The abc, abcd, abcde…conjectures and their implications on uniform boundedness

We start with an introduction to Szpiro’s conjecture and its equivalent formulation, the abc conjecture. Szpiro’s goal in the 1980s was to prove the Mordell conjecture (effectively too!); it was soon observed that the implications of this conjecture and its alternate versions also include landmarks such as Fermat’s Last Theorem, Baker’s theorem, Roth’s theorem, and the non-existence of Siegel zeroes for certain Dirichlet L-functions. We will then highlight recent work by Nicole Looper on new implications in arithmetic dynamics, in particular that the abcd conjecture implies uniform boundedness for preperiodic points of polynomials as well as a weak dynamical Lang’s conjecture for points of small height.

http://people.math.harvard.edu/~demarco/AlgebraicDynamics/