The Density of a family of monogenic number fields

A monogenic polynomial $f$ is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime $q$, using the Chebotarev density theorem, we will show the density of primes $p$, such that $t^q-p$ is monogenic, is bigger or equal than $(q-1)/q$. We will also prove that, when $q=3$, the density of primes $p$, which $\mathbb{Q}(\sqrt[3]{p})$ is non-monogenic, is at least 1/9.Comment: Various corrections. Submitte

Mertens's theorem for splitting primes and more

Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is also quite adaptable, and it is readily combined with well-known results from prime number theory. We demonstrate this by proving a version of the theorem for primes in arithmetic progressions with uniformity in the modulus, as well as a non-abelian analogue of this.Comment: This is a survey pape

On equality of ranks of local components of automorphic representations

We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li, Dvorsky--Sahi, and Kobayashi--Savin. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian unipotent radicals, our proof works uniformly for all of the (classical as well as exceptional) groups under consideration. Our result is an extension of the statement known for several semisimple groups that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal.Comment: Remark 9.6 is modified. A corrigendum will also appear in International Mathematical Research Notice

On the Erdos-Ko-Rado property for finite Groups

Let a finite group $G$ act transitively on a finite set $X$. A subset $S\subseteq G$ is said to be {\it intersecting} if for any $s_1,s_2\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\it weak Erd\H{o}s-Ko-Rado} property, if the cardinality of any intersecting set is at most $|G|/|X|$. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the {\it strong Erd\H{o}s-Ko-Rado} property. In this paper we will investigate the weak and strong Erd\H{o}s-Ko-Rado property and attempt to classify the groups whose all transitive actions have these properties. In particular, we show that a group with the weak Erd\H{o}s-Ko-Rado property is solvable and that a nilpotent group with the strong Erd\H{o}s-Ko-Rado property is product of a $2$-group and an abelian group of odd order.Comment: This is the final version. To appear in the Journal of Algebraic Combinatoric

Quasi-Random profinite groups

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\mathrm{SL}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\mathrm{Sp}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups $\mathrm{SL}_{k}({\mathbb{Z}_p})$ and $\mathrm{Sp}_{2k}(\mathbb{Z}_p)$. We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.Comment: This is the final version. To appear in Glasgow Mathematical Journa

On the chromatic number of structured Cayley graphs

In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings.Comment: arXiv admin note: text overlap with arXiv:1507.0530

On a generalization of the Hadwiger-Nelson problem

For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an edge if and only if $Q(v-w)=1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field $F$ of characteristic zero, the Borel chromatic number of $\mathrm{QG}_Q$ is infinite if and only if $Q$ represents zero non-trivially over $F$. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009.Comment: This is the final version. Accepted in Israel Journal of Mathematic

Minimal dimension of faithful representations for pp-groups

For a group $G$, we denote by $m_{faithful}(G)$, the smallest dimension of a faithful complex representation of $G$. Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the maximal ideal $\mathfrak{p}$. In this paper, we compute the precise value of $m_{faithful}(G)$ when $G$ is the Heisenberg group over $\mathcal{O}/\mathfrak{p}^n$. We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over $\mathcal{O}/\mathfrak{p}^n$ and many of its subgroups. By a theorem of Karpenko and Merkurjev, our result yields the precise value of the essential dimension of the latter finite groups.Comment: Final version. To appear in "Journal of group theory

Kirillov's orbit method and polynomiality of the faithful dimension of pp-groups

Given a finite group $\mathrm{G}$ and a field $K$, the faithful dimension of $\mathrm{G}$ over $K$ is defined to be the smallest integer $n$ such that $\mathrm{G}$ embeds into $\mathrm{GL}_n(K)$. In this paper we address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_q:=\exp(\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q)$ associated to $\mathfrak{g}_q:=\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of $\mathscr{G}_p$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such on each part the faithful dimension of $\mathscr{G}_q$ for $q:=p^f$ is equal to $f g(p^f)$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory

Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields

Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the prime ideal $\mathfrak{p}$ and let $G={\bf G}\left(\mathcal{O}/\mathfrak{p}^n\right)$ be the adjoint Chevalley group. Let $m_f(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone-von Neumann theorem, we determine a lower bound for $m_f(G)$ which is asymptotically the same as the results of Landazuri, Seitz and Zalesskii for split Chevalley groups over $\mathbb{F}_q$. Our result yields a conceptual explanation of the exponents that appear in the aforementioned resultsComment: Final version. To appear in "Groups, Geometry, and Dynamics