Related papers: The canonical subgroup: a "subgroup-free" approach
Here we initiate a program to study relationships between finite groups and arithmetic-geometric invariants in a systematic way. To do this we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock…
We use methods of the general theory of congruence and *congruence for complex matrices--regularization and cosquares-to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices…
For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the…
We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $\delta_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain…
The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is desirable to…
We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational…
Various descending chains of subgroups of a finite permutation group can be used to define a sequence of `basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the…
We develop the fundamental theory to study cubical isometry groups as totally disconnected, locally compact groups. We show how cubical isometries are determined by their local actions and how this can be applied in explicit constructions.…
In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary…
A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve $C$ in a variety of general type is bounded from…
It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define…
We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve…
Let $N$ be a positive integer and $\Gamma$ be a subgroup of $\mathrm{SL}_2(\mathbb{Z})$ containing $\Gamma_1(N)$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$ in $K$. Under some…
The JSJ decomposition and the Makanin-Razborov diagram were proved to be essential in studying varieties over free groups, semigroups and associative algebras. In this paper we suggest a unified conceptual approach to the applicability of…
The theory of the Oukounkov body is a useful tool for studying the asymptotic behaviour of the canonical ring of a line bundle over a projective manifold. In this note, combined with the algebraic reduction, we study the asymptotic…
We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of…
We define and study a class of subshifts of finite type (SFTs) defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. The main…
Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body…
We study topological aspects of the category of abstract Cuntz semigroups, termed Cu. We provide a suitable setting in which we are able to uniformly control how to approach an element of a Cu-semigroup by a rapidly increasing sequence.…
We investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups,…