This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski’s classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre–Fenchel type transforms. During the course of the proof, we obtain a refined structure theorem for the movable cone of curves.

This paper answers a question raised by Grothendieck in 1970 on the “Grothendieck closure” of an integral linear group and proves a conjecture of the first author made in 1980. This is done by a detailed study of the congruence topology of arithmetic groups, obtaining along the way, an arithmetic analogue of a classical result of Chevalley for complex algebraic groups. As an application we also deduce a group theoretic characterization of thin subgroups of arithmetic groups.

We generalize Deligne’s approach to tame geometric class field theory to the case of a relative curve, with arbitrary ramification.

We propose another proof of geometric class field theory for curves by considering blow-ups of symmetric products of curves.

In this paper we prove that for any connected reductive algebraic group $G$ and a large enough prime $l$, there are continuous homomorphisms

$$Gal\left(\bar{\mathbb{Q}}∕\mathbb{Q}\right)\to G\left({\bar{\mathbb{Q}}}_l\right)$$

with Zariski-dense image, in particular we produce the first such examples for ${SL}_n,{Sp}_{2n},{Spin}_n,E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a mod-$l$ representation of $Gal\left(\bar{\mathbb{Q}}∕\mathbb{Q}\right)$ related to the Weyl group of $G$ and use a variation of Stefan Patrikis’ generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.

Let $f$ be a cuspidal eigenform (holomorphic or Maass) for the congruence group ${\mathrm{\Gamma }}_0\left(N\right)$ with $N$ square-free. Let $p$ be a prime and let $\chi$ be a primitive character of modulus $p^{3r}$. We shall prove the Weyl-type subconvex bound

$$L\left(\frac{1}{2}+it,f\otimes \chi \right){\ll }_{f,t,\epsilon }{p}^{r+\epsilon },$$

where $\epsilon >0$ is any positive real number.

We show that certain one-dimensional hypergeometric differential systems underlie objects of the category of irregular mixed Hodge modules, which was recently introduced by Sabbah, and compute the irregular Hodge filtration for them. We also provide a comparison theorem between two different types of Fourier–Laplace transformation for algebraic integrable twistor $\mathcal{D}$-modules.

We show that any polarized K3 surface supports special Ulrich bundles of rank $2$.

We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety $G$ which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. We show that an irreducible curve $W$ in $G$ meets this set Zariski-densely only if $W$ lies in a fiber of the family or is a translate of a Ribet curve by a multiplicative section. We further deduce from this result a proof of the Zilber–Pink conjecture (over number fields) for the mixed Shimura variety attached to the threefold $G$, when the parameter space is the universal one.

Let $F$ be a nonarchimedean local field and let $T$ be a torus over $F$. With ${\mathcal{T}}^{NR}$ denoting the Néron–Raynaud model of $T$, a result of Chai and Yu asserts that the model ${\mathcal{T}}^{NR}{\times }_{{\mathfrak{O}}_F}{\mathfrak{O}}_F∕{\mathfrak{p}}_F^m$ is canonically determined by $\left({Tr}_l\left(F\right),\mathrm{\Lambda }\right)$ for $l\gg m$, where ${Tr}_l\left(F\right)=\left({\mathfrak{O}}_F∕{\mathfrak{p}}_F^l,{\mathfrak{p}}_F∕{\mathfrak{p}}_F^{l+1},ϵ\right)$ with $ϵ$ denoting the natural projection of ${\mathfrak{p}}_F∕{\mathfrak{p}}_F^{l+1}$ on ${\mathfrak{p}}_F∕{\mathfrak{p}}_F^l$, and $\mathrm{\Lambda }:=X_{\ast }\left(T\right)$. In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat–Tits building of a connected reductive group over $F$.

Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most

$$2n{log}_{2}n+4n$$

is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $\frac{n^2}{3}+\frac{2}{3}$ bound by Paz (1984) and an $O\left(n^{3∕2}\right)$ bound of Pappacena (1997).