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Variation in the number of points on elliptic curves and applications to excess rank

Steven J. Miller
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Abstract

Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is sharp by studying y^2 = x^3 + Tx^2 + 1. Lower order terms for such moments in a family are related to lower order terms in the n-level densities of Katz and Sarnak, which describe the behavior of the zeros near the central point of the associated L-functions. We conclude by investigating similar families and show how the lower order terms in the second moment may affect the expected bounds for the average rank of families in numerical investigations.

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Steven J. Miller. 2005-11-30. Variation in the number of points on elliptic curves and applications to excess rank. https://arxiv.org/abs/math/0506461

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