# Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

## TL;DR

This paper introduces a new method using the Weil explicit formula to evaluate discrete series multiplicities for classical groups over integers and classifies certain automorphic representations, simplifying computations in automorphic forms theory.

## Contribution

A novel approach based on the Weil explicit formula for computing automorphic form multiplicities and classifying level one algebraic cusp forms with bounded motivic weight.

## Key findings

- New method for multiplicity evaluation in automorphic forms
- Classification of level one cuspidal automorphic representations with weight ≤ 24
- Simplified computation of Siegel modular cuspform spaces

## Abstract

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ 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 ${\rm GL}_n$, 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 ${\rm Sp}_{2g}(\mathbb{Z})$: we recover all the previously known cases without relying on any, and go further, by a unified and ``effortless'' method.