Related papers: Convolution and square in abelian groups II

In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent…

For a finite non-abelian group $G$ let $\rat(G)$ denote the largest ratio of degrees of two nonlinear irreducible characters of $G$. We prove that the number of non-abelian composition factors of $G$ is bounded above by…

We discuss functions from the edges and vertices of a directed graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced and form an Abelian group. We study this group in two cases:…

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring…

We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit…

We provide an explicit construction of the arboreal Galois group for the postcritically finite polynomial $f(z) = z^2 +c$, where $c$ belongs to some arbitrary field of characteristic not equal to $2$. In this first of two papers, we…

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number…

We compute the central critical value of the triple product $L$-function associated to three cusp forms $f_1,f_2,f_3$ with trivial character for groups $\Gamma_0(N_i)$ with square free levels $N_i$ not all of which are $1$ and weights $k_i$…

Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $\partial X$. We define the critical…

For a finite group $G$, we consider the zeta function $\zeta_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m…

In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential…

In this paper, we compute the essential $l$-dimension of the finite groups of classical Lie type for odd primes $l$ not equal to the defining prime, specifically the general linear groups, the symplectic groups, the orthogonal groups, and…

We classify the almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0 \ltimes_A \mathbb R^{2n-1}$ admitting complex or symplectic structures. The matrix $A\in M(2n-1,\mathbb R )$ encodes the adjoint action of $e_0$ on the abelian ideal…

We count the number of critical points of a modular form with real Fourier coefficients in a $\gamma$-translate of the standard fundamental domain $\mathcal{F}$ (with $\gamma\in \mathrm{SL}_2(\mathbb{Z})$). Whereas by the valence formula…

We define a new invariant for a $p$-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete…

A polynomial $P \in \mathbb{C}[z_1, \ldots, z_d]$ is strongly $\mathbb{D}^d$-stable if $P$ has no zeroes in the closed unit polydisc $\overline{\mathbb{D}}^d.$ For such a polynomial define its spectral density function as…

Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the…

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper…

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $\Gamma$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup…

Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known…