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Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

Volume 196 / 2020

John R. Doyle Acta Arithmetica 196 (2020), 219-268 MSC: Primary 37P35; Secondary 14G05, 37P05, 37P15. DOI: 10.4064/aa180403-16-3 Published online: 11 July 2020


Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $\alpha \to \beta $ if and only if $f(\alpha ) = \beta $. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $\mathbb Q$, while recent work of the author, Faber, and Krumm has provided a detailed study of this question for all quadratic extensions of $\mathbb Q$. In this article, we give a conjecturally complete classification like Poonen’s, but over the cyclotomic quadratic fields $\mathbb Q(\sqrt {-1})$ and $\mathbb Q(\sqrt {-3})$. The main tools we use are dynamical modular curves and results concerning quadratic points on curves.


  • John R. DoyleDepartment of Mathematics and Statistics
    Louisiana Tech University
    Ruston, LA 71272, U.S.A.

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