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Title: Mahler's measure on Abelian varieties.
Author: Fhlathuin, Brid ni.
ISNI:       0000 0001 3463 4851
Awarding Body: University of East Anglia
Current Institution: University of East Anglia
Date of Award: 1995
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This thesis is a study of the integration of proximity functions over certain compact groups. Mean values are found of the ultrametric valuation of certain rational functions associated with a divisor on an abelian variety, and it is shown how these may be expressed in terms of an integral, thus finding the analogue, for an abelian variety, of Mahler's definition of the measure of a polynomial. These integrals are shown to arise in a manner which mimics classical Riemann sums, and their relation with the global canonical height is investigated. It is shown that the measure is a rational multiple of log p. Similar results are given for elliptic curves, taking the divisor to be the identity of the group law, and somewhat stronger mean value theorems proven in this more specific case by working directly with local canonical heights rather than approaching them through related functions. Effective asymptotic formulae for the local height are derived, first for the kernel of reduction of a curve and then, via a detailed analysis of the local reduction of the curve, for the group of rational points. The theory of uniform distribution is used to show that the mean value also takes an integral form in the case of an archimedean valuations, and recent inequalities for elliptic forms in logarithms are used to give error terms for the convergence towards the measure. This is undertaken first for the local height on an elliptic curve, and then, in terms of general theta-functions, on an abelian variety. We then seek to exploit these generalisations of the Mahler measure to yield an alternative method to that of Silverman and Tate for the determining of the global height. The integration over a cyclic group of the laws satisfied locally by the height allows us to reformulate our theorems in a manner conducive to practical application. It is demonstrated how our asymptotic formulae may be used together with an appropriate computer software package, PARI in our case, to calculate the mean value of heights, and, more generally, of rational functions, on an elliptic curve and on abehan varieties of higher genus. Some such calculations are displayed, with comments on their efficacy and their possible future development.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Pure mathematics