Entropy and the canonical height

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

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. 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.

Citation

Einsiedler, M., Everest, G., & Ward, T. (2001). Entropy and the canonical height. Journal of Number Theory, 91(2), 256-273. https://doi.org/10.1006/jnth.2001.2682

• 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
• DOI: https://doi.org/10.1006/jnth.2001.2682