One-level densities in families of Gr\"ossencharakters associated to CM elliptic curves

TL;DR

This paper investigates the low-lying zeros of L-functions associated with CM elliptic curves, revealing a natural decomposition into symplectic and orthogonal subfamilies with explicit lower order terms.

Contribution

It provides a detailed analysis of the family of L-functions attached to CM elliptic curves, identifying their decomposition into symplectic and orthogonal subfamilies and computing explicit lower order density terms.

Findings

Approximately 25% of L-functions have negative root number.

The family decomposes into symplectic and orthogonal subfamilies based on congruence classes.

Explicit lower order terms in the one-level density are computed for each subfamily.

Abstract

We study the low-lying zeros of a family of $L$-functions attached to the CM elliptic curve $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free integer $d$. Specifically, upon writing the $L$-function of $E_d$ as $L(s-\frac12, \xi_d)$ for the appropriate Gr\"ossencharakter $\xi_d$ of conductor $\mathfrak{f}_d$, we consider the collection $\mathcal{F}_d$ of $L$-functions attached to $\xi_{d,k}$, $k \geq 1$, where for each integer $k$, $\xi_{d, k}$ denotes the primitive character inducing $\xi_d^k$. We observe that $25\%$ of the $L$-functions in $\mathcal{F}_d$ have negative root number. $\mathcal{F}_d$ is thus not one of the essentially homogeneous families of the Universality Conjecture of Sarnak, Shin and Templier, with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one-level density in the family of $L$-functions in $\mathcal{F}_{d}$ with conductor at most $K^2 \mathrm N (\mathfrak{f}_d)$, we find that $\mathcal{F}_d$ naturally decomposes into subfamilies: more specifically, a collection of symplectic ($L(s, \xi_{d,k})$ for $k \equiv \alpha \bmod 8$, $\alpha$ even) and orthogonal ($L(s, \xi_{d,k})$ for $k \equiv \alpha \bmod 8$, $\alpha$ odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of $\log (K^2 \mathrm N(\mathfrak{f}_d))$.