Multi-scale Vandermonde test kernels for spectral trace formulas

Abstract

We construct a family of test kernels for use in spectral trace formulas on locally symmetric spaces. The key innovation is the factorization $h_T = g_T \star \widetilde{g}_T$, which simultaneously achieves: (i) automatic positive semi-definiteness of the spectral multiplier $m_{h_T}(\pi) = |m_{g_T}(\pi)|^2 \ge 0$; (ii) $J$-fold moment annihilation via a multi-scale Vandermonde construction, yielding super-polynomial decay of all error terms; (iii) uniform spectral parameter bounds (Master-Bound) $\mathfrak{E}_{\mathrm{tot}}(T) \ll T^{d+1-\delta}$ with $\delta > 0$ depending only on the symmetry order $k$ and the annihilation depth $J \asymp \sqrt{(\log T)/k}$, representing a power saving over the main term $\asymp T^{d+1}$. The cost is a controlled polynomial growth $T^{c_0^2/2+o(1)}$ in the Vandermonde coefficients (with exponent strictly less than 1), which is dominated by the super-polynomial decay of the off-diagonal terms. The construction is axiomatized over two analytic hypotheses -- a Weyl law and Bessel/Airy asymptotics -- making it applicable beyond the classical $\mathrm{GL}(2)$ setting.