144 research outputs found

### Hilbert Irreducibility above algberaic groups

The paper offers versions of Hilbert's Irreducibility Theorem for the lifting
of points in a cyclic subgroup of an algebraic group to a ramified cover. A
version of Bertini Theorem in this context is also obtained.Comment: 22 page

### 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

### A lower bound for the height of a rational function at $S$-unit points

Let $\Gamma$ be a finitely generated subgroup of the multiplicative group
\G_m^2(\bar{Q}). Let p(X,Y),q(X,Y)\in\bat{Q} be two coprime polynomials not
both vanishing at $(0,0)$; let $\epsilon>0$. We prove that, for all
$(u,v)\in\Gamma$ outside a proper Zariski closed subset of $G_m^2$, the height
of $p(u,v)/q(u,v)$ verifies $h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-\epsilon
\max(h(uu),h(v))$. As a consequence, we deduce upper bounds for (a generalized
notion of) the g.c.d. of $u-1,v-1$ for $u,v$ running over $\Gamma$.Comment: Plain TeX 18 pages. Version 2; minor changes. To appear on
Monatshefte fuer Mathemati

### Integral points, divisibility between values of polynomials and entire curves on surfaces

We prove some new degeneracy results for integral points and entire curves on
surfaces; in particular, we provide the first example, to our knowledge, of a
simply connected smooth variety whose sets of integral points are never
Zariski-dense (and no entire curve has Zariski-dense image). Some of our
results are connected with divisibility problems, i.e. the problem of
describing the integral points in the plane where the values of some given
polynomials in two variables divide the values of other given polynomials.Comment: minor changes, two references adde

### Rational points in periodic analytic sets and the Manin-Mumford conjecture

We present a new proof of the Manin-Mumford conjecture about torsion points
on algebraic subvarieties of abelian varieties. Our principle, which admits
other applications, is to view torsion points as rational points on a complex
torus and then compare (i) upper bounds for the number of rational points on a
transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds
for the degree of a torsion point (Masser), after taking conjugates. In order
to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves
contained in an analytic variety supposed invariant for translations by a full
lattice, which is a topic with some independent motivation.Comment: 12 page

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