Nondegeneracy of the half-system matrix $N_{\ell}$ and the completion of Koo–Yoon's explicit construction of class fields over cyclotomic fields

Koo and Yoon (Kyoto J. Math. 2016) constructed, for odd primes $\ell$ and $p$, explicit class fields $K_\mu$ between consecutive ray class fields $k_\mu \subset k_{\mu+1}$ of the cyclotomic field $k=\mathbb{Q}(\zeta_\ell)$ of conductor $2p^\mu$, computed the degree $K_\mu/k_\mu$, and produced Hilbert-style primitive generators as special values of theta-quotients. Their results are conditional on two unproven conjectures and exclude two degenerate cases. The open problem is to settle the two conjectures — $\det(N_\ell)\neq0$ for every odd prime $\ell$, where $N_\ell$ is the $(0,1)$-matrix recording whether $ij \bmod \ell$ falls in the half-system $\{1,\dots,(\ell-1)/2\}$, and $\dim_{\mathbb{Z}/p\mathbb{Z}}(H_1/S_2)=n-1$ for almost all $p$, where $H_1/S_2$ measures the real-unit contribution to consecutive ray class groups — and to complete the primitive-generator construction when $p \mid \det(A_\ell(\mathbf{r},\mathbf{s}))$ and for theta-quotients that vanish or have a pole at the CM point, making the construction unconditional for every odd prime $\ell$ and every odd prime $p\nmid\ell h^+_\ell n$, away from the $\ell$-dependent exceptional set of part (2) below.

Background

A central theme of algebraic number theory is explicit class field theory, the program going back to Hilbert's twelfth problem of writing down concrete generators for the abelian extensions of a number field. The Kronecker–Weber theorem solves this over $\mathbb{Q}$ (every abelian extension of $\mathbb{Q}$ sits inside a cyclotomic field $\mathbb{Q}(\zeta_N)$ with $\zeta_N=e^{2\pi i/N}$), and the theory of complex multiplication handles imaginary quadratic base fields via singular values of modular functions; for general base fields, including the cyclotomic fields themselves, no comparable recipe is known.

For a number field $K$, an integral ideal $\mathfrak{m}$, and the ray subgroup of principal ideals admitting a generator $\equiv 1 \bmod \mathfrak{m}$, the generalized ideal class group is called the ray class group modulo $\mathfrak{m}$, and class field theory attaches to it a finite abelian extension $K_{\mathfrak{m}}/K$, the ray class field. Koo and Yoon study the case $K=k=\mathbb{Q}(\zeta_\ell)$ for an odd prime $\ell$ and the modulus $2p^\mu$, writing $k_\mu$ for the corresponding ray class field and $n=(\ell-1)/2$ so that $[k:\mathbb{Q}]=2n$. Their work proceeds along two complementary lines.

First, algebraically. Let $S_\mu=\{a\in k^{\times}: a\equiv 1 \pmod{2p^\mu}\}$, let $E$ be the unit group of $k$, and set $H_\mu=S_{\mu+1}(S_\mu\cap E)$; class field theory gives $\mathrm{Gal}(k_{\mu+1}/k_\mu)\simeq S_\mu/H_\mu$. The quotient $H_\mu/S_{\mu+1}$ is generated by real units of $k$ (Lemma 4.3 of the paper), such as the circular units $\xi_a=\zeta_{2\ell}^{1-a}(1-\zeta^a)/(1-\zeta)$, where $\zeta=\zeta_\ell$ and $1<a<\ell/2$ — explicitly written-down algebraic units lying in the maximal real subfield. The paper defines an endomorphism $\varphi^{\ast}(a)=\prod_{i=1}^{n}a^{\varphi_i^{-1}}$, where $\varphi_i(\zeta)=\zeta^i$, and lets $K_\mu$ be the class field of $k$ attached to the kernel of the induced endomorphism of $S_\mu/H_\mu$, so that $k_\mu\subset K_\mu\subset k_{\mu+1}$. The degree of $K_\mu/k_\mu$ is then governed by linear algebra modulo $p$ whose pivot structure is encoded in the $n\times n$ matrix $N_\ell=[n_{ij}]$ with $n_{ij}=1$ if the least positive residue of $ij$ modulo $\ell$ lies in the half-system $\{1,\dots,n\}$ and $n_{ij}=0$ otherwise (Lemma 4.4): if $p\nmid \ell h^+_\ell n$ then $\mathrm{Gal}(K_\mu/k_\mu)\simeq(\mathbb{Z}/p\mathbb{Z})^{\mathrm{rank}\,M_\ell(p)}$, where $M_\ell(p)$ is the $(n+1)\times 2n$ matrix over $\mathbb{Z}/p\mathbb{Z}$ whose $(i,j)$-entry is $1$ if $ij^{-1}\bmod\ell$ lies in $\{1,\dots,n\}$ and $0$ otherwise, and if moreover $p\nmid\det(N_\ell)$, equivalently $\mathrm{rank}\,M_\ell(p)=n+1$, then $\mathrm{Gal}(K_\mu/k_\mu)\simeq(\mathbb{Z}/p\mathbb{Z})^{n+1}$; here $h^+_\ell$ is the class number of the maximal real subfield of $k$ (Theorem 4.5). Corollary 4.6 shows that, under the same hypothesis $p\nmid\ell h^+_\ell n$, $K_\mu=k_{\mu+1}$ for all $\mu$ if and only if $\dim_{\mathbb{Z}/p\mathbb{Z}}(H_1/S_2)=n-1$ (the real units contributing a subspace of the maximal possible dimension $n-1$, the unit rank of the maximal real subfield) and $p\nmid\det(N_\ell)$.

Second, analytically. The field $k$ is a CM field (a totally imaginary quadratic extension of the totally real subfield $\mathbb{Q}(\zeta_\ell+\zeta_\ell^{-1})$) with primitive CM-type $\{\varphi_1,\dots,\varphi_n\}$, a consistent choice of half the embeddings that enables a higher-dimensional complex-multiplication theory. Let $z_\ell$ be the associated CM point on the genus-$n$ Siegel upper half-space (the period point of the principally polarized abelian variety with complex multiplication by $\mathbb{Z}[\zeta_\ell]$). The paper forms quotients $\Phi^{[\mathbf{r},\mathbf{s};\mu,i]}(z)$ of products of Riemann theta constants with rational characteristics $\mathbf{r},\mathbf{s}\in\mathbb{Q}^n$ — explicitly computable modular functions on the Siegel space — and applies Shimura's reciprocity law for theta functions with complex multiplication, which describes exactly how Galois automorphisms act on such special values, to compute the Galois action on $\Phi^{[\mathbf{r},\mathbf{s};\mu,i]}(z_\ell)$. The outcome (Theorem 6.4) is an explicit Hilbert-style primitive generator of $K_\mu$ over $k_\mu$ as a suitable $p$-th root of a product of these values, under two hypotheses: $p\nmid \ell h^+_\ell n\cdot\det(A_\ell(\mathbf{r},\mathbf{s}))$, where $A_\ell(\mathbf{r},\mathbf{s})$ is the integer matrix of exponents produced by the reciprocity computation, and the assumption that $z_\ell$ is neither a zero nor a pole of the involved theta-quotient product.

Both lines stop short of a complete theory. In Remark 4.7 the authors report a Maple verification that $\det(N_\ell)\neq0$ for all odd primes $\ell\leq 10000$ and conjecture this for all odd primes $\ell$; and, based on computations for $\ell=5$ (all odd $p\leq101$ except $p=3$) and $\ell=7$ (all odd $p\leq101$), they conjecture that $\dim_{\mathbb{Z}/p\mathbb{Z}}(H_1/S_2)=n-1$ for almost all odd primes $p$, noting that the two conjectures together would yield $\mathrm{Gal}(k_{\mu+1}/k_\mu)\simeq(\mathbb{Z}/p\mathbb{Z})^{n+1}$ for each odd prime $\ell$ and almost all $p$. The closest classical relative of the first conjecture is the Maillet determinant $\det[(i^{-1}j \bmod \ell)]_{1\leq i,j\leq(\ell-1)/2}$, the determinant of the residue-valued multiplication table over the half-system, whose nonvanishing was proved by Carlitz and Olson; but $N_\ell$ is the half-system folding of that table (it records only on which side of $\ell/2$ the residue falls), a different matrix whose nonvanishing is not covered by that theory. The remaining gaps concern the degenerate cases of the analytic construction, which the source leaves untreated.

Problem Statement

Settle the issues left open by Koo–Yoon's construction of class fields over the cyclotomic field $k=\mathbb{Q}(\zeta_\ell)$:

(1) Prove or disprove: $\det(N_\ell)\neq0$ for every odd prime $\ell$, where $n=(\ell-1)/2$ and $N_\ell=[n_{ij}]\in M_n(\mathbb{Z})$ is the $(0,1)$-matrix with $n_{ij}=1$ if and only if the least positive residue of $ij$ modulo $\ell$ lies in $\{1,\dots,n\}$ (verified computationally for $\ell\leq10000$).

(2) Prove or disprove: for each fixed odd prime $\ell$, $\dim_{\mathbb{Z}/p\mathbb{Z}}(H_1/S_2)=n-1$ for all but finitely many odd primes $p$, where $S_\mu=\{a\in k^{\times}: a\equiv1 \pmod{2p^\mu}\}$, $E$ is the unit group of $k$, and $H_\mu=S_{\mu+1}(S_\mu\cap E)$; in the negative direction, exhibit an odd prime $\ell$ with infinitely many exceptional $p$.

(3) Complete the explicit generation in the degenerate cases: (a) construct Hilbert-style primitive generators of $K_\mu$ over $k_\mu$ by theta-quotient special values when $p\mid\det(A_\ell(\mathbf{r},\mathbf{s}))$, with explicit formulas playing the role of Theorem 6.4; (b) for general $\ell$, determine systematically which theta-quotients $\Phi^{[\mathbf{r},\mathbf{s};\mu,i]}(z)$ vanish or have poles at the CM point $z_\ell$, i.e. which characteristics $(\mathbf{r},\mathbf{s})$ are admissible.

The common objective uniting these tasks is an unconditional, explicit construction of the layer between $k_\mu$ and $k_{\mu+1}$ — degrees, Galois groups, and primitive generators as special values of theta-quotients — for each odd prime $\ell$ and every odd prime $p\nmid\ell h^+_\ell n$ outside the $\ell$-dependent exceptional set of part (2), whether or not that set is finite, with the two conjectures of Remark 4.7 settled in the process and the case $p\mid\ell h^+_\ell n$ left outside the scope of the source's standing hypothesis.

References

1. Ja Kyung Koo; Dong Sung Yoon, Construction of class fields over cyclotomic fields, Kyoto Journal of Mathematics 56 (2016), no. 4, 803–829. DOI: 10.1215/21562261-3664923 (arXiv:1203.4662), https://doi.org/10.1215/21562261-3664923
2. Ja Kyung Koo; Dong Sung Yoon, Construction of class fields over imaginary biquadratic fields, Indiana University Mathematics Journal 68 (2019), no. 2, 413–434. DOI: 10.1512/IUMJ.2019.68.7626 (arXiv:1306.6390), https://doi.org/10.1512/IUMJ.2019.68.7626
3. Ja Kyung Koo; Gilles Robert; Dong Hwa Shin; Dong Sung Yoon, On Siegel invariants of certain CM-fields, The Ramanujan Journal 54 (2021), no. 2, 261–284. DOI: 10.1007/s11139-019-00223-3 (arXiv:1508.05602), https://doi.org/10.1007/s11139-019-00223-3
4. Ja Kyung Koo; Dong Hwa Shin; Dong Sung Yoon, On a problem of Hasse and Ramachandra, Open Mathematics 17 (2019), no. 1, 131–140. DOI: 10.1515/math-2019-0013 (arXiv:1608.06705), https://doi.org/10.1515/math-2019-0013
5. Leonard Carlitz; F. R. Olson, Maillet's determinant, Proceedings of the American Mathematical Society 6 (1955), 265–269. DOI: 10.1090/S0002-9939-1955-0069207-2, https://doi.org/10.1090/S0002-9939-1955-0069207-2
6. Youngmi Hur; Zachary Lubberts, Generalizations of the Maillet determinant, preprint arXiv:1407.6802 [math.RA] (2014), https://arxiv.org/abs/1407.6802
7. Goro Shimura, Theta functions with complex multiplication, Duke Mathematical Journal 43 (1976), 673–696. DOI: 10.1215/S0012-7094-76-04353-2, https://doi.org/10.1215/S0012-7094-76-04353-2