发表机构
C. N. Yang Institute for Theoretical Physics, Stony Brook University; Weinberg Institute, The University of Texas at Austin（纽约州立大学石溪分校杨振宁理论物理研究所; 德克萨斯大学奥斯汀分校温伯格研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过尖点限制构造次级不变量，证明带层级结构的Bunke--Naumann不变量在非平凡层级上消失，并利用同伦与模形式方法给出完整证明。
AI 中文摘要
模曲线的全尖点除子上的限制定义了一个次级不变量，其有理不确定性来源于单个整体模形式。对于每个权被四整除的整弱全纯一级模形式 $h$，我们证明在每一个非平凡的 $\Gamma_0(N)$ 层级上，导入值 $[h/2]$ 消失。该构造在完备化之前局部化系数，并且仅在过渡到同伦群之后才进行有理化。在奇素数层级，我们将全尖点谱识别为两个实 Tate 因子的乘积，并构造一个全纯权二修正。该修正产生一个实际的整整体同伦类和一个有理整体源类，满足联合湮灭所需的等式。该等式可传递到每个奇复合层级，而偶层级通过反转二得到。在层级三，Mahowald--Rezk 同伦计算仅留下茎 $8K+3$ 中的周期 $\nu$ 族。一个尖点留数同态检测其次级值，该值在整个周期族中具有精确的阶二。作为一个应用，双次数 $(8k+1,8k'+2)$ 中的乘积的 Bunke--Naumann 次级不变量在过渡到每个非平凡层级后消失。
英文摘要
Restriction to the full cusp divisor of a modular curve defines a secondary invariant whose rational indeterminacy comes from a single global modular form. For every integral weakly holomorphic level-one modular form $h$ of weight divisible by four, we prove that the imported value $[h/2]$ vanishes at every nontrivial $Γ_0(N)$ level. The construction localizes coefficients before completion and rationalizes only after passing to homotopy groups. At odd prime level, we identify the full cusp spectrum as a product of two real Tate factors and construct a single holomorphic weight-two correction. This correction yields an actual integral global homotopy class and a rational global source class satisfying the equality required for joint annihilation. The equality transports to every odd composite level, while even levels follow by inverting two. At level three, the Mahowald--Rezk homotopy calculation leaves only the periodic $ν$ family in stems $8K+3$. A cusp-residue homomorphism detects its secondary value, which has exact order two throughout the periodic family. As an application, the Bunke--Naumann secondary invariants of products in bidegrees $(8k+1,8k'+2)$ vanish after passage to every nontrivial level.
Comments16 pages
