Global discrepancy and small points on elliptic curves

Details Global discrepancy and small points on elliptic curves Global discrepancy and small points on elliptic curves

2005-07-11

arxiv.org/abs/math/0507228v1

Mathematics

Number Theory

33 pages, 1 figure

Scientific paper

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 of algebraic points on E which in a precise sense measures how far the set is from being adelically equidistributed. We then 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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