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 $\text{Cl}(K)=\langle g\rangle\simeq C_{4M}$. 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.