of root numbers among the fibers of these
surfaces. We also prove that these surfaces satisfy a variant of weak-weak
approximation. Our results are conditional on the finiteness of
Tate-Shafarevich groups for elliptic curves over the field of rational numbers.Comment: Latex; 26 pages. To appear in Algebra and Number Theor
Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields
We prove that infinite p-adically discrete sets have Diophantine definitions
in large subrings of some number fields. First, if K is a totally real number
field or a totally complex degree-2 extension of a totally real number field,
then there exists a prime p of K and a set of K-primes S of density arbitrarily
close to 1 such that there is an infinite p-adically discrete set that is
Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number
field over which there exists an elliptic curve of rank 1, then there exists a
set of K-primes S of density 1 and an infinite Diophantine subset of O_{K,S}
that is v-adically discrete for every place v of K. Third, if K is a number
field over which there exists an elliptic curve of rank 1, then there exists a
set of K-primes S of density 1 such that there exists a Diophantine model of Z
over O_{K,S}. This line of research is motivated by a question of Mazur
concerning the distribution of rational points on varieties in a
non-archimedean topology and questions concerning extensions of Hilbert's Tenth
Problem to subrings of number fields.Comment: 17 page
On the -adic closure of a subgroup of rational points on an Abelian variety
In 2007, B. Poonen (unpublished) studied the --adic closure of a subgroup
of rational points on a commutative algebraic group. More recently, J.
Bella\"iche asked the same question for the special case of Abelian varieties.
These problems are --adic analogues of a question raised earlier by B. Mazur
on the density of rational points for the real topology. For a simple Abelian
variety over the field of rational numbers, we show that the actual --adic
rank is at least the third of the expected value
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