Algebraicity of higher elliptic units for number fields with exactly one complex place (the unproven half of Morain's optimal-settings conjecture)

For a number field $\mathbb{K}$ of degree $n\ge 3$ with exactly one complex place, Morain (2026) conjectures that certain finite smoothed products of values of the multiple elliptic Gamma function $G_{n-2}$ , taken at explicitly computable points of $\mathbb{K}$ , are algebraic units in the narrow ray class field $\mathbb{K}^{+}(\mathfrak{f})$ , that they form Galois-conjugate families indexed by the narrow ray class group $\mathrm{Cl}^{+}(\mathfrak{f})$ , and that they obey the reciprocity law $\sigma_{\mathfrak{c}}(u_{\mathfrak{c}'})=u_{\mathfrak{c}\mathfrak{c}'}$ under the Artin map. This algebraicity half of Conjecture 2.2.5 of arXiv:2601.11961 — which generalizes the 2023 conjecture of Bergeron, Charollois and García for complex cubic fields, where only the companion Kronecker limit formula is proven — is open, already in the first case $n=3$ . A proof would give the first proven explicit analytic generation of abelian extensions beyond the classical rational and imaginary quadratic settings, i.e., a case of Hilbert's twelfth problem for signature $(n-2,1)$ .

Background

Hilbert's twelfth problem asks, for a given number field, for explicit analytic functions whose special values generate all of its abelian extensions. The two classical solutions are the field of rational numbers, whose abelian extensions are generated by roots of unity (the Kronecker–Weber theorem), and imaginary quadratic fields, where the theory of complex multiplication (elliptic curves whose endomorphism ring is strictly larger than $\mathbb{Z}$ ), together with the elliptic units built from elliptic functions, produces the ray class fields. For every other type of number field no general construction is known.

The main structural guide beyond these cases is Stark's rank-one conjecture. For a number field $\mathbb{K}$ and an integral ideal $\mathfrak{f}$ , the narrow ray class group $\mathrm{Cl}^{+}(\mathfrak{f})$ is the group of fractional ideals coprime to $\mathfrak{f}$ modulo the principal ideals that admit a totally positive generator (an element positive under every real embedding of $\mathbb{K}$ ) congruent to $1$ modulo $\mathfrak{f}$ ; class field theory identifies $\mathrm{Cl}^{+}(\mathfrak{f})$ with the Galois group of the narrow ray class field $\mathbb{K}^{+}(\mathfrak{f})$ via the Artin map, which sends an ideal class to its Frobenius automorphism. To each class $\mathfrak{c}$ one attaches the partial zeta function $\zeta_{\mathfrak{f}}(\mathfrak{c},s)$ , the Dirichlet series over integral ideals in the class $\mathfrak{c}$ . Stark's conjecture predicts that in a rank-one situation, such as a field with exactly one complex place, suitable first derivatives $\zeta'_{\mathfrak{f}}(\mathfrak{c},0)$ are logarithms of algebraic units (the predicted "Stark units") in an associated abelian extension. Shintani showed that partial zeta values can be evaluated by decomposing the Minkowski space of the field into polyhedral cones indexed by the units ("Shintani cones"), the cone sums being expressible through double sine functions; for real quadratic fields his invariants are themselves conjectured to generate the ray class fields. For fields with exactly one complex place, Espinoza introduced signed Shintani cones, in which the cone decomposition acquires explicit signs.

A newer analytic input is the elliptic Gamma function of Ruijsenaars, a meromorphic function on $\mathbb{C}\times\mathbb{H}^{2}$ ( $\mathbb{H}$ the upper half-plane) defined by a double infinite product and arising in mathematical physics, whose $\mathrm{SL}_{3}(\mathbb{Z})$ -transformation properties were determined by Felder and Varchenko. It sits at level one of a hierarchy of multiple elliptic Gamma functions $G_{r}$ ( $G_{0}$ is the Jacobi theta function, $G_{1}$ the elliptic Gamma function), each defined by a multidimensional Fourier product and enjoying pseudo-periodicity relations that lower the level. Bergeron, Charollois and García (2023) evaluated smoothed products of elliptic Gamma values at explicitly constructed points of a complex cubic field $\mathbb{K}$ and conjectured that the resulting complex numbers are algebraic units in the narrow ray class fields of $\mathbb{K}$ obeying a Shimura-style reciprocity law for the Galois action. They proved a new Kronecker limit formula — the analogue, for these values, of Kronecker's classical second limit formula relating zeta values at $s=0$ to logarithms of elliptic functions — identifying a combination of partial zeta derivatives at $s=0$ with $\log|u|^{2}$ ; the algebraicity and reciprocity statements themselves remain unproven.

Morain (2026) extended this construction from cubic to arbitrary degree. For a number field $\mathbb{K}$ of degree $n\ge 3$ with exactly one complex place, an integral ideal $\mathfrak{f}\neq\mathcal{O}_{\mathbb{K}}$ of norm $q$ with cyclic quotient $\mathcal{O}_{\mathbb{K}}/\mathfrak{f}$ , and — the "optimal setting" hypothesis proper — the property that every totally positive unit of $\mathbb{K}$ is congruent to $1$ modulo $\mathfrak{f}$ (equivalently, $\mathfrak{f}$ divides the ideal $J(\mathbb{K})=\sum_{\varepsilon}(\varepsilon-1)\mathcal{O}_{\mathbb{K}}$ , the sum running over totally positive units $\varepsilon$ ), together with regularity assumptions on auxiliary unit-system invariants (the "contents" and "overflow ideals" attached to sets of fundamental totally positive units) and the assumption that $\mathbb{K}$ has no proper subfield, the paper fixes fundamental totally positive units $\varepsilon_{1},\dots,\varepsilon_{n-2}$ , a class $\mathfrak{c}\in\mathrm{Cl}^{+}(\mathfrak{f})$ represented by an ideal $k\cdot\mathfrak{b}$ , a compatible set of base points $h_{\rho}$ indexed by permutations $\rho$ of the units, and a prime smoothing index $N>n+1$ coprime to $\mathfrak{f}\mathfrak{b}$ . From these data it writes down explicit evaluation points $z_{\rho},\tau_{1,\rho},\dots,\tau_{n-1,\rho}\in\mathbb{K}$ and the smoothed product

$u_{k,\mathfrak{b}}=\prod_{\rho\in\mathfrak{S}_{n-2}}\frac{G_{n-2}(z_{\rho},\tau_{1,\rho},\dots,\tau_{n-1,\rho})^{N}}{G_{n-2}(Nz_{\rho},N\tau_{1,\rho},\dots,N\tau_{n-1,\rho})}$

where $\mathfrak{S}_{n-2}$ is the symmetric group. Extensive computations in degrees $3$ – $6$ show that these values coincide to high precision with algebraic units of $\mathbb{K}^{+}(\mathfrak{f})$ satisfying the expected Kronecker limit formula. The construction is inspired by the rank-one Stark conjecture, the numbers $u_{\mathfrak{c}}$ being expected to be smoothed versions of the conjectural Stark units. The orientation signs $\epsilon_{\rho},\epsilon'_{\rho}\in\{\pm 1\}$ entering the evaluation points are currently fixed ad hoc by matching the Kronecker limit formula and are expected to be expressible through Espinoza's signed Shintani cones.

Problem Statement

Prove the algebraicity half of Conjecture 2.2.5 of Morain (2026), in its stated generality. Let $\mathbb{K}$ be a number field of degree $n\ge 3$ with exactly one complex place, let $(\mathbb{K},\mathfrak{f})$ be an optimal setting in the sense of the source paper (Section 2.1, hypotheses 1–7: $\mathfrak{f}\neq\mathcal{O}_{\mathbb{K}}$ with $\mathcal{O}_{\mathbb{K}}/\mathfrak{f}$ cyclic, unit-system contents equal to $1$ and overflow ideals coprime to the conductor norm with $\rho$ -independent narrow ideal class, $\mathcal{O}_{\mathbb{K}}^{+,\times}=\mathcal{O}_{\mathfrak{f}}^{+,\times}$ , and $\mathbb{K}$ without proper subfields), let $q=\mathcal{N}(\mathfrak{f})$ , and choose the auxiliary data as in the paper: a class $\mathfrak{c}\in\mathrm{Cl}^{+}(\mathfrak{f})$ represented by $k\cdot\mathfrak{b}$ , a prime smoothing index $N>n+1$ coprime to $\mathfrak{f}\mathfrak{b}$ , fundamental totally positive units $\varepsilon_{1},\dots,\varepsilon_{n-2}$ , and a compatible set of base points $(h_{\rho})_{\rho}$ , with evaluation parameters $z_{\rho},\tau_{1,\rho},\dots,\tau_{n-1,\rho}$ defined by equations (16)–(17) of the paper. The conjecture asserts that there exist orientation signs $\epsilon_{\rho},\epsilon'_{\rho}\in\{\pm 1\}$ such that

$u_{k,\mathfrak{b}}=\prod_{\rho\in\mathfrak{S}_{n-2}}\frac{G_{n-2}(z_{\rho},\tau_{1,\rho},\dots,\tau_{n-1,\rho})^{N}}{G_{n-2}(Nz_{\rho},N\tau_{1,\rho},\dots,N\tau_{n-1,\rho})}$

is an algebraic unit of the narrow ray class field $\mathbb{K}^{+}(\mathfrak{f})$ , independent of the choice of fundamental units and of the compatible set of base points, and that: (i) the family $(u_{k,\mathfrak{b}})_{k\in(\mathbb{Z}/q\mathbb{Z})^{\times}}$ consists of units of $\mathbb{K}^{+}(\mathfrak{f})$ that are Galois conjugates over the narrow Hilbert class field of $\mathbb{K}$ ; (ii) $u_{\mathfrak{c}}$ depends only on the class $\mathfrak{c}$ and not on its representation by $k\cdot\mathfrak{b}$ ; (iii) the family $(u_{\mathfrak{c}})_{\mathfrak{c}\in\mathrm{Cl}^{+}(\mathfrak{f})}$ consists of Galois conjugates over $\mathbb{K}$ with the explicit reciprocity law $\sigma_{\mathfrak{c}}(u_{\mathfrak{c}'} )=u_{\mathfrak{c}\mathfrak{c}'}$ for the Artin map $\mathfrak{c}\mapsto\sigma_{\mathfrak{c}}$ . The three statements (i)–(iii) form a single conjecture of the source, all bearing on the same units $u_{\mathfrak{c}}$ . The first case $n=3$ (where the product has a single factor and the construction reduces to that of Bergeron, Charollois and García) is already open. The companion Kronecker limit formula (identity (19) of the source, $N\zeta'_{\mathfrak{f}}(\mathfrak{c},0)-\zeta'_{\mathfrak{f}}(\mathfrak{a}\mathfrak{c},0)=\log|u_{\mathfrak{c}}|^{2}$ , proven for $n=3$ and conjectural for $n\ge 4$ ) is a logically separate statement and is not part of this problem.

References

Pierre L. L. Morain (2026), Computations of higher elliptic units in optimal settings, preprint arXiv:2601.11961 (v2, 22 July 2026), DOI 10.48550/arXiv.2601.11961, https://arxiv.org/abs/2601.11961
Nicolas Bergeron, Pierre Charollois, Luis E. García (2023), Elliptic units for complex cubic fields, preprint arXiv:2311.04110 (v2, 30 November 2023), DOI 10.48550/arXiv.2311.04110, https://arxiv.org/abs/2311.04110
Pierre L. L. Morain (2025), Geometric families of multiple elliptic Gamma functions and arithmetic applications, I, preprint arXiv:2510.16515 (v2, 24 January 2026), DOI 10.48550/arXiv.2510.16515, https://arxiv.org/abs/2510.16515
Pierre L. L. Morain (2026), Geometric families of multiple elliptic Gamma functions and arithmetic applications, II, preprint arXiv:2602.06561, DOI 10.48550/arXiv.2602.06561, https://arxiv.org/abs/2602.06561
Pierre L. L. Morain (2026), Higher elliptic Gamma functions, cohomology of linear groups and higher elliptic units, PhD thesis, Sorbonne Université, HAL tel-05674839, https://hal.science/tel-05674839
Milton Espinoza (2014), Signed Shintani cones for number fields with one complex place, Journal of Number Theory 145, 496–539, DOI 10.1016/j.jnt.2014.06.003, https://doi.org/10.1016/j.jnt.2014.06.003
Giovanni Felder, Alexander Varchenko (2000), The elliptic gamma function and SL(3,Z)⋉Z^3, Advances in Mathematics 156(1), 44–76, DOI 10.1006/aima.2000.1951, https://doi.org/10.1006/aima.2000.1951
Bora Yalkinoglu (2026), A note on Shintani's invariant, Functiones et Approximatio Commentarii Mathematici, DOI 10.7169/facm/250808-31-12, https://doi.org/10.7169/facm/250808-31-12 (preprint arXiv:2408.07309, 2024, https://arxiv.org/abs/2408.07309)
Pierre L. L. Morain (2024), Elliptic Units Above Fields With Exactly One Complex Place, preprint arXiv:2406.06094, DOI 10.48550/arXiv.2406.06094, https://arxiv.org/abs/2406.06094