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

TL;DR

This paper investigates the variability in the number of points on elliptic curves over finite fields, establishing sharp bounds for moments, and explores implications for the average rank in families of elliptic curves.

Contribution

It demonstrates the sharpness of Michel's second moment bound by analyzing specific elliptic curve families and links lower order moments to zero distributions and rank bounds.

Findings

The second moment bound p^2 + O(p^{3/2}) is sharp for certain families.

Lower order terms influence the behavior of zeros near the central point.

Implications for average rank bounds in elliptic curve families.

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.