Low-lying zeros in families of elliptic curve \(L\)-functions over function fields (Q2081331)

In the paper under review, the authors consider the family \(\mathcal{F}_N\) of \(L\)-functions attached to quadratic and cubic twists of degree \(N\) of a fixed elliptic curve given by the minimal Weierstrass equation \(y^2 = x^3 + Ax + B\), where \(A\), \(B \in \mathbb{F}_q[T]\). Recently, \textit{A. Comeau-Lapointe} [J. Number Theory J. 241, 165--197 (2022; Zbl 07575694)] computed the expected value of the powers of trace of the Frobenius over this family: \[ \langle\mathrm{Tr}(\Theta^n)\rangle_{\mathcal{F}_N}= \eta_2(n)+O\left(\frac{1}{q^{(\frac{1}{8}-\varepsilon)N}}+\frac{q^{n/2}}{q^{(1-\varepsilon)N}}+\frac{n^2}{q^{n/4}}\right). \] The authors determine lower order terms in the above estimate. They specifically derive the exact form of the error term \(\frac{n^2}{q^{n/4}}\). The article continues by considering the one-level density of an \(N\times N\) unitary matrix \(U\): \[ Z_\phi(U):=\sum_{j=1}^N\sum_{k\in\mathbb{Z}}\phi\left(N\left(\frac{\theta_j}{2\pi}-k\right)\right), \] where the \(\theta_j\) are the eigenangles of \(U\), and \(\phi\) is any even Schwartz test function. The second result uses the first result to show that as long as \(\mathrm{supp}(\widehat{\phi})\subset(-1,1)\), then the expected value of the one-level density for the family \(\mathcal{F}_N\) matches the orthogonal matrices with a deviation term: \[ \langle Z_{\phi}(\Theta)\rangle_{\mathcal{F}_N}=\int_{O(N)}Z_{\phi}(U)dU+ \frac{\mathrm{dev}_E(\phi)}{N}+O_\varepsilon\left(\frac{1}{N^{2-\varepsilon}}\right). \] Similar formulas are derived for the cubic twists.