Singular Vectors on Manifolds over totally real Number Fields

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $n \ge 2$, we prove that a finite volume complex hyperbolic $n$-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps be…

Singular vectors on manifolds and fractals

We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of $\mathbb{R}^n$, in particular on any analytic submanifold of $\mathbb{R}^n$ of dimension $\ge 2$ which is not contained in a proper rational affine subspace.

On the growth of the number of totally geodesic surfaces in some hyperbolic $3$-manifolds

Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed totally geodesic surfaces in $Γ_{-d}\backslash \mathbb{H}^3$ having area less than $X$.

The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "s…

Hoffmann's conjecture for totally singular forms of prime degree

One of the most significant discrete invariants of a quadratic form $ϕ$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $ϕ$ under scalar extension to arbitrary overfields of $k$. A similarly important, but more accessible variant of this notion is that of the Knebusch splitting pattern of $ϕ$, which captures the isotropy behaviour of $ϕ$ as one passes over a certain prescribed tower of $k$-overfields. In this paper, we determine all possible values of this latter invariant in the case where $ϕ$ is totally singul…

拟线性p-形式与全奇异二次型的Vishik等价与相似性

Vishik equivalence and similarity of quasilinear $p$-forms and totally singular quadratic forms

本文将Vishik等价概念推广到拟线性p-形式，证明Vishik等价的p-形式一般不必相似，并找到若干全奇异二次型族使该问题成立。