{
  "abstract": "Let $K$ be an imaginary quadratic field with fundamental discriminant $D=-5n$, where $n>0$ is odd, and cyclic ideal class group $\\text{Cl}(K)=\\langle g\\rangle\\simeq C_{4M}$. Assume that the ramified prime class above $5$ is the unique involution $g^{2M}$. For the Kronecker eta invariant \\[ F(A)=\\frac{|η(z_A)|^2}{\\sqrt{a_A}}, \\] where $z_A$ and $a_A$ arise from a positive definite quadratic form representing the ideal class $A$, we show that the logarithms of the mirror ratios \\[ R_j=\\frac{F(g^{2M-j})}{F(g^j)},\\qquad 0\\le j<M, \\] are related by an explicit discrete cosine transform to the odd class-character Hecke $L(1)$-values. A single level-$5$ generalized-Weber orbit realizes all of these mirrors. Combining Fourier nonvanishing, lower-power Weber class invariants and a ray-class cancellation argument, we prove \\[ \\mathbb{Q}(R_j)=H^{\\langle c\\rangle},\\qquad R_j^{σ^{2M}}=R_j^{-1},\\qquad \\mathbb{Q}(R_j+R_j^{-1})=H^{\\langle c,σ^{2M}\\rangle}. \\] Here $H$ is the Hilbert class field, $c$ denotes complex conjugation, and $σ$ corresponds to $g$ under the Artin map. Consequently \\[ [\\mathbb{Q}(R_j):\\mathbb{Q}]=4M,\\qquad [\\mathbb{Q}(R_j+R_j^{-1}):\\mathbb{Q}]=2M, \\] and the latter degree becomes $M$ over the real genus field $\\mathbb{Q}(\\sqrt5)$. Thus the degree pattern $2^r,2^{r-1},2^{r-2}$ in the cyclic $2$-power hierarchy is a specialization of a uniform $C_{4M}$ theorem. We give exact low-class-number evaluations and certified computations at $D=-895,-3295,-8495$, separating respectively four, eight and sixteen non-genus Hecke $L(1)$-values and producing corresponding alternating Rogers-Ramanujan singular values. Classical mixed modular equations then give ordinary Rogers--Ramanujan companions at the same arguments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18910",
  "author_total": 1,
  "authors": [
    {
      "name": "Sumit Kumar Jha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cscc96btx8tnf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18910",
  "fulltext_url": null,
  "id": "cscc96btx8tnf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18910",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7221982032",
  "pdf_url": "https://arxiv.org/pdf/2610.18910",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cscc96btx8tnf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.18910",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18910"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cyclic Kronecker Fourier Transforms, Weber Norms, and Rogers-Ramanujan Singular Values",
  "title_text": "Cyclic Kronecker Fourier Transforms, Weber Norms, and Rogers-Ramanujan Singular Values",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/cscc96btx8tnf",
  "venue": "arXiv",
  "versions": []
}