Abstract
Let E be an elliptic curve defined over a number field k. In this paper, we define the “global discrepancy” of a finite set Z ⊂ E(k) 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 padic 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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