We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $\gamma \mapsto ord\left(Res\left({\phi }^{\gamma }\right)\right)$ factors through a function $ordRe{s}_{\phi }\left(·\right)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P{¹}_K$, or on a segment, and the minimal resultant locus is contained in the tree in $P{¹}_K$ spanned by the fixed points and poles...

A rational map ϕ: ℙ¹ → ℙ¹ along with an ordered list of fixed and critical points is called a totally marked rational map. The space $Rat{₂}^{t}m$ of totally marked degree two rational maps can be parametrized by an affine open subset of (ℙ¹)⁵. We consider the natural action of SL₂ on $Rat{₂}^{t}m$ induced from the action of SL₂ on (ℙ¹)⁵ and prove that the quotient space $Rat{₂}^{t}m/SL₂$ exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over ℤ[1/2].

The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...