arXiv subjects

Laura Walton

Publications and source records attributed to Laura Walton.

4 recordsLinked to original sources

arXiv2020

Local arboreal representations

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups of the extensions of $K$ generated by the solutions to $f^n(z)=a$. The behavior depends upon $v(c)$, and we find that it shifts dramatically as $v(c)$ crosses a certain value: $0$ in the case $p \nmid \ell$, and $-p/(p-1)$ in the case $p=\ell$.

math.NT↗

arXiv2019

An alternate proof of idempotent relations among periodic points and quotients

We give a short proof of an idempotent relation formula for counting periodic points of endomorphisms defined over finite fields. The original proof of this result, due to Walton, uses formal manipulation of arithmetic zeta functions, whereas we deduce the result directly from a related theorem of Kani and Rosen.

math.NT↗

arXiv2018

Arboreal representations for rational maps with few critical points

Jones conjectures the arboreal representation of a degree two rational map will have finite index in the full automorphism group of a binary rooted tree except under certain conditions. We prove a version of Jones' Conjecture for quadratic and cubic polynomials assuming the $abc$-Conjecture and Vojta's Conjecture. We also exhibit a family of degree $2$ rational maps and give examples of degree $3$ polynomial maps whose arboreal representations have finite index in the appropriate group of tree automorphisms.

math.NT↗

arXiv2017

Counting periodic points over finite fields

Let $V$ be a quasiprojective variety defined over $\mathbb{F}_q$, and let $ϕ:V\rightarrow V$ be an endomorphism of $V$ that is also defined over $\mathbb{F}_q$. Let $G$ be a finite subgroup of $\operatorname{Aut}_{\mathbb{F}_q}(V)$ with the property that $ϕ$ commutes with every element of $G$. We show that idempotent relations in the group ring $\mathbb{Q}[G]$ give relations between the periodic point counts for the maps induced by $ϕ$ on the quotients of $V$ by the various subgroups of $G$. We also show that if $G$ is abelian, periodic point counts for the endomorphism on $V/G$ induced by $ϕ$ are related to periodic point counts on $V$ and all of its twists by $G$.

math.NT↗