Related papers: On the rank varieties of some simple modules for s…

This paper focuses on the rank varieties for modules over a group algebra $\mathbb{F}E$ where $E$ is an elementary abelian $p$-group and $p$ is the characteristic of an algebraically closed field $\mathbb{F}$. In the first part, we give a…

We show that the simple modules of the Rouquier blocks of symmetric groups, in characteristic $p$ and having $p$-weight $w$ with $w < p$, have a common complexity $w$, and that when $p$ is odd, $D^{(p+1,1^{p-1})}$ has complexity 1, while…

We concern the VIGRE's conjecture; namely the complexity of a Specht module is the p-weight of the corresponding partition if and only if the partition is not p by p. In abelian defect case, we calculate the cohomological variety of the…

Let $p$ be a prime. In this paper, we compute complexities of some simple modules of symmetric groups labelled by two-part partitions. Most of the simple modules considered here are contained in the $p$-blocks with non-abelian defect…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

We study the category $\mathcal{M}_{\mathfrak{sl}(m|1)}(k|k)$ of $\mathcal U(\mathfrak h)\text{-free}$ $\mathcal U(\mathfrak{sl}(m|1))$-modules of rank $k$ in each parity (rank $(k|k)$), where $k\in\mathbb{Z}_{\geq1}$. We construct an…

Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{sl}(V)$ of maximal dimension and let $\mathfrak{n} \subset \mathfrak{p}$ be the corresponding nilradical. In this paper we classify the set of $\mathfrak{sl}(V)$-modules whose…

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for $p$-elementary abelian groups $E_r$ of rank $r$ over a field of characteristic $p > 0$, we introduce the notions of modules with constant $d$-radical rank…

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. For $\pi$ an admissible smooth representation of $\operatorname{GL}_2(K)$ over $\mathbb{F}$ satisfying…

A finite abelian $p$-group having an automorphism $x$ such that $1+\ldots+x^{p-1}=0$, can be viewed as a module over an appropriate discrete valuation ring $\mathcal{O}$ containing $\mathbb{Z}_p$ (the ring of $p$-adic integer). This yields…

We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a finite set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$…

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal…

We investigate integral forms of simple modules of symmetric groups over fields of characteristic $0$ labelled by hook partitions. Building on work of Plesken and Craig, for every odd prime $p$, we give a set of representatives of the…

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to…

We define a rank variety for a module of a noncocommutative Hopf algebra $A = \Lambda \rtimes G$ where $\Lambda = k[X_1, ..., X_m]/(X_1^{\ell}, ..., X_m^{\ell})$, $G = ({\mathbb Z}/\ell{\mathbb Z})^m$, and $\text{char} k$ does not divide…

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the…

Let $A$ be an abelian variety defined over a number field $K$. If $\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let $A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction modulo $\mathfrak{p}$. We prove…

Let A be a geometrically simple abelian variety over a number field k, let X be a subgroup of A(k) and let P be an element of A(k). We prove that if P belongs to X modulo almost all primes of k then P already belongs to X.

We introduce the $N=2$ Lie conformal superalgebras ${\frak {K}}(p)$ of Block type, and classify their finite irreducible conformal modules for any nonzero parameter $p$. where $p$ is a nonzero complex number. In particular, we show that…

The rank rk(G) of a profinite group G is the supremum of d(H), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure…