Replication Descent and J-Finality of Replicable Functions
Abstract
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 Γ0(N), then its nth replicate is invariant under Γ0(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.
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BibTeX
@misc{culf2026replication,
title = {{Replication Descent and $J$-Finality of Replicable Functions}},
author = {Eric Culf and Abdellah Sebbar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08243},
eprint = {2609.08243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08243},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Replication Descent and J-Finality of Replicable Functions
AU - Eric Culf
AU - Abdellah Sebbar
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08243
UR - https://arxiv.org/abs/2609.08243
AB - 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 nth 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.
ER -