Number Theory Seminar

Tuesday, Nov. 11
Vijay Sookdeo, U of R
An introduction to Bombieri's proof of an upper bound for the degree of an L-function of exponential sums

ABSTRACT:

Let f(x) is a polynomial defined over the rationals. Write f^n(x) for the n-th iterated of f(x) and suppose a and b are not preperiodic for f(x). In 1993, J. Silverman proved that for infinitely many primes p, there exist an n such that f^n(a)=b modulo p. This theorem is the dynamical analogue of Siegel's Theorem for finitely many integer points on a curve of genus g > 0. In 2007, Thomas Tucker asked the more general question of whether there are infinitely many primes p such that there is an n for which f^n(a)=f^n(b) modulo p. It was shown that this question has a positive answer when f(x) has no periodic critical point and when infinitely many of the points (f^n(a),f^n(b)) lie on a curve. We will discuss a recent improvement to this result. More specifically, Tucker's question has a positive answer for any polynomial f(x) for which f(c) is not a critical point for any critical point c.