Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves

TL;DR

This paper investigates the dynamics of surjective endomorphisms on projective bundles over elliptic curves, establishing the Kawaguchi--Silverman conjecture for these bundles and related cases using bundle transition functions.

Contribution

It proves the Kawaguchi--Silverman conjecture for projective bundles on elliptic curves and split bundles on certain varieties, linking dynamical and geometric properties.

Findings

Proves Kawaguchi--Silverman conjecture for projective bundles on elliptic curves.

Extends the conjecture to split bundles on varieties with finitely generated Mori cone.

Uses transition functions to relate bundle geometry to dynamical behavior.

Abstract

We study the dynamics of surjective endomorphisms of projective bundles on elliptic curves and relate their dynamical properties to the geometry of the bundle. As an application we prove the Kawaguchi--Silverman conjecture for projective bundles on elliptic curves, thereby completing the conjecture for all projective bundles on curves. Our approach is to use the transition functions of the bundles. This allows us to further prove the conjecture for projective split bundles on a smooth projective variety with finitely generated Mori cone.