**On the Continuity of the Geometric Side of the Trace Formula **

* Tobias Finis, Erez Lapid *

**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 [10]. The geometric side is decomposed according to the following equivalence relation on $G(\mathbb{Q})$: *γ* _{1}∼*γ* _{2} if *γ* _{1} and *γ* _{2} are conjugate in $G(\overline{\mathbb{Q}})$ 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.