1,649 research outputs found
Arithmetic Quotients of the Complex Ball and a Conjecture of Lang
We prove that various arithmetic quotients of the unit ball in
are Mordellic, in the sense that they have only finitely many rational points
over any finitely generated field extension of . In the previously
known case of compact hyperbolic complex surfaces, we give a new proof using
their Albanese in conjunction with some key results of Faltings, but without
appealing to the Shafarevich conjecture. In higher dimension, our methods allow
us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in
addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel
compactifications of Picard modular surfaces of some precise levels related to
the discriminant of the imaginary quadratic fields.Comment: 21 pages, final versio
On the Hilbert Property and the Fundamental Group of Algebraic Varieties
We review, under a perspective which appears different from previous ones,
the so-called Hilbert Property (HP) for an algebraic variety (over a number
field); this is linked to Hilbert's Irreducibility Theorem and has important
implications, for instance towards the Inverse Galois Problem.
We shall observe that the HP is in a sense `opposite' to the Chevalley-Weil
Theorem, which concerns unramified covers; this link shall immediately entail
the result that the HP can possibly hold only for simply connected varieties
(in the appropriate sense). In turn, this leads to new counterexamples to the
HP, involving Enriques surfaces. We also prove the HP for a K3 surface related
to the above Enriques surface, providing what appears to be the first example
of a non-rational variety for which the HP can be proved.
We also formulate some general conjectures relating the HP with the topology
of algebraic varieties.Comment: 24 page
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