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Entropy and the canonical height

Einsiedler, M.; Everest, G.; Ward, T.

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M. Einsiedler

G. Everest

T. Ward


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.


Einsiedler, M., Everest, G., & Ward, T. (2001). Entropy and the canonical height. Journal of Number Theory, 91(2), 256-273.

Journal Article Type Article
Publication Date Dec 1, 2001
Deposit Date Oct 12, 2012
Publicly Available Date Oct 19, 2012
Journal Journal of Number Theory
Print ISSN 0022-314X
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 91
Issue 2
Pages 256-273


Accepted Journal Article (267 Kb)

Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of number theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of number theory, 91/2, 2001, 10.1006/jnth.2001.2682

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