320 research outputs found
Modularity of Calabi-Yau varieties
In this paper we discuss recent progress on the modularity of Calabi-Yau
varieties. We focus mostly on the case of surfaces and threefolds. We will also
discuss some progress on the structure of the L-function in connection with
mirror symmetry. Finally, we address some questions and open problems.Comment: Further references adde
Model theory of special subvarieties and Schanuel-type conjectures
We use the language and tools available in model theory to redefine and
clarify the rather involved notion of a {\em special subvariety} known from the
theory of Shimura varieties (mixed and pure)
On the distribution of rational points on ramified covers of abelian varieties
We prove new results on the distribution of rational points on ramified
covers of abelian varieties over finitely generated fields of
characteristic zero. For example, given a ramified cover , where
is an abelian variety over with a dense set of -rational points, we
prove that there is a finite-index coset such that
is disjoint from .
Our results do not seem to be in the range of other methods available at
present; they confirm predictions coming from Lang's conjectures on rational
points, and also go in the direction of an issue raised by Serre regarding
possible applications to the Inverse Galois Problem. Finally, the conclusions
of our work may be seen as a sharp version of Hilbert's irreducibility theorem
for abelian varieties.Comment: 38 pages. Title changed. Introduction and abstract improved. No other
changes. Comments more than welcome
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