16 research outputs found
Asymptotic distribution of values of isotropic quadratic forms at -integral points
We prove an analogue of a theorem of Eskin-Margulis-Mozes: suppose we are
given a finite set of places over containing the archimedean
place and excluding the prime , an irrational isotropic form
of rank on , a product of -adic intervals , and
a product of star-shaped sets.
We show that unless and is split in at least one place,
the number of -integral vectors
satisfying simultaneously for is
asymptotically given by as
goes to infinity, where is the product of Haar measures
of the -adic intervals .
The proof uses dynamics of unipotent flows on -arithmetic homogeneous
spaces; in particular, it relies on an equidistribution result for certain
translates of orbits applied to test functions with a controlled growth at
infinity, specified by an -arithmetic variant of the -function
introduced in the work of Eskin, Margulis, Mozes, and an -arithemtic version
of a theorem of Dani-Margulis.Comment: The article will appear in Journal of Modern Dynamics. In the revised
version, several typos have been fixed. Moreover, another preprint
(arXiv:1605.02436) has been incorporate