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Multi-scale Vandermonde test kernels for spectral trace formulas

Stefan Horvath

TL;DR

This work addresses the challenge of designing test kernels for spectral trace formulas that are simultaneously positive, exhibit rapid off-diagonal decay, and control the diagonal contribution. The authors construct a PSD kernel by factorizing $h_T = g_T \star \widetilde{g}_T$, where $g_T$ is built from a multi-scale Vandermonde framework to achieve $J$-fold moment annihilation, yielding super-polynomial decay of off-diagonal terms. Under Weyl-law and Bessel/Airy-type hypotheses, they prove a Master-Bound $\mathfrak{E}_{\mathrm{tot}}(T) \ll T^{d+1-\delta}$ with a positive $\delta$ depending only on the symmetry order $k$ and annihilation depth $J \asymp \sqrt{(\log T)/k}$, while the Vandermonde coefficients grow like $T^{c_0^2/2+o(1)}$ with $c_0<\sqrt{2}$. The construction yields a stable, mass-positive proxy space and is admissible for standard trace formulas (e.g., Kuznetsov/Bruggeman) due to Paley–Wiener-type properties and absolute convergence of orbital integrals. Overall, the approach delivers a rigourous method to attain stronger power savings in spectral mean-value estimates across a broad class of locally symmetric spaces.

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 $m_{h_T}(π) = |m_{g_T}(π)|^2 \ge 0$; (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.

Multi-scale Vandermonde test kernels for spectral trace formulas

TL;DR

This work addresses the challenge of designing test kernels for spectral trace formulas that are simultaneously positive, exhibit rapid off-diagonal decay, and control the diagonal contribution. The authors construct a PSD kernel by factorizing , where is built from a multi-scale Vandermonde framework to achieve -fold moment annihilation, yielding super-polynomial decay of off-diagonal terms. Under Weyl-law and Bessel/Airy-type hypotheses, they prove a Master-Bound with a positive depending only on the symmetry order and annihilation depth , while the Vandermonde coefficients grow like with . The construction yields a stable, mass-positive proxy space and is admissible for standard trace formulas (e.g., Kuznetsov/Bruggeman) due to Paley–Wiener-type properties and absolute convergence of orbital integrals. Overall, the approach delivers a rigourous method to attain stronger power savings in spectral mean-value estimates across a broad class of locally symmetric spaces.

Abstract

We construct a family of test kernels for use in spectral trace formulas on locally symmetric spaces. The key innovation is the factorization , which simultaneously achieves: (i) automatic positive semi-definiteness of the spectral multiplier ; (ii) -fold moment annihilation via a multi-scale Vandermonde construction, yielding super-polynomial decay of all error terms; (iii) uniform spectral parameter bounds (Master-Bound) with depending only on the symmetry order and the annihilation depth , representing a power saving over the main term . The cost is a controlled polynomial growth 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 setting.
Paper Structure (16 sections, 17 theorems, 15 equations)

This paper contains 16 sections, 17 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Statement of main results
  4. Notation and analytic hypotheses
  5. The symmetry order and spectral measure
  6. The base kernel
  7. The Vandermonde system
  8. Moment asymptotics
  9. Vandermonde factorization and the annihilation system
  10. Vandermonde conditioning and stability
  11. Factorization and the proxy/product spaces
  12. Super-polynomial off-diagonal decay
  13. Global error accounting and Master-Bound
  14. Admissibility for trace formulas
  15. Summary
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k\ge 1$, and let $h_{\mathrm{base}}\in C_c^\infty((0,\infty))$ satisfy H1--H4 below. Then for every there exist unique coefficients $\alpha_0,\ldots,\alpha_J\in\mathbb{R}$ with $\max_j|\alpha_j|\le T^{c_0^2/2+o(1)}$ such that the multi-scale proxy satisfies the $J$-fold moment annihilation:

Theorems & Definitions (39)

  • Theorem 1.1: Existence and efficiency
  • Theorem 1.2: PSD and spectral mass
  • Theorem 1.3: Super-polynomial off-diagonal decay
  • Theorem 1.4: Global Master-Bound
  • Definition 2.1: Symmetry order
  • Remark 2.4: Verification for $\mathop{\mathrm{GL}}\nolimits(2)$
  • Definition 2.5: Base kernel
  • Proposition 3.1: Moment matrix structure
  • proof
  • Lemma 3.2: Factorization
  • ...and 29 more