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International Mathematics Research Notices (2005) 2005:3791-3834, doi:10.1155/IMRN.2005.3791
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Copyright © 2005 Hindawi Publishing Corporation. All rights reserved.

Global discrepancy and small points on elliptic curves

Matthew Baker and Clayton Petsche

Let E be an elliptic curve defined over a number field k. In this paper, we define the "global discrepancy" of a finite set Formula which in a precise sense measures how far the set is from being adelically equidistributed. We prove an upper bound for the global discrepancy of Z in terms of the average canonical height of points in Z. We deduce from this inequality a number of consequences. For example, we give a new and simple proof of the Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves. We also prove a non-Archimedean version of the Szpiro-Ullmo-Zhang theorem which takes place on the Berkovich analytic space associated to E. We then prove some quantitative "non-equidistribution" theorems for totally real or totally p-adic small points. The results for totally real points imply similar bounds for points defined over the maximal cyclotomic extension of a totally real field.


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