Root Number Equidistribution for Self-Dual Automorphic Representations on $GL_N$

Let $F$ be a totally real field. We study the root numbers $\epsilon(1/2, \pi)$ of self-dual cuspidal automorphic representations $\pi$ of $\mathrm{GL}_{2N}/F$ with conductor $\mathfrak n$ and regular integral infinitesimal character $\lambda$. If $\pi$ is orthogonal, then $\epsilon(1/2, \pi)$ is known to be identically one. We show that for symplectic representations, the root numbers $\epsilon(1/2, \pi)$ equidistribute between~$\pm 1$ as $\lambda \to \infty$, provided that there exists a prime dividing $\mathfrak n$ with power $>N$.We also study conjugate self-dual representations with respect to a CM extension $E/F$, where we obtain a similar result under the assumption that $\mathfrak n$ is divisible by a large enough power of a ramified prime and provide evidence that equidistribution does not hold otherwise. In cases where there are known to be associated Galois representations, we deduce root number equidistribution results for the corresponding families of $N$-dimensional Galois representations. The proof generalizes a classical argument for the case of $\mathrm{GL}_2/\mathbb Q$ by using Arthur's trace formula and the endoscopic classification for quasisplit classical groups similarly to a previous work (arxiv:2212.12138). The main new technical difficulty is evaluating endoscopic transfers of the required test functions at central elements.