发表机构
Institute for Quantum Computing, University of Waterloo; Department of Mathematics and Statistics, University of Ottawa（滑铁卢大学量子计算研究所; 渥太华大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明复制下降法则：同余不变可复制函数的复制保持同余对称性，其指数被模整除的复制必为J，从而给出J-终结性的免分类证明，并刻画有限阶完全可复制函数的终端函数。
AI 中文摘要
我们证明复制将同余对称性沿一个显式的层级塔向下传递。若一个在上半平面全纯的归一化可复制函数 $f$ 在 $\u0393_0(N)$ 下不变，则其第 $n$ 次复制在 $\u0393_0(N/(N,n))$ 下不变。因此，每个指数被 $N$ 整除的复制都是归一化模不变量 $J=j-744$。这给出了同余不变可复制函数的 $J$-终结性的直接且免分类的证明。该论证仅使用复制恒等式和同余子群的一个基本生成定理；它既不需要完全可复制性，也不需要对傅里叶系数的算术假设。然后我们将下降法则应用于有限复制阶的完全可复制函数。它们的复制塔具有一个典范的终端复制，且唯一可能的终端函数是 $J$、$q^{-1}$ 和 $q^{-1}+q$。此外，一个有限阶完全可复制函数是 $J$-终结的，当且仅当其终端复制是 $J$，并且任何超越平移的对称性都迫使这一选择。
英文摘要
We show that replication carries congruence symmetry down an explicit level tower. If a normalized replicable function $f$, holomorphic on the upper half-plane, is invariant under $Γ_0(N)$, then its $n$th replicate is invariant under $Γ_0(N/(N,n))$. Thus every replicate whose index is divisible by $N$ is the normalized modular invariant $J=j-744$. This gives a direct and classification-free proof of $J$-finality for congruence-invariant replicable functions. The argument uses only the replication identities and an elementary generation theorem for congruence subgroups; it requires neither complete replicability nor arithmetic hypotheses on the Fourier coefficients. We then apply the descent law to completely replicable functions of finite replication order. Their replication towers have a canonical terminal replicate, and the only possible terminal functions are $J$, $q^{-1}$, and $q^{-1}+q$. Moreover, a finite-order completely replicable function is $J$-final precisely when its terminal replicate is $J$, and any symmetry beyond translations forces this alternative.
