Variation of periods modulo p in arithmetic dynamics

Let φ : V → V be a self-morphism of a quasiprojective variety defined over a number field K and let P ∈ V (K) be a point with infinite orbit under iteration of φ. For each prime p of good reduction, let mp(φ, P ) be the size of the φ-orbit of the reduction of P modulo p. Fix any ǫ > 0. We show that for almost all primes p in the sense of analytic density, the orbit size mp(φ, P ) is larger than (logNK/Qp) . Introduction Let φ : PQ −→ PQ be a morphism of degree d defined over Q and let P ∈ P(Q) be a point with infinite forward orbit Oφ(P ) = { P, φ(P ), φ(P ), . . . } . For all but finitely many primes p, we can reduce φ to obtain a morphism φ̃p : P N Fp −→ PNFp whose degree is still d. We write mp(φ, P ) for the size of the orbit of the reduced point P̃ = P mod p, mp(φ, P ) = #Oφ̃p(P̃ ). (For the remaining primes we define mp(φ, P ) to be ∞.) Using an elementary height argument (see Corollary 10), one can show that mp(φ, P ) ≥ d log log p+O(1) for all p, but this is a very weak lower bound for the size of the mod p orbits. Our principal results say that for most primes p, we can do (almost) exponentially better. In the following result, we write δ(P) for the Date: February 1, 2008 (Draft 1). 1991 Mathematics Subject Classification. Primary: 11G35; Secondary: 11B37, 14G40, 37F10.