AI 中文总结
研究幂零覆盖上热核及幂零商中闭测地线渐近性，利用有理弗洛凯 - 布洛赫理论等，通过正罗克兰估计等论证给出逐点高阶热核展开式，还产生幂零切博塔廖夫型现象，计算了相关模型。
AI 中文摘要
我们建立了幂零覆盖上热核以及紧致双曲曲面幂零商固定中心类中素闭测地线的所有阶长时间渐近展开式。确切的格点侧输入来自配套论文的有限维有理弗洛凯 - 布洛赫理论：有理基里洛夫限制给出精确的有限维纤维，广义皮特利克泛函给出精确的傅里叶反演和归一化迹恒等式。在有理参数\(p/q\)处分解是精确的，纤维被积函数的波动仅由\(q\)控制。对于一般幂零模型，通过正罗克兰估计、普兰切尔 - 梅林公式和迹级阶平衡论证，系数加权谱和在每个固定阶都是合理的。该方法给出了真正局部的、逐点高阶热核展开式，与通常给出首项或积分埃奇沃思型渐近性的方法不同。相同的表示理论量控制闭测地线渐近性的首项，产生幂零切博塔廖夫型现象。计算了海森堡模型的一阶修正项，并通过四次振荡器的预解式和热核演算表示了恩格尔模型。
英文摘要
We establish all order long-time asymptotic expansions for heat kernels on nilpotent coverings and for prime closed geodesics in fixed central classes of nilpotent quotients of compact hyperbolic surfaces. The exact lattice-side input is the finite-dimensional rational Floquet-Bloch theory of the companion paper: rational Kirillov restrictions give exact finite-dimensional fibers, and a generalized Pytlik functional gives exact Fourier-inversion and normalized-trace identities. At a rational parameter $p/q$ the decomposition is exact, and the fluctuation of the fiber integrand is controlled only by $q$. Hence the large-denominator comparison with the smooth Kirillov or Schrödinger normal form is uniform on the rational support of the Pytlik functional; irrational parameters do not enter the rigorous trace argument. For general nilpotent models, coefficient-weighted spectral sums are justified to every fixed order by positive Rockland estimates, the Plancherel-Mellin formula, and a trace-level order-balance argument. In contrast with approaches which usually give leading terms or integrated Edgeworth-type asymptotics, the method gives genuinely local, pointwise higher-order heat-kernel expansions. The same representation-theoretic quantity governs the leading term in the closed-geodesic asymptotics, producing a nilpotent Chebotarev-type phenomenon. The Heisenberg model is computed to the first correction term, and the Engel model is represented through the resolvent and heat-kernel calculus of the quartic oscillato
Comments84 pages. Analytic companion to arXiv:2607.12069. Draws on and substantially revises the application part of the longer preprint arXiv:2509.16848
