1,289 research outputs found
Diophantine Geometry as Galois Theory in the Mathematics of Serge Lang
A remark about the role of Galois theory in Diophantine geometry as reflected
in the work of Serge Lang. An entry in `The mathematical contributions of Serge
Lang.
The arithmetic of Prym varieties in genus 3
Given a curve of genus 3 with an unramified double cover, we give an explicit
description of the associated Prym-variety. We also describe how an unramified
double cover of a non-hyperelliptic genus 3 curve can be mapped into the
Jacobian of a curve of genus 2 over its field of definition and how this can be
used to do Chabauty- and Brauer-Manin type calculations for curves of genus 5
with an unramified involution. As an application, we determine the rational
points on a smooth plane quartic with no particular geometric properties and
give examples of curves of genus 3 and 5 violating the Hasse-principle. We also
show how these constructions can be used to design smooth plane quartics with
specific arithmetic properties. As an example, we give a smooth plane quartic
with all 28 bitangents defined over Q(t).Comment: 21 page
Kummer sandwiches and Greene-Plesser construction
In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method
constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in
, starting from a special one-parameter family of K3 varieties
known as the quartic Dwork pencil. We show that certain K3 double covers
obtained from the three-parameter family of quartic Kummer surfaces associated
with a principally polarized abelian surface generalize the relation of the
Dwork pencil and the quartic mirror family. Moreover, for the three-parameter
family we compute a formula for the rational point-count of its generic member
and derive its transformation behavior with respect to -isogenies of the
underlying abelian surface.Comment: 27 pages; minor typos corrected in version
The Brauer-Manin obstruction on Kummer varieties and ranks of twists of abelian varieties
Let r > 0 be an integer. We present a sufficient condition for an abelian
variety A over a number field k to have infinitely many quadratic twists of
rank at least r, in terms of density properties of rational points on the
Kummer variety Km(A^r) of the r-fold product of A with itself. One consequence
of our results is the following. Fix an abelian variety A over k, and suppose
that for some r > 0 the Brauer-Manin obstruction to weak approximation on the
Kummer variety Km(A^r) is the only one. Then A has a quadratic twist of rank at
least r. Hence if the Brauer-Manin obstruction is the only one to weak
approximation on all Kummer varieties, then ranks of twists of any
positive-dimensional abelian variety are unbounded. This relates two
significant open questions.Comment: 12 pages; final versio
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