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.Full text not available from this repository.
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.
|Faculty \ School:||Faculty of Science > School of Mathematics|
|Depositing User:||Vishal Gautam|
|Date Deposited:||18 Mar 2011 14:48|
|Last Modified:||22 Feb 2017 01:35|
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