ARITHMETIC DYNAMICS ON ALGEBRAIC CURVES

Abstract

In this thesis we proved the classical Northcott's theorem: there are only finitely many preperiodic points in orbits of rational functions with degree 2 or more. We also prove a classical theorem on number of integral points in an orbit: outside an exceptional class, the number of integral points in an orbit is finite if the rational function has degree 2 or more. The number of integral points can be arbitrarily large if its resultant is unbounded. This leads to a consideration of rational functions with minimal resultants in some sense. We classify such rational functions over $\mathbb Q$ and provide an algorithm for testing affine minimality. This also serves as an improvement over the current state of art algorithm. As an application, we show that rational functions of type $(az^2+b)/z\in \mathbb Q(z), a,b\in \mathbb Z$ and $b$ square-free cannot have more than 3 consecutive integral points in its orbit.