The arithmetic of curves defined by iteration

Wade Hindes¹

¹ Department of Mathematics Brown University Providence, RI 02912, U.S.A.

Acta Arithmetica, Tome 169 (2015) no. 1, pp. 1-27.

Résumé

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 $C_n:y^2=f^n(x)$ and their quadratic twists (here $f^n$ denotes the $n$th 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 for the Galois groups of the fourth iterate of rational quadratic polynomials $x^2+c$, using techniques in the theory of rational points on curves. Moreover, we show that the Hall–Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when $f(x)=x^2+3$. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.

Détail

DOI : 10.4064/aa169-1-1

Keywords: size galois groups iterates quadratic polynomial parametrized certain rational points curves their quadratic twists here denotes nth iterate nbsp end study arithmetic curves global finite fields translating key problems arithmetic polynomial iteration geometric framework point view has several dynamical applications instance establish maximality theorem galois groups fourth iterate rational quadratic polynomials using techniques theory rational points curves moreover hall lang conjecture integral points elliptic curves implies serre type finite index result these dynamical galois groups conjectural bounds mordell curves predict index still unknown finally provide evidence these curves defined iteration have geometrical significance construct family curves whose rational points completely determine whose geometrically simple jacobians have complex multiplication positive rank