Configuration of the Crucial Set for a Quadratic Rational Map

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $\varphi(z) \in K(z)$ have degree two. We describe the crucial set of $\varphi$ in terms of the multipliers of $\varphi$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $\varphi$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set

Computing algebraic numbers of bounded height

We describe an algorithm for listing all elements of bounded height in a given number field

Apollonian circle packings of the half-plane

We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive real numbers that are the roots of quadratic polynomials with rational coefficients. This is reflected in a close connection between Apollonian circle packings and continued fractions which allows us to completely classify such packings up to similarity.Comment: 35 pages; final version -- minor improvements to the exposition from the first versio

Quadratic points on dynamical modular curves

Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over quadratic fields, extending previous work of Poonen, Faber, and the authors.Comment: Improvements to the exposition; stronger version of what is now Proposition 4.