We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation with initial data below $L^2$. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of the NLS equation almost surely for the initial data in the support of the canonical Gaussian measures on $H^s\left(\mathbb{T}\right)$ for each $s>-\frac{1}{3}$, and global well-posedness for each $s>-\frac{1}{12}$.

We prove that every algebraic curve $X/\overline{\mathbf{Q}}$ is birational over $\mathbf{C}$ to a Teichmüller curve. This result is a corollary of our main theorem, which asserts that most finite-index subgroups of $SL(2,\mathbf{Z})$ are Veech groups.

In this article, we prove the following conjecture by Lubotzky. Let $G={\mathbb{G}}_0\left(K\right)$, where $K$ is a local field of characteristic $p\ge 5$ and where ${\mathbb{G}}_0$ is a simply connected, absolutely almost simple $K$-group of $K$-rank at least 2. We give the rate of growth of

$${\rho }_x\left(G\right):=|\{\Gamma \subseteq G|\Gamma \text{ a lattice in }G,\mathrm{vol}(G/\Gamma )\le x\}/\sim |,$$

where ${\Gamma }_1\sim {\Gamma }_2$ if and only if there is an abstract automorphism $\theta$ of $G$ such that ${\Gamma }_2=\theta \left({\Gamma }_1\right)$. We also study the rate of subgroup growth $s_x(\Gamma )$ of any lattice $\Gamma$ in $G$. As a result, we show that these two functions have the same rate of growth, which proves Lubotzky’s conjecture. Along the way, we also study the rate of growth of the number of equivalence classes of maximal lattices in $G$ with covolume at most $x$.

We describe all the discrete subgroups of $Ad\left({\mathbb{G}}_0\right)\left(F\right)⋊Aut\left(F\right)$ that act transitively on the set of vertices of $\mathfrak{B}=\mathfrak{B}(F,{\mathbb{G}}_0)$, the Bruhat–Tits building of a pair $(F,{\mathbb{G}}_0)$ of a characteristic $0$ nonarchimedean local field, and a simply connected, absolutely almost simple $F$-group if $\mathfrak{B}$ is of dimension at least $4$. In fact, we classify all such maximal subgroups. We show that there are exactly eleven families of such subgroups and explicitly construct them. Moreover, we show that four of these families act simply transitively on the vertices. In particular, we show that there is no such action if either the dimension of the building is larger than $7$, if $F$ is not isomorphic to ${\mathbb{Q}}_p$ for some prime $p$, or if the building is associated to ${\mathbb{SL}}_{n,D_0}$, where $D_0$ is a noncommutative division algebra. Along the way we also give a new proof of the Siegel–Klingen theorem on the rationality of certain Dedekind zeta functions and $L$-functions.