One-level densities in families of Grössencharakters associated to CM elliptic curves

Abstract: We study the low-lying zeros of a family of $L$-functions attached to the CM elliptic curves $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free integer $d$. Writing the $L$-function of $E_d$ as $L(s-\frac12, \xi_d)$ for the appropriate Gr\"ossencharakter $\xi_d$ of conductor $\mathfrak{f}d$, the family $\mathcal{F}_d$ is defined as the family of $L$-functions attached to the Gr\"ossencharakters $\xi{d,k}$, where for each integer $k \geq 1$, $\xi_{d, k}$ denotes the primitive character inducing $\xi_d^k$. We observe that $25\%$ of the functions in $\mathcal{F}d$ have negative root number. This makes the symmetry type of the family (unitary, symplectic or orthogonal) somewhat mysterious, as none of the symmetry types would lead to this proportion. We give an asymptotic expression for the one-level density in the family of $L$-functions in $\mathcal{F}{d}$ with conductor at most $K^2 \mathrm{N} (\mathfrak{f}d)$, and find that $\mathcal{F}_d$ breaks down into two natural subfamilies; namely, a symplectic family ($L(s, \xi{d,k})$ for $k$ even) and an orthogonal family ($L(s, \xi_{d,k})$ for $k$ odd). For $k$ odd, $\mathcal{F}_d$ is in fact a subfamily of the automorphic forms of fixed level $4 \mathrm{N} (\mathfrak{f}_d )$, and even weight $k+1$, and this larger family also has orthogonal symmetry. Finally, we compute explicit lower order terms in decreasing powers of $\log (K^2 \mathrm{N} (\mathfrak{f}_d) )$ for each case.