Entropy and the Canonical Height,

Einsiedler, M., Everest, G. and Ward, T. (2001) Entropy and the Canonical Height,. Journal of Number Theory, 91 (2). pp. 256-273.

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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.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: Vishal Gautam
Date Deposited: 18 Mar 2011 14:48
Last Modified: 05 Nov 2013 14:03
URI: https://ueaeprints.uea.ac.uk/id/eprint/19696
DOI: 10.1006/jnth.2001.2682

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