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

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.

14 pages (6 pages are appendices of calculations for additional families). Updated version with some minor typos corrected. A shorter version (minus the appendices) will appear in Comptes Rendus Mathematiques

References in corpus (4)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• On the frequency of vanishing of quadratic twists of modular L-functions
• Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves
• Rank distribution in a family of cubic twists

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• Effective equidistribution and the Sato-Tate law for families of elliptic curves
• An identity for sums of polylogarithm functions
• Lower-Order Biases Second Moments of Dirichlet Coefficients in Families of $L$-Functions
• Determining Optimal Test Functions for Bounding the Average Rank in Families of $L$-Functions
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• Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions
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• Fredholm Theory and Optimal Test Functions for Detecting Central Point Vanishing Over Families of L-functions
• Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves
• Low-lying Zeros of Number Field $L$-functions
• Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters
• An elliptic curve test of the L-Functions Ratios Conjecture