Root Number Equidistribution for Self-Dual Automorphic Representations on $GL_N$
Rahul Dalal, Mathilde Gerbelli-Gauthier
TL;DR
The paper studies root numbers $\ε(1/2, \pi)$ for self-dual automorphic representations on $\mathrm{GL}_N$ over a totally real field and establishes equidistribution between $\pm 1$ in the symplectic case under mild level conditions, extending the classical $\mathrm{GL}_2$ weight-aspect results to higher rank via Arthur’s trace formula and endoscopic classification. It develops a comprehensive framework combining newvector theory, Rankin–Selberg integrals, twisted traces, and carefully constructed test functions $\widetilde{E}^\infty_\mathfrak{n}$ and $\widetilde{C}^\infty_\mathfrak{n}$ to count representations by root numbers, while controlling contributions from oldforms and central elements through Shalika germs and stable endoscopic transfer. The arguments rely on endoscopic results (Art13, Mok15) and a detailed analysis of central-term transfers, with a conditionality discussion tied to the (unpublished) weighted fundamental lemma, now addressed in related work. In addition to the automorphic results, the authors derive equidistribution statements for associated families of $N$-dimensional Galois representations under a Langlands correspondence, tying the analytic statistics to arithmetic objects and indicating potential applications in non-abelian Iwasawa theory and the arithmetic of higher-dimensional abelian varieties. The methods showcase how deep trace-formula techniques, refined shape theory, and Germ calculus can extend classical parities to high rank, providing a blueprint for analyzing level-aspect and more general endoscopic phenomena in automorphic families. The work thus advances understanding of parity phenomena in automorphic $L$-functions and their arithmetic consequences, with implications for the distribution of root numbers in high-rank families and for the construction of symplectic Galois representations with prescribed conductors and Hodge–Tate weights.
Abstract
Let $F$ be a totally real field. We study the root numbers $ε(1/2, π)$ of self-dual cuspidal automorphic representations $π$ of $\mathrm{GL}_{2N}/F$ with conductor $\mathfrak n$ and regular integral infinitesimal character $λ$. If $π$ is orthogonal, then $ε(1/2, π)$ is known to be identically one. We show that for symplectic representations, the root numbers $ε(1/2, π)$ equidistribute between~$\pm 1$ as $λ\to \infty$, 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.