We study product theorems for matrix spaces. In particular, we prove the following theorems.

Theorem 1. For all $\varepsilon>0$, there is $\delta>0$ such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$ is a finite set, then either $A$ intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3|>|A|^{1+\delta}$.

Theorem 2. Let $A$ be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$ is contained in a virtually abelian subgroup, or $|A^3|>c|A|^{1+\delta}$ for some absolute constant $\delta>0$.

Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$ is the $3$-fold product set of $A$.

One of the main obstacles in applying converse theorems to prove new cases of functoriality is that of stability of $\gamma$-factors for a certain class of $L$-functions obtained from the ‘Langlands–Shahidi’ method, where the $\gamma$-factors are defined inductively by means of ‘local coefficients’. The problem then becomes that of stability of local coefficients upon twisting the representation by a highly ramified character. In this paper we first establish that the inverses of certain local coefficients are, up to an abelian $\gamma$-factor, genuine Mellin transforms of partial Bessel functions of the type we analysed in our previous paper. The second main result is then the resulting stability of the local coefficients in this situation, which include all the cases of interest for functoriality. Hopefully, the analysis given here will open the door to a proof of the general stability and the equality of $\gamma$-factors obtained from different methods through integration over certain quotient spaces whose generic fibres are closed. They do not seem to have been studied before in any generality.

Nous démontrons des théorèmes d'immersion holomorphe dans un espace projectif complexe pour des variétés kählériennes non compactes et des laminations par variétés complexes qui admettent un fibré en droites holomorphe strictement positif. En particulier, nous montrons que sur une lamination compacte par surfaces de Riemann, les fonctions méromorphes séparent les points si et seulement si aucun cycle feuilleté n'est homologue à $0$.

We prove holomorphic immersion theorems in a finite dimensional complex projective space for kählerian non-compact manifolds and for laminations by complex manifolds that carry a line bundle of positive curvature. In particular, we prove that on a Riemann surfaces lamination of a compact space, the space of meromorphic functions separates points if and only if every foliation cycle is non homologous to $0$.

The pseudovariety $\mathbf{DG}$ of all finite monoids all of whose regular $\mathcal{D}$-classes are subgroups is shown to be local, that is, it is verified that the pseudovariety $g\mathbf{DG}$ of finite categories generated by $\mathbf{DG}$ coincides with the pseudovariety $\ell\mathbf{DG}$ of all finite categories whose local monoids all belong to $\mathbf{DG}$. Yet more general statements of this kind are deduced, yielding results such as that, for every prime number $p$, the pseudovariety $\mathbf{DG}_p$ of all finite monoids all of whose regular $\mathcal{D}$-classes are $p$-groups is local, or that the pseudovarieties $\mathbf{DG}_{\mathrm{sol}}$ and $\mathbf{DG}_{\mathrm{nil}}$ of all finite monoids all of whose regular \mbox{$\mathcal{D}$-classes} are, respectively, solvable groups and nilpotent groups are local.

Nous proposons une façon simple de compter le nombre de $G$-chtoucas de Drinfeld avec modifications arbitraires, inspirée par les travaux de Kottwitz sur les variétés de Shimura.

We propose a simple method to count the number of Drinfeld $G$-shtukas with arbitrary modifications which is inspired by works by Kottwitz on Shimura varieties.