Lower Order Terms for the One-Level Density of Elliptic Curve L-Functions

TL;DR

This paper calculates the finite conductor 1-level density of elliptic curve L-functions using ratios conjectures, revealing lower order terms that explain zero statistics but not the discrepancy near the critical point.

Contribution

It provides a formula for the 1-level density including lower order terms for finite conductors, highlighting limitations near the critical point.

Findings

Lower order terms model zero statistics for moderate conductors.

Discrepancy near the critical point is not explained by current models.

More accurate models are needed for zeros close to the critical point.

Abstract

It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.