# Lower order terms in the 1-level density for families of holomorphic cuspidal newforms

## TL;DR

This paper investigates the subtle lower order terms in the 1-level density of zeros for families of holomorphic cuspidal newforms, revealing family-dependent corrections that depend on arithmetic properties and extending the understanding of universality in zero distributions.

## Contribution

It derives a simplified explicit formula for GL(2) L-functions that isolates family-dependent lower order terms, highlighting differences based on arithmetic features like CM and rank.

## Key findings

- Lower order correction terms vary with family properties.
- Explicit formula simplifies comparison of zero distributions.
- Biases in Fourier coefficients relate to family rank and CM status.

## Abstract

The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting behavior. We break this universality and find family dependent lower order correction terms in many cases; these lower order terms have applications ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) L-functions which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the moments of the Fourier coefficients lambda_f(p). Our formula highlights the differences that we expect to exist from families whose Fourier coefficients obey different laws (for example, we expect Sato-Tate to hold only for non-CM families of elliptic curves). Further, by the work of Rosen and Silverman we expect lower order biases to the Fourier coefficients in families of elliptic curves with rank over Q(T); these biases can be seen in our expansions. We analyze several families of elliptic curves and see different lower order corrections, depending on whether or not the family has complex multiplication, a forced torsion point, or non-zero rank over Q(T).