In arithmetic dynamics, the smoothness and irreducibility of the dynatomic modular curves $Y_1(n)$ and $Y_0(n)$ have frequently been studied for the polynomial family $f_c(x)=x^d+c$ in both char 0 and positive char $p$, but less is known about the dynamical behavior of other families. I am studying the smoothness and irreducibility of $Y_1(n)$ and $Y_0(n)$ for families of the form $f_c(x)=g^p+bx+c$ defined over a field $k$ of positive char $p$, with $g\in k[x,c]$. For example, it has been shown that for such a dynamical system, $Y_1(1)$ is smooth, and for $n\geq2$, we have the following: (i) when $b\neq1$, $Y_1(n)$ is smooth if and only if $b^n\neq1$, and (ii), when $b=1$, $Y_1(n)$ is smooth if and only if $p\nmid n$, except in the case $p=d=2$ where $Y_1(2)$ is smooth, where $d=\deg(f_c)$. More generally, I am interested in asking what dynamical properties of this family can be attributed to them all having a unique critical point at infinity.
Some dynatomic modular curves in positive characteristic
Colette La Pointe, CUNY Graduate CenterAuthors: Colette La Pointe
2023 AWM Research Symposium
Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]