29 research outputs found
Arithmetic groups with isomorphic finite quotients
Two finitely generated groups have the same set of finite quotients if and
only if their profinite completions are isomorphic. Consider the map which
sends (the isomorphism class of) an S-arithmetic group to (the isomorphism
class of) its profinite completion. We show that for a wide class of
S-arithmetic groups, this map is finite to one, while the the fibers are of
unbounded size.Comment: Minor revisions. To appear in Journal of Algebr
Integer points and their orthogonal lattices
Linnik proved in the late 1950's the equidistribution of integer points on
large spheres under a congruence condition. The congruence condition was lifted
in 1988 by Duke (building on a break-through by Iwaniec) using completely
different techniques. We conjecture that this equidistribution result also
extends to the pairs consisting of a vector on the sphere and the shape of the
lattice in its orthogonal complement. We use a joining result for higher rank
diagonalizable actions to obtain this conjecture under an additional congruence
condition.Comment: 15 pages, and 6 pages of appendix, (Appendix by Ruixiang Zhang
Simultaneous supersingular reductions of CM elliptic curves
We study the simultaneous reductions at several supersingular primes of
elliptic curves with complex multiplication. We show -- under additional
congruence assumptions on the CM order -- that the reductions are surjective
(and even become equidistributed) on the product of supersingular loci when the
discriminant of the order becomes large. This variant of the equidistribution
theorems of Duke and Cornut-Vatsal is an(other) application of the recent work
of Einsiedler and Lindenstrauss on the classification of joinings of
higher-rank diagonalizable actions.Comment: 46 pages. Revised according to the referee's comment
Integer points on spheres and their orthogonal grids
The set of primitive vectors on large spheres in the euclidean space of
dimension d>2 equidistribute when projected on the unit sphere. We consider
here a refinement of this problem concerning the direction of the vector
together with the shape of the lattice in its orthogonal complement. Using
unipotent dynamics we obtained the desired equidistribution result in dimension
d>5 and in dimension d=4,5 under a mild congruence condition on the square of
the radius. The case of d=3 is considered in a separate paper.Comment: 20 pages. To appear in the Journal of the London Mathematical Societ
On metric diophantine approximation in matrices and Lie groups
We study the diophantine exponent of analytic submanifolds of the space of m
by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis.
We identify a family of algebraic obstructions to the extremality of such a
submanifold, and give a formula for the exponent when the submanifold is
algebraic and defined over the rationals. We then apply these results to the
determination of the diophantine exponent of rational nilpotent Lie groups.Comment: Theorem 3.4 was modified. To appear in Comptes Rendus Mathematiqu
Planes in four space and four associated CM points
To any two-dimensional rational plane in four-dimensional space one can
naturally attach a point in the Grassmannian Gr(2,4) and four lattices of rank
two. Here, the first two lattices originate from the plane and its orthogonal
complement and the second two essentially arise from the accidental local
isomorphism between SO(4) and SO(3)xSO(3). As an application of a recent result
of Einsiedler and Lindenstrauss on algebraicity of joinings we prove
simultaneous equidistribution of all of these objects under two splitting
conditions