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 $X\to 𝒜$ be a non-constant morphism from a curve $X$ to an abelian variety $𝒜$, all defined over a number field $k$. Suppose that $X$ is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on $X$ to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that $𝒜\left(k\right)$ and $Ш(𝒜/k)$ are finite.

A curve $X$ over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space $Y$ is trivial if and only if $Y$ is a subspace of ${\mathbf{P}}^1$. As a consequence, $X$ has locally a trivial class group if and only if the stable reduction of $X$ has only rational components.

Let $\ell \ge 7$ be a prime, $C$ be the non-singular projective curve defined over $\mathbb{Q}$ by the affine model $y(1-y)=x^{\ell }$, $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$, and $\phi :C\to J$ the albanese embedding with $\infty$ as base point. Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. Taking care of a case not covered in [12], we show that $\phi \left(C\right)\cap J_{tors}\left(\overline{\mathbb{Q}}\right)$ consists only of the image under $\phi$ of the Weierstrass points of $C$ and the points $(x,y)=(0,0)$ and $(0,1)$, where $J_{tors}$ denotes the torsion points of $J$.

Let $X$ be a curve over a field $k$ with a rational point $e$. We define a canonical cycle ${\Delta }_e\in Z^2(X^3{)}_{\mathrm{hom}}$. Suppose that $k$ is a number field and that $X$ has semi-stable reduction over the integers of $k$ with fiber components non-singular. We construct a regular model of $X^3$ and show that the height pairing $⟨{\tau }_{*}\left({\Delta }_e\right),{\tau }_{*}^{\text{'}}\left({\Delta }_e\right)⟩$ is well defined where $\tau$ and ${\tau }^{\text{'}}$ are correspondences. The paper ends with a brief discussion of heights and $L$-functions in the case that $X$ is a modular curve.

In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb{Q}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb{Q}$ can be arbitrarily large, where $A$ is an abelian variety, with $dimA$ bounded by a constant depending only on $p$.

The Schottky-Jung proportionality theorem, from which the Schottky relation for theta functions follows, is proved for Mumford curves, i.e. curves defined over a non-archimedean valued field which are parameterized by a Schottky group.