arXiv subjects
Publications and source records attributed to Kristian Holm.
Let $n \geqslant 2$. We prove that, up to conjugation, $\mathrm{Sp}_{2n} (\mathbf{Z})$ is the unique lattice in $\mathrm{Sp}_{2n} (\mathbf{R})$ of the smallest covolume.
Let $\mathbb{K} = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number field of class number $1$ and $\mathcal{O}_{\mathbb{K}}$ its ring of integers. We study a family of Hecke $L$-functions associated to angular characters on the non-zero ideals of $\mathcal{O}_{\mathbb{K}}$. Using the powerful Ratios Conjecture (RC) due to Conrey, Farmer, and Zirnbauer, we compute a conditional asymptotic for the average $1$-level density of the zeros of this family, including terms of lower order than the main term in the Katz-Sarnak Density Conjecture coming from random matrix theory. We also prove an unconditional result about the $1$-level density, which agrees with the RC prediction when our test functions have Fourier transforms with support in $(-1,1)$.
We use a method developed by Björklund and Gorodnik to show a central limit theorem (as $T$ tends to $\infty$) for the counting functions $\# \left( Λ\cap Ω_T \right)$ where $Λ$ ranges over the space $Y_{2d}$ of symplectic lattices in $\mathbb{R}^{2d}$ ($d \geqslant 4$). Here $\lbrace Ω_T \rbrace_T$ is a certain family of bounded domains in $\mathbb{R}^{2d}$ that can be tessellated by means of the action of a diagonal semigroup contained in $\mathrm{Sp}(2d, \mathbb{R})$. In the process we obtain new $L^p$ bounds on a certain height function on $Y_{2d}$ originally introduced by Schmidt.
Following Södergren, we consider a collection of random variables on the space $X_n$ of unimodular lattices in dimension $n$: Normalizations of the angles between the $N = N(n)$ shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions evaluated at these random variables in the regime where $N$ tends to infinity with $n$ at the rate $N = o \left( n^{1/6} \right)$. Our main result is that as $n \longrightarrow \infty$, these random variables exhibit a joint Poissonian and Gaussian behaviour.