Monograph

#### Degree

Doctor of Philosophy

Mathematics

Hall, Chris J.

#### Abstract

We prove that for every $n \ge 10$ there are at most finitely many values $c \in \mathbb{Q}$ such that the quadratic polynomial $x^2 + c$ has a point $\alpha \in \mathbb{Q}$ of period $n$. We achieve this by proving that for these values of $n$, every $n$-th dynatomic modular curve has genus at least two.

#### Summary for Lay Audience

Number theory studies properties of integers $(\ldots, -2,-1,0,1,2, \ldots)$ and rational numbers (numbers which are fractions''). While it is easy to come up with good enough'' approximations to solutions to polynomial equations using a computer, it can be much (and sometimes much, much) harder to find exact solutions which are integers or rational numbers.

A dynamical system is some function $f$ whose outputs and inputs come from the same set $S$. Whenever this is the case, we can iterate'' $f$. By this we mean that we can start from some input $a \in S$ and apply $f$ to $a$ to get output $f(a)$, and then apply $f$ to $f(a)$ to get output $f(f(a))$, and then apply $f$ to $f(f(a))$ to get $f(f(f(a)))$ and so on. For many, things start to get interesting when you perform some number of iterations and get back the original input $a$ you started with. When this happens $a$ is called a periodic point for $f$, and the smallest number of iterations which gets you back to $a$ is called the period.

In arithmetic dynamics, we look at dynamical systems from the point of view of number theory. The present thesis is concerned with showing that for the polynomial dynamical system $f_c (x) = x^2 +c$ where $c$ is some rational number, periodic points which are rational numbers are extremely rare. A big conjecture in arithmetic dynamics states that there are no periodic points of period $4$ or larger for $f_c (x)$. In some sense, this thesis shows that the number of possible exceptions to the big conjecture is finite. Previously, this was only known for points of period $5$, $6$, $7$, and $9$. Here we show that this also holds for any period at least $10$.