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

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

2005-06-22

arxiv.org/abs/math/0506461v2

C. R. Math. Rep. Acad. Sci. Canada 27 (2005), no. 4, 111--120

Mathematics

Number Theory

14 pages (6 pages are appendices of calculations for additional families). Updated version with some minor typos corrected. A

Scientific paper

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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