| Abstract: This dissertation studies the affine dynatomic modular curves $Y_1(n)$ and $Y_0(n)$, which parametrize unicritical polynomial maps $f_c(x)=x^d+c$ of degree $d>2$ together with their periodic orbits of period $n$, as well as their smooth projective completions $X_1(n)$ and $X_0(n)$. The central objective is to determine for which primes $p$ the curve $X_0(n)$ remains geometrically irreducible despite having bad reduction modulo $p$. Morton (1996) algebraically characterized the primes of bad reduction for these curves using the discriminant of the primitive parabolic factor $D_{n,n}=\operatorname{disc}_c(\Delta_{n,n}(c))$. For the quadratic family $f_c(x)=x^2+c$, Doyle et al. (2019) showed that $X_0(n)$ remains geometrically irreducible modulo $p$ whenever an odd prime $p$ divides $D_{n,n}$ exactly once. In this dissertation, we extend this framework to arbitrary degrees $d>2$. Specifically, we prove that if $p \nmid d$ and the $p$-adic valuation satisfies
$$
v_p(D_{n,n}) \le (3d-4)/2,
$$
then $X_0(n)$ is geometrically irreducible in characteristic $p$. Under this condition, up to $3d-4$ finite branch points may collide modulo $p$, yet geometric irreducibility is nevertheless preserved. The proof combines complex dynamics and the combinatorial structure of the degree-$d$ Multibrot set to establish the required transitivity of the associated monodromy action. |
Mathematics Seminars and Colloquia