# The preperiodic points of self-morphisms on semi-abelian varieties

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.