arXiv subjects

Laney Williams

Publications and source records attributed to Laney Williams.

2 recordsLinked to original sources

arXiv2025

Counting $C_2 \wr S_4$ fields with a power saving error term

Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|\Delta_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^{\alpha}(\log X)^\beta$. Previously, Kl\"uners proved Malle's conjecture for $G=C_2 \wr S_4$. His proof gives a power savings of $O(X^{7/8})$. We improve Kl\"uners' result by establishing a stronger power saving error term for the count of such fields. Specifically, we show $N_8(C_2\wr S_4,X)=CX+O(X^{3/4-1/30})$. Additionally, we obtain new bounds on $N_8(G,X)$ for the groups $S_4$, $C_2^3 \rtimes S_4$, $GL_2 (\mathbb{F}_3)$, and $Q_8\rtimes S_4$ as permutation subgroups of $S_8$.

math.NT↗

arXiv2025

The average genus number for pure fields of prime degree

Let $\ell\geq 5$ be prime. Let $\mathcal{F}_\ell$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\ell]{a})$ of degree $\ell$, ordered by the absolute value of their discriminant. In 2018, Benli proved a counting theorem for $\mathcal{F}_\ell$, generalizing a previous theorem of Cohen and Morra when $\ell=3$. We prove that the proportion of pure fields of degree $\ell$ with genus number one is asymptotic to $(A_\ell \log X)^{-1}$ and that the average genus number for pure fields of degree $\ell$ is asymptotic to $B_\ell(\log X)^{\ell-1}$. Both $A_\ell$ and $B_\ell$ are expressed explicitly as a product over primes.

math.NT↗