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The elliptic Mahler measure

Published online by Cambridge University Press:  24 October 2008

G. R. Everest  and
Bríd Ní Fhlathúin
G. R. Everest
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ e-mail: g.everesta@uea.ac.uk
Bríd Ní Fhlathúin
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ e-mail: g.everesta@uea.ac.uk
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Extract

In this paper, we are going to introduce an elliptic analogue of the classical Mahler measure of an integral polynomial. The measure is shown to vanish if and only if the roots of the polynomial are attached to division points of the curve. This is the exact analogue of the statement that the Mahler measure of an integral polynomial vanishes if and only if all of its roots are division points of the circle (in other words, roots of unity). The proof of this result exploits an integral representation for the local canonical heights on the elliptic curve. The integral is that of a simple polynomial function over the complete curve in the appropriate valuation. Attention then shifts to the calculation of local integrals of arbitrary rational functions on elliptic curves. Results are proved which show these integrals may be computed as effective limits of Riemann sums. Finally, consideration is given to analogous behaviour for abelian varieties. We give a method, in principle, for computing the global canonical height of a rational point on an abelian variety denned over an algebraic number field, once again exploiting the integral representation of the local heights. The methods used include recent inequalities for linear forms in elliptic and abelian logarithms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 13 - 25
Copyright
Copyright © Cambridge Philosophical Society 1996

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