Height argument for post critically finite $z^2+c$

Question

I know that if we require $c$ to be rational the only post-critically finite maps of the form $z^2+c$ have $c = 0$ or $c=-1$. Is there a height argument for why this is true? It seems there must be, as heights are closely related to preperiodic points. I would also be interested in real-values or complex values and the associated proofs.

Answer 1

Do you know about canonical heights? If not, first look them up, then here's some material. Let $K$ be a number field and $f(z)\in K(z)$ a polynomial of degree $d\ge2$. The critical height of $f$ is $$ h^{\text{crit}}(f) = \sum_{\alpha\in\operatorname{Crit}(f)} \hat h_f(\alpha). $$ A fairly recent theorem of Patrick Ingram says that $$ h^{\text{crit}}(f) = h(f) +O(1), $$ where the $O(1)$ depends only on $d$. (Here $h(f)$ is the height of $f$ in the moduli space of maps of degree $d$. For $f_c(z)=z^2+c$, you can just take $h(f_c)=h(c)$. In particular, $$ \text{$f$ is PCF}\quad\Longleftrightarrow\quad h^{\text{crit}}(f)=0. $$ Hence $$ \text{$f$ is PCF}\quad\Longrightarrow\quad \text{$h(f)$ is bounded.} $$

There was earlier work proving this for just polynomials, and earlier works proving somewhat more directly (but still using heights) that the set of PCF maps is a set of bounded height. But you can get those from the references in Ingram's papers:

• Ingram, Patrick, The critical height is a moduli height, Duke Math. J. 167 (2018), 1311-1346

• Ingram, Patrick, A finiteness result for post-critically finite polynomials, Int. Math. Res. Not. IMRN 3 (2012), 524-543