Stark's unit conjecture for the theta-null generators of class fields of primitive CM fields: integrality of the theta-null Stark units $\epsilon(R)$

For a primitive CM field K, Sixin Zeng's 2006 preprint `Notes On Hilbert's 12th Problem' constructs an analytic function $\eta_K$ on a moduli space of CM abelian varieties via a singular theta lift and claims that the quotients $\epsilon(R) = \eta(R)/\eta(\mathcal{O}_K)$, formed from the normalized theta-null values $\eta(R)$ and indexed by ideal classes $R$, are algebraic numbers in the Hilbert class field satisfying the reciprocity transformations predicted by Stark's conjecture. The paper explicitly leaves open the strongest — and central — part: whether these $\epsilon(R)$ are algebraic integers or units. The paper also raises, as further speculative questions, whether $\eta_K$ admits an explicit arithmetic description (a Dedekind-$\eta$-like infinite product — which the author himself argues should not be expected — or a canonical algebraic definition with descent and integral models); these are context for, not part of, the target. The problem asks for a proof or refutation of the unit-integrality assertion for these theta-null values over primitive CM fields.

Background

Hilbert's twelfth problem asks for explicit generators of the abelian extensions (class fields) of a number field, generalizing Kronecker's description of abelian extensions of $\mathbb{Q}$ by roots of unity and of imaginary quadratic fields by singular values of modular functions. A CM field is a totally imaginary quadratic extension of a totally real number field; primitive' here refers to the setting of the source's moduli problem (CM types not induced from proper subfields). Classical complex multiplication (CM) theory constructs class fields of CM fields from special values of automorphic forms, but does not produce a single analytic object whose special values generate all class fields in the way the Dedekind eta function and elliptic units (algebraic units constructed from singular values of modular functions) do for imaginary quadratic fields. Stark's conjecture (in the rank-one refined form relevant here) predicts, for each ideal class $R$ of such a field $K$, a Stark unit' $\epsilon(R)$: an algebraic unit in the Hilbert class field $K_0$ of $K$ — the maximal unramified abelian extension of $K$ — characterized analytically by $\log|\epsilon(R_1)/\epsilon(R_2)|$ equaling $\delta_K(R_1)-\delta_K(R_2)$, the difference of the coefficients of the order-one terms at $s=0$ of the partial zeta functions $\zeta_K(s,R_1)$, $\zeta_K(s,R_2)$ (the Dirichlet series attached to the ideal classes), and characterized algebraically by a reciprocity law: the Galois group $\mathrm{Gal}(K_0/K)$, identified with the ideal class group, acts by $\epsilon(R)^{(\mathfrak{a}^{-1},K)} = \epsilon(R\mathfrak{a})\,\epsilon(\mathfrak{a})^{-1}$. The Dedekind eta function $\eta$ is the classical infinite product $q^{1/24}\prod_{n\ge1}(1-q^n)$ on the upper half plane; its multiplicative structure underlies elliptic units and, via the Borcherds lift (a singular theta lift whose output has an infinite-product expansion), the arithmetic of Hilbert modular surfaces. Zeng's construction defines $\eta_K(w;\mathfrak{a})$ as a theta-null — the value of an explicit theta function at the origin — on moduli spaces of cone-polarized' abelian varieties with CM by $K$ (cone-polarization being the source's term for polarization data on the abelian variety indexed by a cone attached to the CM type), related across types by a proposed mirror symmetry for abelian varieties', and claims the difference identity $\delta_K(R_1) - \delta_K(R_2) = \log|\eta_K(R_1)/\eta_K(R_2)|$ plus the algebraicity-in-$K_0$ and Shimura-reciprocity behavior of $\epsilon(R) = \eta(R)/\eta(\mathcal{O}_K)$, where $\eta(R)$ is the normalized theta-null value attached to $R$ (the Shimura reciprocity law is the classical CM rule describing Galois action on special values of theta functions). The unit assertion, however, is left open, and the paper identifies the obstruction: unlike the one-variable classical case, there is no Dedekind-like infinite product for $\eta_K$ — the regulator $R_F$ — the covolume of the lattice given by logarithms of the units of the totally real subfield $F$, whose unit group is infinite — prevents exponentiating Fourier expansions — so the $q$-expansion principle (algebraic-integer values from integral Fourier coefficients) has no purchase. In the adjacent, well-developed literature, integrality of Stark-type units is known over totally real base fields: the Brumer–Stark conjecture (a $p$-adic analog asserting that a Brumer–Stark element is a unit times a Gauss-sum-type factor) was proved by Dasgupta–Kakde away from $p=2$, completed over $\mathbb{Z}$ by Dasgupta–Kakde–Silliman–Wang, and made explicit by Dasgupta and by Dasgupta–Kakde's construction of Brumer–Stark units giving explicit class field theory for totally real fields; Dasgupta's earlier work connects Shintani-type multiple-Gamma machinery (higher-dimensional analogues of the Gamma function occurring in special values of partial zeta functions — the same circle of special values in which $\eta_K$ sits) to Gross–Stark units (the $p$-adic limit units predicted by Gross's refinement of Stark's conjecture for totally real fields); Popescu proved Rubin–Stark type statements for special CM extensions of totally real fields. All of these treat CM extensions sitting above a totally real base; none addresses Stark units over a CM base field, which is exactly Zeng's setting, and none produces any eta-like generating function there.

Problem Statement

Let $K$ be a primitive CM field of degree $2n$ with Hilbert class field $K_0$, and let $\eta_K(w;\mathfrak{a})$ denote the theta-null on the cone-polarized CM moduli space $M_n(\mathfrak{a})$ constructed by the singular theta lift in Zeng's framework, evaluated at the CM point $w\in\mathcal{H}^n$. For an ideal class $R$ with type ideal $\mathfrak{a}$, define, following the source, the normalized theta-null value and its quotient

$$\eta(R) \;=\; \frac{\eta_K(w;\mathfrak{a})}{N(\mathfrak{a})\prod_{i=1}^{n}\mathrm{Im}(w_i)}, \qquad \epsilon(R) \;=\; \frac{\eta(R)}{\eta(\mathcal{O}_K)}.$$

Let $\delta_K(R)$ be the coefficient of the order-one term at $s=0$ of the partial zeta function $\zeta_K(s,R)$. The source claims the analytic characterization $\delta_K(R_1)-\delta_K(R_2)=\log|\eta_K(R_1)/\eta_K(R_2)|$ for the theta-null quotients and claims, as its Theorem 8.2, that $\epsilon(R)$ is an algebraic number in $K_0$ satisfying the reciprocity law below; together these characterizations determine the quotients $\epsilon(R_1)/\epsilon(R_2)$ up to roots of unity. The problem is to prove or refute, for primitive CM fields $K$, the remaining — `integral' — assertion of Stark's conjecture for these values: that each quotient $\epsilon(R)$ is a unit — an algebraic integer whose inverse is again an algebraic integer — in the Hilbert class field $K_0$, with the Galois action

$$\epsilon(R)^{(\mathfrak{a}^{-1},K)} \;=\; \epsilon(R\mathfrak{a})\,\epsilon(\mathfrak{a})^{-1}.$$

A refutation would exhibit a primitive CM field $K$ and class $R$ for which the corresponding $\epsilon(R)$ fails to be an algebraic integer or unit, together with a rigorous identification of the algebraic number involved.

References

1. Sixin Zeng, Notes On Hilbert's 12th Problem, arXiv math/0605101 (2006), DOI 10.48550/arXiv.math/0605101, https://arxiv.org/abs/math/0605101
2. Samit Dasgupta, Shintani zeta functions and Gross–Stark units for totally real fields, Duke Mathematical Journal 143 (2008), no. 2, DOI 10.1215/00127094-2008-019, https://doi.org/10.1215/00127094-2008-019
3. Samit Dasgupta and Mahesh Kakde, On the Brumer–Stark Conjecture, arXiv 2010.00657 (2020); Annals of Mathematics 197 (2023), no. 1, DOI 10.4007/annals.2023.197.1.5, https://arxiv.org/abs/2010.00657
4. Samit Dasgupta, Mahesh Kakde, Jesse Silliman, and Jiuya Wang, The Brumer–Stark Conjecture over Z, arXiv 2310.16399 (2023), DOI 10.48550/arXiv.2310.16399, https://arxiv.org/abs/2310.16399
5. Samit Dasgupta and Mahesh Kakde, Brumer–Stark units and explicit class field theory, Duke Mathematical Journal 173 (2024), no. 8, DOI 10.1215/00127094-2023-0039, https://doi.org/10.1215/00127094-2023-0039
6. Cristian D. Popescu, On the Rubin–Stark Conjecture for a special class of CM extensions of totally real number fields, Mathematische Zeitschrift 247 (2004), no. 3, DOI 10.1007/s00209-003-0624-x, https://doi.org/10.1007/s00209-003-0624-x
7. Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Inventiones Mathematicae 132 (1998), 491–562, DOI 10.1007/s002220050232, https://doi.org/10.1007/s002220050232