Related papers: Lifting subgroups of $\mathrm{PSL}_2$ to $\mathrm{…

Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of ${\rm PSL_2}(K)$ to be discrete, where either $K=\mathbb{Q}_p$ or $G$…

We present a practical algorithm which, given a non-archimedean local field $K$ and any two elements $A,B\in {\rm SL_2}(K)$, determines after finitely many steps whether or not the subgroup $\langle A, B \rangle\le {\rm SL_2}(K)$ is…

We consider the group $SL_2(K)$, where $K$ is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of $SL_2 (K)$ is larger than the depth of the corresponding Langlands parameter,…

It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low…

The Serre conjecture II predicts that every torsor under a semisimple, simply connected, algebraic group over a field of cohomological dimension at most 2 and of degree of imperfection at most 1 has a rational point. We generalize this…

If the characteristic of a field $k$ is odd any infinitesimal group scheme of $PGL_{2,k}$ lifts to $SL_{2,k}$. In this paper, we prove that this is not true in characteristic $2$ and we give a complete description, up to isomorphism, of…

The aim of this paper is to describe all definable subgroups of SL2(K), for K a p-adically closed field. We begin by giving some "frame subgroups" which contain all nilpotent or solvable subgroups of SL2(K). A complete description is givien…

Let $G$ be a split reductive group over a local field $\bK$, and let $G((t))$ be the corresponding loop group. In \cite{GK} we have introduced the notion of a representation of (the group of $\bK$-points) of $G((t))$ on a pro-vector space.…

We complete the description of automorphism groups of all nonsplit extensions of elementary abelian $2$-groups by $PSL_2(q)$, with $q$ odd, for an irreducible induced action. An application of this result to the theory of $\pi$-submaximal…

Let A be a locally compact group topologically generated by d elements and let k>d. Consider the action, by pre-composition, of Aut(F_k) on the set of marked, k-generated, dense subgroups D_{k,A} := {h:F_k --> A | h(F_k) is dense in A}. We…

We show that any lattice in $\mathrm{SL}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product $\mathbb{Z}^2*\mathbb{Z}$. To our knowledge, the subgroups we construct give the…

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are…

The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such group explicitly by lifting from Maass cusp forms…

Farre, Pozzetti and Viaggi proved that any (d-k)-hyperconvex subgroup of PSL(d,C) is virtually isomorphic to a convex cocompact Kleinian group and that its k-th simple root critical exponent is at most 2. We show that a (d-k)-hyperconvex…

It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary…

Let $G$ be the simple algebraic group $SL_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. In this paper, we find the second cohomology of all irreducible representations of $G$

We describe all of the irreducible polynomial $\mathbb{F}_p\mathrm{SL}_2(p^r)$ representations which lift to $(\mathbb{Z}/p^s\mathbb{Z})\mathrm{SL}_2(p^r)$ representations for $s>1$, observing that they almost never do. We also show that…

We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups $(\mathrm{Sp}_{2n}, \mathrm{O}(V))$ defined over a local nonarchimedean field…

For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on…

We consider lifting of mod p representations to mod p^2 representations in the setting of representations of (i) finite groups; (ii) absolute Galois groups of abstract fields; and (iii) absolute Galois groups of local and global fields.