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PreprintCC BY 4.0arXiv

On the occurrence of congruence multiplicities between Ramanujan's theta functions

  • Shane Chern
  • Nicolas Allen Smoot
  • Dazhao Tang
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Abstract

Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions φ(−q) and ψ(q). In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for φ(−q) and ψ(q), which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups Γ0(6) and Γ0(10). In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.

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Cite

BibTeXRIS
BibTeX
@misc{chern2026occurrence,
  title = {{On the occurrence of congruence multiplicities between Ramanujan's theta functions}},
  author = {Shane Chern and Nicolas Allen Smoot and Dazhao Tang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02592},
  eprint = {2609.02592},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02592},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the occurrence of congruence multiplicities between Ramanujan's theta functions
AU  - Shane Chern
AU  - Nicolas Allen Smoot
AU  - Dazhao Tang
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02592
UR  - https://arxiv.org/abs/2609.02592
AB  - Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions φ(−q) and ψ(q). In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for φ(−q) and ψ(q), which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups Γ_0(6) and Γ_0(10). In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.
ER  -