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 $\xi_d^k$. As observed by Devin for the particular case $d=1$, the average root number over the family $\mathcal{F}_d$ is $\frac14$, which makes the symmetry type of the family (unitary, symplectic or orthogonal) somehow mysterious, as none of the symmetry types would lead to this average value. We 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 $4 N (\mathfrak{f}_d )$, and even weight $k+1$, and this larger family also has orthogonal symmetry, as proven by Iwaniec, Luo, Sarnak. This is joint work with L. Devin and E. Waxman.

Bio: Chantal David is a French Canadian mathematician, working in analytic number theory and arithmetic statistics. She earned her doctorate at McGill University under the supervision of Ram Murty. Her work has been recognized by the 2013 Krieger-Nelson Prize, given annually by the CMS to an outstanding female researcher in mathematics, and by a 2022 Concordia University Award, which recognizes leadership, training of students and overall influence of her scholarly work.