Related papers: Asymptotic behavior of modular representations ove…

We classify, up to isomorphism, the $\mathbb{Z}_pG$-modules of rank $1$ (i.e., the quotients of $\mathbb{Z}_pG$) for $G$ cyclic of order $p$, where $\mathbb{Z}_p$ is the ring of $p$-adic integers. This allows us in particular to determine…

We introduce the concept of quasirational relation modules for discrete and pro-$p$ presentations of discrete and pro-$p$ groups and show that aspherical presentations and their subpresentations are quasirational. In the pro-$p$-case…

The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic…

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having…

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We…

In order to describe the asymptotic behaviour of a vanishing period in a one parameter family we introduce and use a very simple algebraic structure : regular geometric (a,b)-modules generated (as left $\A-$modules) by one element. The idea…

Let p > 2 be a prime number and let N be any rational integer. We consider the p-class groups Cl(L), Cl(M) of the fields $L:=Q(N^{1/p})$ and $M:=Q(N^{1/p},\mu_p)$, by comparison with the p-torsion groups T(L) and T(M) of the abelian…

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear…

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change,…

Let $A$ be a commutative ring, let $k\in\mathbb{Z}^+$, and let $\vec{s}=(n_1,\dots,n_k)\in(\mathbb{Z}^+)^k$ with $n=\min_a(n_a)-1$. We attach to $\vec{s}$ a diagonal simplicial tensor module $X_\bullet(\vec{s};A)$ whose $p$-simplices are…

There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of…

Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension $K/\Q$. Let $p$ be an odd prime split in $K/\Q$ and $l\neq p$ an odd unramified prime. We prove…

In this article we show how certain irreducible unitary representation $ \Pi_\lambda $ of the twisted Heisenberg group $ \He_\lambda^n(\C)$ leads to the twisted modulation spaces $ M_\lambda^{p,q}(\R^{2n}).$ These $ \Pi_\lambda $ also turn…

In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties…

In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the…

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group…

We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and…

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

Groups with a non-cyclic Sylow $p$-subgroup have too many representations over a field of characteristic~$p$ to describe them fully. A~natural question arises, whether the world of representations coming from algebraic varieties with a…

It is proven that for any representation over a field of characteristic 0 of the non-abelian semidirect product of a cyclic group of prime order p and the group of order 3 the corresponding algebra of polynomial invariants is generated by…