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

Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.

Let $F$ be a field with characteristic $\neq 2$. We show that $F$ is a nonrigid field if and only if certain small 2-groups occur as Galois groups over $F$. These results provide new "automatic realizability" results for Galois groups over…

Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\varepsilon$-fixed points. Then there…

We introduce the notion of a bicollapsible 2-complex. This allows us to generalize the hyperbolicity of one-relator groups with torsion to a broader class of groups with presentations whose relators are proper powers. We also prove that…

Let $\mathrm{G}$ be a symplectic or a split orthogonal group over a local non-archimedean field $\mathrm{F}$. A prime $\ell$ is called banal with respect to $\mathrm{G}$ if it does not divide the cardinality of the $k$-points of…

We investigate closed subgroups $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose modulo-$2$ images coincide with the image $\mathfrak{S}_{2g + 1} \subseteq \mathrm{Sp}_{2g}(\mathbb{F}_2)$ of $S_{2g + 1}$ or the image $\mathfrak{S}_{2g +…

Let K be an algebraically closed field. We prove that a polynomial K-derivation $D$ in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is…

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the…

The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,\C) or $A_1$. This leads to new types of both…

Using the dilaton scalar and axion pseudoscalar fields we construct a number of scalars and differential forms which are symmetric under the $\mathbf{Z}_2$-subgroup of the group $SL(2, \mathbf{R})$. These invariants enable us to establish…

Let k be a local field and G the set of k-points of a connected semisimple algebraic k-group of rank one. We describe all torsion-free discrete subgroups of G\times G acting properly discontinuously on G by left and right multiplication. To…

We develop the theory of $L^2$-torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a…

We define consistent finite-superfields reductions of the $N=1,2$ super-KP hierarchies via the coset approach we already developped for reducing the bosonic KP-hierarchy (generating e.g. the NLS hierarchy from the $sl(2)/U(1)-{\cal KM}$…

We prove upper bounds for the average size of the $\ell$-torsion $\text{Cl}_K[\ell]$ of the class group of $K$, as $K$ runs through certain natural families of number fields and $\ell$ is a positive integer. We refine a key argument, used…

In this article we provide simple and provable bounds on the size and shape of the locus of discrete subgroups of $\mathsf{PSL}(2,\mathbb{C})\cong \operatorname{Isom}^+(\mathbb{H}^3)$ which split as a free product of cyclic groups…

Let $G:=G_2(K)$ be a simple algebraic group of type $G_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. Let $\sigma$ denote a standard Frobenius automorphism of $G$ such that $G_\sigma\cong G_2(q)$ with $q\geq 4$.…

We discuss the following topics: n-dimensional local fields and adelic groups; harmonic analysis on local fields and adelic groups for two-dimensional schemes (function spaces, Fourier transform, Poisson formula); representations of…

Let F be a local field of characteristic 0. The Breuil-Schneider conjecture for GL_2(F) predicts which locally algebraic representations of this group admit an integral structure. We extend the methods of [K-dS12], which treated smooth…

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$,…