Entropy and the Canonical Height

Abstract

The height of an algebraic number in the sense of Diophantine geometry is a measure of arithmetic complexity. There is a well-known relationship between the entropy of automorphisms of solenoids and classical heights. We consider an elliptic analogue of this relationship, which involves two novel features. First we consider the introduction of a notion of entropy for sequences of transformations. Second, we consider the recognition of canonical local heights as integrals over the closure of the torsion subgroup of the curve (an elliptic Jensen formula). A sequence of transformations is defined for which there is a canonical arithmetically defined quotient whose entropy is the canonical height and in which the fibre entropy is accounted for by canonical local heights at primes of singular reduction, yielding a dynamical interpretation of singular reduction. This system is related to local systems whose entropy coincides with the canonical local height up to sign. The proofs use transcendence theory, a strong form of Siegel's theorem, and an elliptic analogue of Jensen's formula.