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 O and the maximal ideal p. In this paper, we compute the precise value of m faithful (G) when G is the Heisenberg group over O/p n . We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over O/p n and many of its subgroups. By a theorem of Karpenko and Merkurjev [7, Theorem 4.1], our result yields the precise value of the essential dimension of the latter finite groups.

We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense of [31]. Our theorem is an analogue of the results previously obtained by Howe [16], Li [21], Dvorsky-Sahi [9], and Kobayashi-Savin [19]. 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 (see [12], [30]) that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal.

Abstract. Inspired by Gowers' seminal paper [6], we will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of SL k (Z/(p n Z)) and Sp 2k (Z/(p n Z)). Our method also delivers a lower bound for the minimal degree of a faithful representation of 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 SL k (Zp) and Sp 2k (Zp). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F , let QG Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u, w ∈ 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 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 [6].

Let a finite group G act transitively on a finite set X . A subset S ⊆ G is said to be intersecting if for any s 1 , s 2 ∈ S, the element s −1 1 s 2 has a fixed point. The action is said to have the weak Erdős-Ko-Rado (EKR) property, if the cardinality of any intersecting set is at most |G|/|X |. If, moreover, any maximum intersecting set is a coset of a point stabilizer, the action is said to have the strong EKR property. In this paper, we will investigate the weak and strong EKR property and attempt to classify groups in which all transitive actions have these properties. In particular, we show that a group with the weak EKR property is solvable and that a nilpotent group with the strong EKR property is the direct product of a 2-group and an abelian group of odd order.