Entropy and the canonical height

M. Einsiedler, G. Everest, T. Ward

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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 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 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 local canonical height up to sign. The proofs use transcendence theory, a strong form of Siegel's theorem, and an elliptic analogue of Jensen's formula.
Original languageEnglish
Pages (from-to)256-273
Number of pages18
JournalJournal of Number Theory
Issue number2
Publication statusPublished - 2001

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