For a rational point of algebraic variety defined over a number field, the height is an
important quantity. In the field of Diophantine geometry, there are some applications of
studying the growth rate of the height under iteration of self-morphisms on the variety.
On such growth rate, Kawaguchi and Silverman defined the arithmetic degree of an orbit
and conjectured that if the orbit is Zariski-dense, the arithmetic degree is equal to the
(first)-dynamical degree.
Recently, we proved Kawaguchi-Silverman conjecture for any self-morphisms of semi-abelian
varieties, and also proved that if the self-morphism on semi-abelian variety satisfy some
condition, there is a rational point b such that a rational point a is preperiodic if and
only if a-b is a torsion point.
In this talk, we start with the definition of the height function, introduce Kawaguchi-
Silverman conjecture with some examples, and give the outline of the proof of Kawaguchi-
Silverman conjecture and the equivalent condition of the preperiodicity.
This is a joint work with Yohsuke Matsuzawa in University of Tokyo.