Entropy and the Canonical Height

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. Firstly, the introduction of a notion of entropy for sequences of transformations. Secondly, 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. 2000 Mathematics Subject Classification. 11G07, 37B40. The first author acknowledges the support of EPSRC postdoctoral award GR/M49588, the second thanks Jonathan Lubin and Joe Silverman for the AMS Sectional meeting on Arithmetic Dynamics at Providence, RI, 1999. 1 2 M. EINSIEDLER, G. EVEREST, AND T. WARD