|Location:||MSRI: Simons Auditorium|
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.No Notes/Supplements Uploaded No Video Files Uploaded