Multi-scale Vandermonde test kernels for spectral trace formulas
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913011192561664 |
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| author | Horvath, Stefan |
| author_facet | Horvath, Stefan |
| contents | 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 $m_{h_T}(π) = |m_{g_T}(π)|^2 \ge 0$; (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) $\mathfrak{E}_{\mathrm{tot}}(T) \ll T^{d+1-δ}$ with $δ> 0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11205 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multi-scale Vandermonde test kernels for spectral trace formulas Horvath, Stefan Number Theory 11F72 (primary), 11F25, 42B20 (secondary) 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 $m_{h_T}(π) = |m_{g_T}(π)|^2 \ge 0$; (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) $\mathfrak{E}_{\mathrm{tot}}(T) \ll T^{d+1-δ}$ with $δ> 0$ 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. |
| title | Multi-scale Vandermonde test kernels for spectral trace formulas |
| topic | Number Theory 11F72 (primary), 11F25, 42B20 (secondary) |
| url | https://arxiv.org/abs/2602.11205 |