12 research outputs found
Splitting fields of elements in arithmetic groups
We prove that the number of unimodular integral matrices in a norm ball whose
characteristic polynomial has Galois group different than the full symmetric
group is of strictly lower order of magnitude than the number of all such
matrices in the ball, as the radius increases. More generally, we prove a
similar result for the Galois groups associated with elements in any connected
semisimple linear algebraic group defined and simple over a number field .
Our method is based on the abstract large sieve method developed by Kowalski,
and the study of Galois groups via reductions modulo primes developed by Jouve,
Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice
point counting result, and a non-concentration phenomenon for lattice points in
algebraic subvarieties of the group variety, both established previously by the
authors. The results answer a question posed by Rivin and by Jouve, Kowalski
and Zywina, who have considered Galois groups of random products of elements in
algebraic groups.Comment: submitte
Best possible rates of distribution of dense lattice orbits in homogeneous spaces
The present paper establishes upper and lower bounds on the speed of
approximation in a wide range of natural Diophantine approximation problems.
The upper and lower bounds coincide in many cases, giving rise to optimal
results in Diophantine approximation which were inaccessible previously. Our
approach proceeds by establishing, more generally, upper and lower bounds for
the rate of distribution of dense orbits of a lattice subgroup in a
connected Lie (or algebraic) group , acting on suitable homogeneous spaces
. The upper bound is derived using a quantitative duality principle for
homogeneous spaces, reducing it to a rate of convergence in the mean ergodic
theorem for a family of averaging operators supported on and acting on
. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of in the
automorphic representation on . We show that the rate
is best possible when the representation in question is tempered, and show that
the latter condition holds in a wide range of examples
Manin's and Peyre's conjectures on rational points and adelic mixing
Let X be the wonderful compactification of a connected adjoint semisimple
group G defined over a number field K. We prove Manin's conjecture on the
asymptotic (as T\to \infty) of the number of K-rational points of X of height
less than T, and give an explicit construction of a measure on X(A),
generalizing Peyre's measure, which describes the asymptotic distribution of
the rational points G(K) on X(A). Our approach is based on the mixing property
of L^2(G(K)\G(A)) which we obtain with a rate of convergence.Comment: to appear in Ann. Sci. Ecole Norm. Su
Rigidity of group actions on homogeneous spaces, III
Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice
{\Gamma} acts on the left strictly conservatively. The following rigidity
results are obtained: morphisms, factors and joinings defined apriori only in
the measurable category are in fact algebraically constrained. Arguing in an
elementary fashion we manage to classify all the measurable {\Phi} commuting
with the {\Gamma}-action: assuming ergodicity, we find they are algebraically
defined