Hecke Operators on Drinfeld Cusp Forms

In this paper, we study the Drinfeld cusp forms for $\Gamma_1(T)$ and $\Gamma(T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for $\Gamma_1(T)$ of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for $\Gamma_1(T)$ of large weights, and not for $\Gamma(T)$ even of small weights. The Hecke eigenvalues on cusp forms for $\Gamma(T)$ with small weights are determined and the eigenspaces characterized.Comment: 28 pages To appear in JN

An Equivalence Relation on A Set of Words of Finite Length

In this work, we study several equivalence relations induced from the partitions of the sets of words of finite length. We have results on words over finite fields extending the work of Bacher (2002, Europ. J. Combinatorics, {\bf 23}, 141-147). Cardinalities of its equivalence classes and explicit relationships between two words are determined. Moreover, we deal with words of finite length over the ring $\mathbb{Z}/N\mathbb{Z}$ where $N$ is a positive integer. We have arithmetic results parallel to Bacher's.Comment: 16 page

Ramanujan Graphs on Cosets of PGL2(Fq)PGL_2(\mathbb{F}_q)

In this paper we study Cayley graphs on \PGL_2(\mathbb F_q) mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of \PGL_2(\mathbb F_q) of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here

NATHANSON’S HEIGHTS AND THE CSS CONJECTURE FOR CAYLEY GRAPHS

Abstract. Let G be a finite directed graph, β(G) the minimum size of a subset X of edges such that the graph G ′ = (V, E � X) is directed acyclic and γ(G) the number of pairs of nonadjacent vertices in the undirected graph obtained from G by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan [CSS07] proved that if G is triangle-free, then β(G) ≤ γ(G). They conjectured a sharper bound (so called the “CSS conjecture”) that β(G) ≤ γ(G). Nathanson and Sullivan verified this conjecture for the 2 directed Cayley graph Cay(Z/NZ, EA) whose vertex set is the additive group Z/NZ and whose edge set EA is determined by EA = {(x, x + a) : x ∈ Z/NZ, a ∈ A} when N is prime in [NS07] by introducing “height”. In this work, we extend the definition of height and the proof of CSS conjecture for Cay(Z/NZ, EA) to any positive integer N. 1