Periodic points, multiplicities, and dynamical units.

Abstract

at ProvidenceLet φ (z) e C [z] be a polynomial of degree at least 2. The fixed points of the iterates of φ have been widely studied since the time of Julia and Fatou in order to analyze the dynamical System associated to φ. (See, for example, [2].) If φ (ζ) has coefficients in a number field K, these periodic points will generate interesting algebraic extensions of K. More precisely, if α e K is a periodic point for φ, then the map σ : α ι-> φ (α) will frequently induce a field automorphism of K(v) over K. In other words, there is often an element σ e Aut(K(aC)/K) whose action on the generator α is given by the polynomial φ. For a discussion of when φ induces such a σ, see for example [10], [11], [12].The fields generated by periodic points of polynomials, or more generally rational functions φ (z) e K(z), are thus of interest because one is able to express part of the action of the Galois group in terms of explicit polynomial actions. For example, if φ (ζ) = z d , then the periodic points are roots of unity, and one knows that the action of the Galois group on a root of unity is given by a polynomial map of the form z -> z*. Similarly, let [«] : E -> E be the multiplication-by-w map on a CM elliptic curve. Then the periodic points are points of finite order on E, and the action of the Galois group on these points can be expressed in terms of certain rational functions coming from the group law on E.In these last two examples, an important discovery was that in the fields generated by periodic points, one can use the periodic points to produce units. The units arising from roots of unity are called cyclotomic units and are quite classical. The units coming from points of finite order on CM elliptic curves are known s elliptic units and were originally constructed by Robert [18]. (And recently, units have similarly been constructed using division points on Jacobians of genus two curves [3], [7].) It would be hard to overstate the importance of cyclotomic and elliptic units in modern algebraic number theory and arithmetic geometry.