On the Continuity of the Geometric Side of the Trace Formula

We extend the geometric side of Arthur’s non-invariant trace formula for a reductive group G defined over ℚ$\mathbb {Q}$ continuously to a natural space C(G(A)1)$\mathcal {C}(G(\mathbb {A})^{1})$ 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(ℚ)$G(\mathbb {Q})$: γ1∼γ2 if γ1 and γ2 are conjugate in G(ℚ¯)$G(\overline {\mathbb {Q}})$ and their semisimple parts are conjugate in G(ℚ)$G(\mathbb {Q})$. All terms in the resulting decomposition are continuous linear forms on the space C(G(A)1)$\mathcal {C}(G(\mathbb {A})^{1})$, and can be approximated (with continuous error terms) by naively truncated integrals.