arXiv subjects
Publications and source records attributed to William Y. Chen.
In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and applications of these ideas to the Inverse Galois Problem, Markoff triples and the arithmetic of Fourier coefficients for noncongruence modular forms.
We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group $S$ admits a non-congruence presentation (as conjectured in [CLT24]), and that if $S$ has a non-trivial Schur multiplier, then it admits a smooth cover (as conjectured in [CFLZ]).
We show that the configuration space of four unordered points in $\mathbb{C}$ with barycenter 0 is isomorphic to the space of triples $(E,Q,ω)$, where $E$ is an elliptic curve, $Q\in E^\circ$ a nonzero point, and $ω$ a nonzero holomorphic differential on $E$. At the level of fundamental groups, our construction unifies two classical exceptional exact sequences involving the braid group $B_4$: namely, the sequence $1\rightarrow F_2\rightarrow B_4\rightarrow B_3\rightarrow 1$, where $F_2$ is a free group of rank 2, related to Ferrari's solution of the quartic, and the sequence $1\rightarrow \mathbb{Z} \rightarrow B_4\rightarrow\operatorname{Aut}^+(F_2)\rightarrow 1$ of Dyer-Formanek-Grossman.
In this paper we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group $F_2$. This shows that a "baby" version of the Wiegold conjecture fails for $F_2$, and provides counterexamples to two conjectures in the theory of noncongruence subgroups of $\text{SL}_2(\mathbb{Z})$. Our main result explicitly produces, for every prime power $q\ge 7$, the groups $\text{SL}_3(\mathbb{F}_q)$ and $\text{SU}_3(\mathbb{F}_q)$ as characteristic quotients of $F_2$. Our strategy is to study specializations of the Burau representation for the braid group $B_4$, exploiting an exceptional relationship between $F_2$ and $B_4$ first observed by Dyer, Formanek, and Grossman. Weisfeiler's strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but it is not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of $\text{SL}_3(\mathbb{F}_q)$ and $\text{SU}_3(\mathbb{F}_q)$. We also show that our examples of $\text{PSL}_3(\mathbb{F}_q)$ and $\text{PSU}_3(\mathbb{F}_q)$ are minimal in the sense that no group of the form $\text{PSL}_2(\mathbb{F}_q)$ is a characteristic quotient of $F_2$.
Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli spaces of elliptic curves with $PSL_2(\mathbb{F}_\ell)$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder, about Markoff triples modulo $\ell$.