Periodic points modulo p as p varies
by
Joseph H. Silverman
Brown University

Let F : V -> V be a morphism of a quasiprojective variety defined over a number field K and let R be a point in V(K) having infinite orbit under iteration of F. For each prime p of good reduction, let M(p) denote the length of the orbit of R modulo p, i.e., the number of distinct iterates F^i(R) mod p. Let e > 0. We sketch a proof that for a set of primes of analytic density 1, the orbit length M(p) is greater than (log Norm p)^(1-e). We also conjecture a stronger estimate.

Date received: April 25, 2008