The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group \(G\) over \(\mathbf {Z}\), and provide numerical applications in absolute rank \(\leq 8\). Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of \(\mathrm{GL}_{n}\) over \(\mathbf {Q}\) (\(n\) arbitrary) whose motivic weight is \(\leq 24\).

In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on \(\mathrm{GL}_{n}\), and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of \(G\), for an appropriate test function.

Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for \(\mathrm{Sp}_{2g}(\mathbf {Z})\): we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.

本文的目的是双重的。首先，我们引入了一种新方法，用于评估在\(\mathbf {Z}\)上的分割经典群\(G\)的 1 级自守形式空间中给定离散级数表示的重数，并提供数值应用绝对排名\(\leq 8\)。其次，我们证明了\(\mathrm{GL}_{n}\)在\(\mathbf {Q}\ ) 上的一级尖尖代数自守表示的分类结果（\(n\)任意），其动机权重是\(\leq 24\)。

在这两种情况下，关键要素是基于 Weil 显式公式的经典方法，它可以反驳\(\mathrm{GL}_{n}\)上某些一级代数尖点形式的存在，并且我们推本文将进一步阐述。对于适当的测试函数，我们使用这些消失结果来获得对 \(G\)迹线公式的几何边的椭圆部分的可以说是“毫不费力”的计算。

这些结果对计算\(\mathrm{Sp}_{2g}(\mathbf {Z})\)的（可能是向量值）Siegel 模尖形空间的维数有影响：我们恢复了之前的所有不依赖任何已知案例，并通过统一且“轻松”的方法走得更远。