Let $UY_{n}(q)$ be a Sylow $p$-subgroup of an untwisted Chevalley group $Y_{n}(q)$ of rank $n$ defined over $\mathbb{F}_{q}$ where $q$ is a power of a prime $p$. We partition the set $\text{Irr}(UY_{n}(q))$ of irreducible characters of $UY_{n}(q)$ into families indexed by antichains of positive roots of the root system of type $Y_{n}$. We focus our attention on the families of characters of $UY_{n}(q)$ which are indexed by antichains of length $1$. Then for each positive root $\unicode[STIX]{x1D6FC}$ we establish a one-to-one correspondence between the minimal degree members of the family indexed by $\unicode[STIX]{x1D6FC}$ and the linear characters of a certain subquotient $\overline{T}_{\unicode[STIX]{x1D6FC}}$ of $UY_{n}(q)$. For $Y_{n}=A_{n}$ our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of $\text{Irr}(UE_{i}(q))$, $6\leqslant i\leqslant 8$, and $\text{Irr}(UF_{4}(q))$.

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group $\operatorname{PSL}(2,\mathbb{Z})$, including the following statements: the ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weights four and six, denoted by $E_{4}$ and $E_{6}$; the smallest-weight cusp form $\unicode[STIX]{x1D6E5}$ has weight twelve and can be written as a polynomial in $E_{4}$ and $E_{6}$; and the Hauptmodul $j$ can be written as a multiple of $E_{4}^{3}$ divided by $\unicode[STIX]{x1D6E5}$. The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups $\unicode[STIX]{x1D6E4}_{0}(N)^{+}$ with square-free level $N$, which are related to ‘Monstrous moonshine conjectures’. Certain aspects of our results are generated from extensive computer analysis; as a result, many of the space-consuming results are made available on a publicly accessible web site. However, we do present in this article specific results for certain low-level groups.

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $\mathbb{Q}$ with good reduction away from 3, up to $\mathbb{Q}$-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.