arXiv subjects

Erick Knight

Publications and source records attributed to Erick Knight.

3 recordsLinked to original sources

arXiv2019

On the $\zeta_3$-Pell equation

Let $K = \mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a primitive root of unity. In this paper we study the distribution of integers $\alpha \in \mathcal{O}_K$ for which the norm equation $N_{K(\sqrt[3]{\alpha})/K}(\mathbf{x}) = \zeta_3$ is solvable for integers $\mathbf{x} \in \mathcal{O}_{K(\sqrt[3]{\alpha})}$. The analogous question for $\zeta_2 = -1$ is the well-known negative Pell equation. We also address the natural generalization of Stevenhagen's conjecture on the negative Pell equation in this setting.

math.NT↗

arXiv2019

The negative Pell equation

By applying methods recently developed by A. Smith with regards to Goldfeld's conjecture, we show that the density of square-free integers $d$ in $[1, N]$ for which the negative Pell equation $x^2 - dy^2 = -1$ has a solution is as predicted by the model of Stevenhagen.

math.NT↗

arXiv2017

p-adic Jacquet-Langlands correspondence and patching

We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.

math.NT↗