PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Abstract

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.

Authors

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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image