On the continuity of the geometric side of the trace formula

TL;DR

This paper extends the geometric side of Arthur's trace formula to a broader class of test functions, ensuring continuity and approximation properties, thus enhancing the analytical framework for automorphic forms.

Contribution

It provides a continuous extension of the geometric side of the trace formula to non-compactly supported test functions, with a detailed decomposition and approximation method.

Findings

The geometric side is shown to be continuous on a larger test function space.

Decomposition according to conjugacy classes and semisimple parts is established.

Terms can be approximated by truncated integrals with continuous error control.

Abstract

We extend the geometric side of Arthur's non-invariant trace formula for a reductive group $G$ defined over $\mathbb{Q}$ continuously to a natural space $\mathcal{C}(G(\mathbb{A}^1))$ of test functions which are not necessarily compactly supported. The analogous result for the spectral side was obtained in [MR2811597]. The geometric side is decomposed according to the following equivalence relation on $G(\mathbb{Q})$: $\gamma_1\sim\gamma_2$ if $\gamma_1$ and $\gamma_2$ are conjugate in $G(\bar{\mathbb{Q}})$ and their semisimple parts are conjugate in $G(\mathbb{Q})$. All terms in the resulting decomposition are continuous linear forms on the space $\mathcal{C}(G(\mathbb{A})^1)$, and can be approximated (with continuous error terms) by naively truncated integrals.