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

## 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 |
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Faculty \ School: | Faculty of Science > School of Mathematics |

Related URLs: | |

Depositing User: | Vishal Gautam |

Date Deposited: | 18 Mar 2011 14:48 |

Last Modified: | 15 Dec 2022 01:48 |

URI: | https://ueaeprints.uea.ac.uk/id/eprint/19696 |

DOI: | 10.1006/jnth.2001.2682 |

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