Seminar

Let  F be a rational function of degree > 1 over a  number field or function field K and let z be a point that is not preperiodic.   Ingram and Silverman conjecture that for all but finitely many positive integers (m,n), there is a prime p such that z has exact preperiodic m and exact period n (we call this pair (m,n) the portrait of z modulo p).

  We present some counterexamples to this conjecture and show that a generalized form of abc implies -- one that is true for function fields -- implies  that these are the only counterexamples.  We also present  a connection with Douady-Thursto-Hubbard rigidity.  This represents joint work with several authors.