Cyclic Kronecker Fourier Transforms, Weber Norms, and Rogers-Ramanujan Singular Values
Abstract
Let K be an imaginary quadratic field with fundamental discriminant D=−5n, where n>0 is odd, and cyclic ideal class group Cl(K)=⟨ g⟩≃ C4M. Assume that the ramified prime class above $5$ is the unique involution g2M. For the Kronecker eta invariant F(A)=|η(zA)|2/√(aA), where zA and aA arise from a positive definite quadratic form representing the ideal class A, we show that the logarithms of the mirror ratios Rj=(F(g2M−j))/F(gj), 0≤ 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 ℚ(Rj)=H⟨ c⟩, Rjσ2M=Rj−1, ℚ(Rj+Rj−1)=H⟨ c,σ2M⟩. Here H is the Hilbert class field, c denotes complex conjugation, and σ corresponds to g under the Artin map. Consequently [ℚ(Rj):ℚ]=4M, [ℚ(Rj+Rj−1):ℚ]=2M, and the latter degree becomes M over the real genus field ℚ(√5). Thus the degree pattern 2r,2r−1,2r−2 in the cyclic $2$-power hierarchy is a specialization of a uniform C4M 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.
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BibTeX
@misc{jha2026cyclic,
title = {{Cyclic Kronecker Fourier Transforms, Weber Norms, and Rogers-Ramanujan Singular Values}},
author = {Sumit Kumar Jha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18910},
eprint = {2610.18910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18910},
}
RIS
TY - UNPB
TI - Cyclic Kronecker Fourier Transforms, Weber Norms, and Rogers-Ramanujan Singular Values
AU - Sumit Kumar Jha
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.18910
UR - https://arxiv.org/abs/2610.18910
AB - Let K be an imaginary quadratic field with fundamental discriminant D=−5n, where n>0 is odd, and cyclic ideal class group Cl(K)=⟨ g⟩≃ C_4M. Assume that the ramified prime class above $5$ is the unique involution g^2M. For the Kronecker eta invariant F(A)=|η(z_A)|^2/√(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=(F(g^2M−j))/F(g^j), 0≤ 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 ℚ(R_j)=H^⟨ c⟩, R_j^σ^2M=R_j^−1, ℚ(R_j+R_j^−1)=H^⟨ c,σ^2M⟩. Here H is the Hilbert class field, c denotes complex conjugation, and σ corresponds to g under the Artin map. Consequently [ℚ(R_j):ℚ]=4M, [ℚ(R_j+R_j^−1):ℚ]=2M, and the latter degree becomes M over the real genus field ℚ(√5). 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.
ER -