COMPLEX DYNAMICS AND INVARIANT FORMS MOD p

Complex dynamical systems on the Riemann sphere do not possess " invariant forms ". However there exist non-trivial examples of dynamical systems, defined over number fields, satisfying the property that their reduction modulo ℘ possesses " invariant forms " for all but finitely many places ℘. The paper completely characterizes those dynamical systems possessing the latter property. 1. Motivation and statement of the Theorem Let k be an algebraically closed field; in our applications k will be either the complex field C or the algebraic closure of a finite field. Let k(t) be the field of rational functions in the variable t and let σ(t) ∈ k(t) be a non-constant rational function. By an invariant form of weight ν ∈ Z for σ(t) we mean a rational function f (t) ∈ k(t) such that (1.1) f (σ(t)) = dσ dt (t) −ν · f (t). If we consider the ν−tuple differential form ω := f (t) · (dt) ν then Equation 1.1 can be written as σ * ω = ω which is actually what justifies our terminology. We shall sometimes refer to ω itself as being an invariant form of weight ν ∈ Z. Our first remark is that if k = C and σ, viewed as a self map P 1 → P 1 of the complex projective line, has degree ≥ 2 then there are no non-zero invariant forms for σ of non-zero weight. Indeed if Equation 1.1 holds for some f = 0 and ν = 0 then the same equation holds with σ replaced by the n−th iterate σ n for any n. Now this equation for the iterates implies that any finite non-parabolic periodic point of σ is either a zero or a pole of f ; cf. [10], p. 99 for the definition of parabolic periodic points. On the other hand by [10], pp. 47 and 143, there are infinitely many non-parabolic periodic points, a contradiction. Although invariant forms don't exist over the complex numbers there exist, nevertheless , interesting examples of complex rational functions with coefficients in number fields whose reduction mod (almost all) primes admit invariant forms. To explain this let F ⊂ C be a number field (always assumed of finite degree over the rationals) and let ℘ be a place of F (always assumed finite). Let O ℘ denote the valuation ring of ℘, let κ ℘ be the …