Explicit analytic generation of ray class fields beyond imaginary quadratic fields: Siegel–Ramachandra invariants over imaginary biquadratic and CM fields

Hilbert's twelfth problem asks for explicit analytic functions whose special values generate all abelian extensions of a given number field; it is solved only for the rationals (cyclotomic functions) and for imaginary quadratic fields (complex multiplication, realized concretely by Siegel–Ramachandra invariants). Koo and Yoon (Indiana Univ. Math. J. 68 (2019) 413–434) made the first unconditional advance beyond degree two: over an imaginary biquadratic field $K=\mathbb{Q}(\sqrt{-d_1},\sqrt{-d_2})$, products of Siegel–Ramachandra invariants of the two imaginary quadratic subfields generate the ray class fields $K_{(Np^{\mu+1})}$ over $K_{(Np^{\mu})}$ at every sufficiently large $p$-power level $\mu\ge\mu_0$, with an effective criterion at the initial step. Beyond what the source proves, two open targets remain: (i) completing the generation theorem to every imaginary biquadratic field and every modulus, removing the source's congruence-type, ramified-prime, $p=2$, $N=2$ and excluded-unit hypotheses and the provable first-layer index gap; and (ii) the generation conjecture for general CM fields (Koo–Robert–Shin–Yoon, Conjecture 6.4), whose missing analytic input is an analogue of the second Kronecker limit formula.

Background

Hilbert's twelfth problem (1900) asks, for a number field $K$, which analytic functions and algebraic numbers are needed to construct all of its abelian extensions (finite Galois extensions of $K$ with abelian Galois group). Class field theory identifies the finite targets: for a modulus $\mathfrak{m}$ (a formal product of an integral ideal and some real places of $K$), the ray class field $K_{\mathfrak{m}}$ is the maximal abelian extension of $K$ with ramification controlled by $\mathfrak{m}$, and every finite abelian extension of $K$ sits inside some $K_{\mathfrak{m}}$; the case $\mathfrak{m}=1$ gives the Hilbert class field. For $K=\mathbb{Q}$ the answer is given by the Kronecker–Weber theorem (roots of unity, i.e. special values of the exponential function), and for imaginary quadratic $K$ by the theory of complex multiplication: singular values of modular functions at CM points, together with torsion coordinates of elliptic curves with complex multiplication, generate all ray class fields. Here a CM field is a totally imaginary quadratic extension of a totally real field, the imaginary quadratic fields being the degree-2 case. For all other number fields the problem is open, and Mazur's Bull. AMS survey (2011) describes the state of the art; for CM fields, torsion points of CM abelian varieties describe the maximal abelian extension only implicitly, not through a single canonical family of explicitly computable analytic functions in Hilbert's sense.

The concrete elliptic-era tool is the Siegel function $g_{\mathbf{r}}(\tau)$, an explicitly given infinite product on the complex upper half-plane that is a modular unit (a modular function with no zeros or poles on the upper half-plane). Ramachandra (1964) proved that suitable products of the singular values $g_{\mathfrak{f}}(C)$ — the Siegel–Ramachandra invariants, indexed by the classes $C$ of the ray class group of an imaginary quadratic field modulo an ideal $\mathfrak{f}$ — generate the ray class field; Jung, Koo and Shin (2011) later showed that, apart from small-level exceptions, a single invariant is a primitive generator, i.e. that one value already generates the full ray class field over the base field. The nonvanishing statement behind generation comes from the second Kronecker limit formula, which expresses the relevant character sums $\sum_C \chi(C)\ln|g_{\mathfrak{f}}(C)|$ as values of Hecke $L$-functions at $s=1$, known to be nonzero.

Koo and Yoon (arXiv:1306.6390; published as Indiana Univ. Math. J. 68 (2019) 413–434) carried this program to imaginary biquadratic fields, i.e. degree-4 fields $K=\mathbb{Q}(\sqrt{-d_1},\sqrt{-d_2})$ with two imaginary quadratic subfields $K_1, K_2$ and one real quadratic subfield $K_3=\mathbb{Q}(\sqrt{d_1d_2})$; these are CM fields whose unit group acquires a fundamental unit $\varepsilon_0$ (a generator of the infinite cyclic part of $\mathcal{O}_{K_3}^{\times}$), in addition to the roots of unity. Writing $K_{(Np^{\mu})}$ for the ray class field of $K$ modulo $Np^{\mu}$ ($N\ge 1$, $p$ an odd prime with $\gcd(N,p)=1$, $\mu\ge 0$), they define $\widetilde{K^{1,2}_{N,p,\mu}} = K_{(Np^{\mu})}(K_1)_{(Np^{\mu+1})}(K_2)_{(Np^{\mu+1})}$, which by their Corollary 4.3 is generated over $K_{(Np^{\mu})}$ by a product of two Siegel–Ramachandra invariants, one from each imaginary quadratic subfield. Their key instrument is Shimura's reciprocity law, which computes the Galois action on special values of Siegel modular functions (here: genus-2 modular functions evaluated at diagonal CM points $\mathrm{diag}(\theta_1,\theta_2)$) through an explicit action of the idèle group of $K$ on the function field. Their comparison of the Galois group $\mathrm{Gal}(K_{(Np^{\mu+1})}/K_{(Np^{\mu})})$ with the norm-image subgroup shows: at the initial step $\mu=0$ (Theorem 4.7), $[K_{(Np)}:\widetilde{K^{1,2}_{N,p,0}}]=m_{p,3}/n_0$ for $N\ne 1$ (with extra unit-index factors when $N=1$), where $m_{p,3}=p-(\tfrac{d_1d_2}{p})$ is a Legendre-symbol quantity and $n_0$ is the multiplicative order of a sign-adjusted $\varepsilon_0$ modulo $Np$; hence $K_{(Np)}=\widetilde{K^{1,2}_{N,p,0}}$ if and only if $N\ne 1$ and $n_0=m_{p,3}$. For higher steps (Theorem 5.4), writing $\ell_0$ for the order of $\varepsilon_0$ modulo $Np\mathcal{O}_K$ and $\mu_0$ for the largest $\mu$ with $\varepsilon_0^{\ell_0}\equiv 1 \pmod{Np^{\mu}\mathcal{O}_K}$, the index $[K_{(Np^{\mu+1})}:\widetilde{K^{1,2}_{N,p,\mu}}]$ equals $p$ for $\mu<\mu_0$ and $1$ for $\mu\ge\mu_0$; thus the two Siegel–Ramachandra invariants generate the full ray class field at every sufficiently large $p$-power level (Corollary 5.5). Both $n_0$ and $\mu_0$ are effectively computable from $\varepsilon_0$, so the generation criteria in the paper are algorithmic. The theorems, however, carry hypotheses: throughout, $-d_1\equiv 1 \pmod 4$, $-d_2\equiv 2,3 \pmod 4$, $\gcd(d_1,d_2)=1$, and $K_1,K_2\ne\mathbb{Q}(\sqrt{-1}),\mathbb{Q}(\sqrt{-3})$; Theorem 4.7 additionally assumes $N\ne 2$ and $p\nmid d_1d_2$; only odd $p$ are treated, so no 2-power ladder is given; and for $N=1$ the initial step provably has index $>1$, so the first layer above the Hilbert class field is never generated by the two invariants.

The declared generalization target of the same research program is higher-degree CM fields. Koo, Robert, Shin and Yoon (Ramanujan J. 54 (2021) 261–284) constructed Siegel invariants $\Theta_{\mathfrak{f}}(C)$ for a CM field $K$ whose CM-type and polarization data satisfy their Assumption 5.1 (the complex torus $\mathbb{C}^g/\Psi(\mathcal{O}_{K^*})$ admits the structure of a principally polarized abelian variety, and $(K^*)^*=K$, i.e. the CM type is primitive) — special values of an explicit genus-$g$ Siegel modular function built from theta constants (even theta functions on the Siegel upper half-space $\mathbb{H}_g$, defined by their Fourier series) — and proved by Shimura reciprocity that $\Theta_{\mathfrak{f}}(C)$ lies in the ray class field $K_{\mathfrak{f}}$ with the correct Galois translation property $\Theta_{\mathfrak{f}}(C)^{\sigma_{\mathfrak{f}}(D)}=\Theta_{\mathfrak{f}}(CD)$, where $\mathfrak{f}=\mathfrak{f}_0\mathcal{O}_K$ ranges over the moduli induced by proper nontrivial integral ideals $\mathfrak{f}_0$ of $\mathcal{O}_{K_0}$ — $K_0$ being either $\mathbb{Q}$ or an imaginary quadratic subfield of both $K$ and $K^*$, and necessarily $K_0=\mathbb{Q}$ when $g\ge 2$, so that in higher degree $\mathfrak{f}=N\mathcal{O}_K$ for a positive integer $N$ — $C$ over the ray class group $\mathrm{Cl}(\mathfrak{f})$, and $\sigma_{\mathfrak{f}}:\mathrm{Cl}(\mathfrak{f})\to\mathrm{Gal}(K_{\mathfrak{f}}/K)$ is the Artin reciprocity map. Writing $K^*$ for the reflex field of the CM type (the CM field generated by the type-norm products $\prod_i a^{\varphi_i}$, $a\in K$, as $\varphi_i$ runs over the embeddings composing the type), the type-norm map $\varphi(a)=\prod_i a^{\varphi_i}$ induces a homomorphism $\widetilde{\varphi}:\mathrm{Cl}(\mathfrak{f})\to\mathrm{Cl}(\mathfrak{f}^*)$ of ray class groups; via $\sigma_{\mathfrak{f}}$, the subgroup $\ker(\widetilde{\varphi})$ cuts out an intermediate field of $K_{\mathfrak{f}}/K$, and Conjecture 6.4 asserts precisely that $\Theta_{\mathfrak{f}}(C)$ is a primitive generator of this intermediate field. But whether such an invariant actually generates the field — their Conjecture 6.4 — remains open, with numerical support only for $K=\mathbb{Q}(\zeta_5)$. The obstruction is analytic: the standard sufficient condition for a family with these two properties to generate primitively is the nonvanishing of the character sums $\sum_{C\in\mathrm{Cl}(\mathfrak{f})} \chi(C)\ln|\Psi_{\mathfrak{f}}(C)|$ for every nontrivial character $\chi$ of $\mathrm{Cl}(\mathfrak{f})$, where $\Psi_{\mathfrak{f}}(C)$ denotes a generic member of such a family (a Siegel family); over imaginary quadratic fields the second Kronecker limit formula supplies this nonvanishing, but no analogue is known for CM fields, so the criterion is unproven in the CM setting. Notably, even in the classical imaginary quadratic setting, Koo and Yoon's proof of Schertz's conjecture (that norms of Siegel–Ramachandra invariants generate all finite abelian extensions) is conditional on such a nonvanishing input.

Problem Statement

Construct explicit analytic generators for all ray class fields of an imaginary biquadratic field — and, in the general form posed by Koo–Robert–Shin–Yoon, of a CM field — thereby realizing Hilbert's twelfth problem beyond the imaginary quadratic case. Concretely, with the notation above:

(1) Biquadratic completion. For every imaginary biquadratic field $K=\mathbb{Q}(\sqrt{-d_1},\sqrt{-d_2})$ — without the restrictions $-d_1\equiv 1 \pmod 4$, $-d_2\equiv 2,3 \pmod 4$, $\gcd(d_1,d_2)=1$, and $K_i\notin\{\mathbb{Q}(\sqrt{-1}),\mathbb{Q}(\sqrt{-3})\}$ imposed in Koo–Yoon — and for every modulus $\mathfrak{m}$ of $K$ (including 2-power moduli, primes $p\mid d_1d_2$ ramified in $K$, and the case $N=2$), exhibit explicitly computable modular units (Siegel functions, Siegel–Ramachandra invariants of the quadratic subfields, or genus-2 analogues) whose singular values generate the ray class field $K_{\mathfrak{m}}$ over $K$, and prove the generation. This includes closing the gap where the existing method provably fails to generate: at the first $p$-step above the Hilbert class field ($N=1$, $\mu=0$) the index $[K_{(p)}:\widetilde{K^{1,2}_{1,p,0}}]$ is always $>1$, so new invariants must be found that remove this unit obstruction, in keeping with the requirement of exact generation at every modulus.

(2) CM generation conjecture. Prove or refute Conjecture 6.4 of Koo–Robert–Shin–Yoon: for a CM field $K$ whose CM-type and polarization data satisfy their Assumption 5.1, the Siegel invariant $\Theta_{\mathfrak{f}}(C)$ is a primitive generator of the fixed field of $\ker(\widetilde{\varphi})$ inside the ray class field $K_{\mathfrak{f}}$, for every modulus $\mathfrak{f}=\mathfrak{f}_0\mathcal{O}_K$ induced by a proper nontrivial ideal $\mathfrak{f}_0$ of $\mathcal{O}_{K_0}$ (as constructed in the source) and every class $C\in\mathrm{Cl}(\mathfrak{f})$. The decisive missing input is an analogue for CM fields of the second Kronecker limit formula — nonvanishing of the character sums $\sum_C \chi(C)\ln|\Psi_{\mathfrak{f}}(C)|$ for every nontrivial ray class character $\chi$.

These two targets form one objective: a complete, hypothesis-free, explicit analytic class field theory, in the sense of Hilbert's twelfth problem, for the first families of number fields beyond the imaginary quadratic case. A solution must establish exact generation (equality of fields, not mere inclusion) and state precisely, in terms of the arithmetic of $K$ and the modulus, at which levels equality holds and which additional explicit values close any remaining gap.

References

1. J. K. Koo and D. S. 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, https://doi.org/10.1512/iumj.2019.68.7626 (preprint: arXiv:1306.6390, https://arxiv.org/abs/1306.6390)
2. J. K. Koo, G. Robert, D. H. Shin and D. S. Yoon, On Siegel invariants of certain CM-fields, The Ramanujan Journal 54 (2021), no. 2, 261–284. DOI 10.1007/s11139-019-00223-3, https://doi.org/10.1007/s11139-019-00223-3 (preprint: arXiv:1508.05602, https://arxiv.org/abs/1508.05602)
3. J. K. Koo and D. S. Yoon, Construction of ray-class fields by smaller generators and their applications, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 147 (2017), no. 4, 781–812. DOI 10.1017/S0308210516000263, https://doi.org/10.1017/S0308210516000263 (preprint: arXiv:1407.5713, https://arxiv.org/abs/1407.5713)
4. J. K. Koo, D. H. Shin and D. S. Yoon, Siegel families with application to class fields, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 148 (2018), no. 4, 751–771. DOI 10.1017/S030821051700052X, https://doi.org/10.1017/S030821051700052X (preprint: arXiv:1601.00404, https://arxiv.org/abs/1601.00404)
5. J. K. Koo and D. S. Yoon, On the Schertz conjecture, Proceedings of the Edinburgh Mathematical Society (Series 2) 62 (2019), no. 3, 837–845. DOI 10.1017/S0013091518000895, https://doi.org/10.1017/S0013091518000895 (preprint: arXiv:1610.01311, https://arxiv.org/abs/1610.01311)
6. S. Choi, Singular moduli of rth roots of modular functions, Open Mathematics 21 (2023), Article No. 20220609. DOI 10.1515/math-2022-0609, https://doi.org/10.1515/math-2022-0609
7. H. Y. Jung, J. K. Koo, D. H. Shin and D. S. Yoon, Form class groups and class fields of CM-fields, Results in Mathematics 78 (2023), no. 1, Article No. 8. DOI 10.1007/s00025-022-01773-7, https://doi.org/10.1007/s00025-022-01773-7 (preprint: arXiv:1912.08128, https://arxiv.org/abs/1912.08128)
8. J. K. Koo, D. H. Shin and D. S. Yoon, Adelic framed form class groups and explicit class field theory, preprint (2026). arXiv:2608.04873, https://arxiv.org/abs/2608.04873
9. K. Ramachandra, Some applications of Kronecker's limit formulas, Annals of Mathematics (2) 80 (1964), 104–148. DOI 10.2307/1970494, https://doi.org/10.2307/1970494
10. H. Y. Jung, J. K. Koo and D. H. Shin, Ray class invariants over imaginary quadratic fields, Tohoku Mathematical Journal (2) 63 (2011), 413–426. DOI 10.2748/tmj/1318338949, https://doi.org/10.2748/tmj/1318338949
11. B. Mazur, How can we construct abelian Galois extensions of basic number fields?, Bulletin of the American Mathematical Society 48 (2011), 155–209. DOI 10.1090/s0273-0979-2011-01326-x, https://doi.org/10.1090/s0273-0979-2011-01326-x