We study the value distribution of the Epstein zeta function $E_n(L,s)$ for $0<s<\frac{n}{2}$ and a random lattice $L$ of large dimension $n$. For any fixed $c\in (1/4,1/2)$ and $n\to \infty$, we prove that the random variable $V_n^{-2c}{E}_n(\cdot ,cn)$ has a limit distribution, which we give explicitly (here $V_n$ is the volume of the $n$-dimensional unit ball). More generally, for any fixed $\epsilon >0$, we determine the limit distribution of the random function $c\mapsto V_n^{-2c}{E}_n(\cdot ,cn)$, $c\in [1/4+\epsilon ,1/2-\epsilon ]$. After compensating for the pole at $c=1/2$, we even obtain a limit result on the whole interval $[1/4+\epsilon ,1/2]$, and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function $h_n\left(L\right)$ of the flat torus ${\mathbb{R}}^n/L$: the random variable $n\left\{h_n\right(L)-(log\left(4\pi \right)-\gamma +1\left)\right\}+logn$ has a limit distribution as $n\to \infty$, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice $L\subset {\mathbb{R}}^n$ for which $E_n(L,s)$ has no zeros in $(0,\infty )$.

We construct holomorphic families of proper holomorphic embeddings of ${\mathbb{C}}^k$ into ${\mathbb{C}}^n$ ($0<k<n-1$), so that for any two different parameters in the family, no holomorphic automorphism of ${\mathbb{C}}^n$ can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of ${\mathbb{C}}^n$, we derive the existence of families of holomorphic ${\mathbb{C}}^{\ast }$-actions on ${\mathbb{C}}^n$ ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of ${\mathbb{C}}^{\ast }$-actions on ${\mathbb{C}}^n$ (with prescribed linear part at a fixed point).

We classify compact complex surfaces which contain a Zariski-open subset whose universal covering is the cylinder $\mathbb{D}\times \mathbb{C}$.

We introduce new methods from $\mathfrak{p}$-adic integration into the study of representation zeta functions associated to compact $p$-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of $p$-adic analytic pro-$p$ groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact $p$-adic analytic groups, we exhibit a link between the relevant $\mathfrak{p}$-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension.

Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups ${SL}_3\left(\mathfrak{o}\right)$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$, and of the groups ${SU}_3(\mathfrak{O},\mathfrak{o})$, where $\mathfrak{O}$ is an unramified quadratic extension of $\mathfrak{o}$. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type $A_2$. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type $A_2$ defined over number fields.