One half log discriminant

TL;DR

This paper presents a geometric proof for computing a generalized Mahler integral through equidistribution of preperiodic points in a dynamical system linked to elliptic curves, connecting number theory and dynamical systems.

Contribution

It introduces a geometric approach to evaluate Mahler integrals using dynamical systems associated with elliptic curves, expanding methods in number theory and dynamics.

Findings

Established a geometric proof for Mahler integral computation.

Linked equidistribution of preperiodic points to Mahler integrals.

Applied dynamical systems on elliptic curves to number theory problems.

Abstract

We give a geometric proof that one may compute a particular generalized Mahler integral using equidistribution of preperiodic points of a dynamical system on the sphere. The dynamical system is associated to the multiplication by 2 map on an elliptic curve over a number field K with Weierstrass equation y^2 = P(x) (a Lattes dynamical system).