The selberg trace formula has two forms: one is for a setting of a semi simple group $G$ and its cocompact subgroup $\Gamma$, and relates the geometric and spectral side of the canonical automorphic representation of $G$, which is normally we are talking about, another is for a Laplacian operator on a Riemannian surface, and relates the spectrum of it with the length of geodesic curves, which seems not a special case of the first form, although of course they have much similarity. I am just wondering if the second form can be reduced from the first form, what is the relation of them?
2026-02-22 21:45:13.1771796713
on Selberg trace formula
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