arXiv subjects
Publications and source records attributed to Chloe Stewart.
In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen--Franks groups associated with these graphs in several cases.
By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\ell]{D})$, $D\in\mathbb{Z}$, of odd prime degree $\ell$. Let $\mathcal{M}_\ell$ denote the collection of isomorphism classes of metacyclic fields of degree $\ell(\ell-1)$. Write $N_\ell(X)=\#\{K\in\mathcal{M}_\ell: |Δ_K|\leq X\}$ for the associated counting function, where $Δ_K$ denotes the discriminant of $K$. We show $$N_\ell(X)\sim A_\ell X^{\frac{1}{(\ell-1)^2}}(\log X)^{\ell-2}\,,$$ for an explicit constant $A_\ell$. We express $A_\ell$ as a rational number times a product over primes of a degree $\ell$ polynomial in $1/p$.
Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. The image of a non-constant morphism from a curve $T$ to $\mathcal{M}_g$ is a curve in $\mathcal{M}_g$. By work of González Díez and Harvey, for every integer $g \geq 3$, there exists a complete curve in $\mathcal{M}_g$. Here we generalize the construction to produce new complete curves in $\mathcal{M}_g$. We also find a formula for the genus of each curve $T$ using Galois theory for function fields.