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arXiv · 1512.08753

On the continuity of the geometric side of the trace formula

Tobias Finis·Erez Lapid
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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.

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Tobias Finis, Erez Lapid. 2015-12-29. On the continuity of the geometric side of the trace formula. https://doi.org/10.1007/s40306-016-0176-x

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