= Heat trace
{title2=$Z_M(t)=\operatorname{Tr}(e^{-t\Delta_M})$}
For the <positive Laplace-Beltrami operator> on a closed compact <Riemannian manifold>, the <heat trace> is $Z_M(t)=\sum_j e^{-t\lambda_j}=\int_M K_M(t,x,x)\,dV(x)$. It records every <eigenvalue> with multiplicity. In a finite free isometric quotient it equals $|U|^{-1}\sum_{u\in U}\int_NK_N(t,x,ux)\,dV(x)$. The orbital integrals form a <class function> of the ambient finite isometry group, so <Gassmann equivalence> gives equal quotient <heat traces> and hence <Sunada theorem>.
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