Related papers: There are at most finitely many singular moduli th…

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every…

We study $\mathcal I$-maximal eventually different families of functions from the set of natural numbers into itself where $\mathcal I$ is an arbitrary ideal on the set of natural numbers that includes the ideal of all finite sets…

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we prove that for an arbitrary finite group scheme G, and for any fixed integer n > 0, there are only finitely many…

We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional…

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more…

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be weakly $S$-prime if…

In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in…

Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the…

We propose to associate to a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequence in terms of the Mellin…

Let $k$ be an algebraically closed field and let $b$ and $n$ be integers with $n\geq 3$ and $1\leq b \leq n-1.$ Consider the moduli space $X$ of hypersurfaces in $\mathbb{P}^n_k$ of fixed degree $l$ whose singular locus is at least…

We introduce the notion of a strong minuscule element, and prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root. In addition, we enumerate the strong…

We give density results for automorphic representations of Hilbert modular groups. In particular, we show that there are infinitely many automorphic representations that have a prescribed discrete series factor at some (but not all) real…

In this paper we prove the existence of an infinite dimensional graded super-module for the finite sporadic Thompson group $Th$ whose McKay-Thompson series are weakly holomorphic modular forms of weight $\frac 12$ satisfying properties…

We show that in the extended modular group PGL(2,Z) there are exactly seven finite subgroups up to conjugacy; three subgroups of size 2, one subgroup each of size 3, 4, and 6, and the trivial subgroup of size 1.

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We…

Let $G$ be a finite group and let $F$ be a finite field of characteristic $2$. We introduce \emph{$F$-special subgroups} and \emph{$F$-special elements} of $G$. In the case where $F$ contains a $p$th primitive root of unity for each odd…

In Finite Group Modular Representation Theory, the basic objects are the indecomposable and simple modules. This paper offers a new classification of these objects that refines the Green Theory Classification of indecomposable and simple…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

For certain negative rational numbers k0, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter k = k0. It was shown by de…