For a number field K\mathbb{K} of degree d≥3d\geq 3 with exactly one complex place (an ATR field), Morain (2024), extending the cubic-case construction of Bergeron–Charollois–García (2023), evaluates geometric variants of Nishizawa's multiple elliptic Gamma functions at algebraic points of K\mathbb{K}, averages the resulting products over units, and smooths them. He conjectures (Conjecture 11 of arXiv:2406.06094, restated more cleanly in a 2025–2026 series of papers and a PhD thesis) that these averaged products are, up to an explicit root-of-unity factor m(r,N)m(r,N), algebraic units in the ray class fields K+(f)\mathbb{K}^{+}(\mathfrak{f}), that they satisfy an explicit reciprocity law under the Artin map, and that they fulfill a Kronecker-type limit formula expressing derivatives at s=0s=0 of the partial zeta functions of K\mathbb{K}. Only the limit formula in the cubic case is proved; the algebraicity and reciprocity statements — even for complex cubic fields — remain open, supported so far by numerical evidence in degrees 3 to 6. The problem is to prove or refute these conjectures.

Background

Hilbert's twelfth problem asks for an explicit class field theory: a direct analytic construction, by special values of distinguished transcendental functions, of the abelian extensions of a number field, in the way roots of unity give the abelian extensions of Q\mathbb{Q} (Kronecker–Weber theorem) and values of the elliptic modular and Weber functions at imaginary quadratic arguments generate the abelian extensions of imaginary quadratic fields (complex multiplication). Beyond these known cases no general analytic solution is available, and the next natural testing ground is the class of ATR fields ('almost totally real': all embeddings real except exactly one complex conjugate pair).

For an ATR field K\mathbb{K}, Stark's rank-one conjecture predicts the existence of algebraic units ('Stark units') in ray class fields of K\mathbb{K} — the finite abelian extensions K+(f)\mathbb{K}^{+}(\mathfrak{f}) parametrized by the narrow ray class group of an integral ideal f\mathfrak{f} (the modulus) — whose logarithms are governed by the derivatives at s=0s=0 of partial zeta functions. The partial zeta function attached to a modulus f\mathfrak{f} and an ideal class [b][\mathfrak{b}] is ζf([b],s)=∑I∈[b]N(I)−s\zeta_{\mathfrak{f}}([\mathfrak{b}],s)=\sum_{I\in[\mathfrak{b}]}\mathcal{N}(I)^{-s}, and an identity expressing such derivatives (or values) through the logarithm of an explicit analytic quantity is called a Kronecker limit formula, after the classical case of Eisenstein series over Q(i)\mathbb{Q}(i).

The analytic engine proposed for this program comes from mathematical physics: the elliptic Gamma function of Ruijsenaars, studied by Felder and Varchenko, which satisfies modular transformation identities under SL3(Z)\mathrm{SL}_{3}(\mathbb{Z}) involving theta functions, and its higher-rank generalizations, the multiple elliptic Gamma functions GrG_{r} introduced by Nishizawa, which transform under SLr+2(Z)\mathrm{SL}_{r+2}(\mathbb{Z}). Morain constructs 'geometric' variants of the GrG_{r} enjoying such transformation laws with explicit Bernoulli-rational-function correction terms, and then 'arithmetic' smoothed versions indexed by ideals; these are direct analogues of the Sczech Eisenstein cocycles — group cocycles for SLd(Z)\mathrm{SL}_{d}(\mathbb{Z}) built from smoothed Bernoulli functions that compute partial zeta values of totally real fields — which provide the cohomological model for the construction.

Bergeron, Charollois and García (2023) carried out the rank-one case: for a complex cubic field K\mathbb{K} (degree 3, one complex place), they evaluated a smoothed elliptic Gamma function at algebraic points determined by a choice of fundamental units and ideals, proved the resulting Kronecker limit formula relating the smoothed evaluations to ζf′([b],0)\zeta_{\mathfrak{f}}'([\mathfrak{b}],0), and conjectured that the evaluations are algebraic units in K+(f)\mathbb{K}^{+}(\mathfrak{f}) related to the conjectural Stark unit. That algebraicity conjecture remains open even in this cubic case.

Morain (2024) generalized the scheme to ATR fields of degree d=r+2d=r+2 (numerically, degrees 3 through 6) using the GrG_{r} functions. The evaluation points are packaged in 'admissible vectors' hh; two integer invariants of the chosen system of fundamental units control the construction: the content λ~\tilde\lambda (the determinant index of the order generated by the units) and the overflow t~\tilde{t} (measuring how far the Z\mathbb{Z}-module spanned by the units is from being closed under multiplication). When t~>1\tilde{t}>1 the computations require t~ r\tilde{t}^{\,r} function evaluations, so Morain introduces 'target ideals' and 'helper ideals' to choose admissible vectors forcing the effective overflow tt to be 1, at the cost of enlarging a level ll. The main statement, Conjecture 11, is formulated under technical hypotheses — denoted (∗)(*) and (∗∗)(**) — which require, among other things, that f\mathfrak{f} be a power of a degree-one prime, that the content λ~\tilde\lambda equal 1 for every ordering of the units, that certain coprimality conditions hold, and that the evaluation parameters avoid the real subfield in even degree. Under these hypotheses the conjecture predicts: (i) a Kronecker-type limit formula (formula (16) of the paper) equating N(a)ζf′([b],0)−ζf′([ab],0)\mathcal{N}(\mathfrak{a})\zeta_{\mathfrak{f}}'([\mathfrak{b}],0)-\zeta_{\mathfrak{f}}'([\mathfrak{a}\mathfrak{b}],0) with the logarithm of the absolute value squared of an averaged product of smoothed arithmetic GrG_{r} functions; (ii) independence of the choice of admissible vectors up to a root of unity of explicit order m(r,N)m(r,N) (with m(r,2)=4m(r,2)=4, m(r,N)=Nm(r,N)=N for 3≤N≤r+33\leq N\leq r+3, m(r,N)=1m(r,N)=1 for N>r+3N>r+3, the bound r+3r+3 being predicted by the von Staudt–Clausen theorem on denominators of Bernoulli numbers); (iii) that the product raised to the power m(r,N)m(r,N) is an algebraic unit uL,au_{L,\mathfrak{a}} of the ray class field K+(f)\mathbb{K}^{+}(\mathfrak{f}), mapped to the unit circle by every embedding above a real place of K\mathbb{K}; and, in the remarks, an explicit reciprocity law uL,aσc=uLc−1,au_{L,\mathfrak{a}}^{\sigma_{\mathfrak{c}}}=u_{L\mathfrak{c}^{-1},\mathfrak{a}} for c\mathfrak{c} in the ray class group.

The 2024 paper has been superseded by a series of three papers by the same author (parts I and II appeared in 2025 and 2026, part III in progress) and by his PhD thesis, which develop the analytic framework (geometric families of multiple elliptic Gamma functions with SLn(Z)\mathrm{SL}_{n}(\mathbb{Z}) transformation laws, attached Bernoulli-rational-function cocycles, smoothed higher Dedekind sums with uniformly bounded denominators, and partial modular symbols) and restate the main conjecture (Conjecture III.38 of the thesis), with a simplified form in 'optimal settings' (Conjecture 2.2.5 of arXiv:2601.11961, July 2026): the averaged product is an algebraic unit ucu_{\mathfrak{c}} in K+(f)\mathbb{K}^{+}(\mathfrak{f}), independent of auxiliary choices, with conjugates over the narrow Hilbert class field given by varying the class c\mathfrak{c}, explicit Artin reciprocity σc(uc′)=ucc′\sigma_{\mathfrak{c}}(u_{\mathfrak{c}'})=u_{\mathfrak{c}\mathfrak{c}'}, and a Kronecker limit formula Nζf′(c,0)−ζf′(ac,0)=log⁡∣uc∣2N\zeta_{\mathfrak{f}}'(\mathfrak{c},0)-\zeta_{\mathfrak{f}}'(\mathfrak{a}\mathfrak{c},0)=\log|u_{\mathfrak{c}}|^{2}. All of these arithmetic statements remain conjectural; the recent work proves only the surrounding analytic and cohomological machinery.

Problem Statement

Prove or refute the conjectural arithmetic of higher elliptic units above ATR fields, in the following source-faithful form (Conjecture 11 of arXiv:2406.06094, in its current successor formulations: Conjecture III.38 of the thesis and the simplified Conjecture 2.2.5 of arXiv:2601.11961). Let K\mathbb{K} be a number field of degree d=r+2≥3d=r+2\geq 3 with exactly one complex place, let f≠OK\mathfrak{f}\neq\mathcal{O}_{\mathbb{K}} be a modulus such that K+(f)\mathbb{K}^{+}(\mathfrak{f}) is totally complex and every unit of K\mathbb{K} congruent to 11 modulo f\mathfrak{f} has norm +1+1, and let fundamental units, ideals b,a\mathfrak{b},\mathfrak{a}, smoothing data, and admissible evaluation vectors be chosen under the hypotheses of the conjecture (the (∗)(*),(∗∗)(**) conditions of the 2024 paper, or the optimal-settings hypotheses of the 2026 paper). Then the averaged product of NN-smoothed arithmetic multiple elliptic Gamma functions GrG_{r} evaluated at the associated algebraic points of K\mathbb{K}:

(i) satisfies the Kronecker-type limit formula N(a)ζf′([b],0)−ζf′([ab],0)=1#Zf1log⁡∣∏(n,ε)∈Zf1Ir,f,b,a∣2\mathcal{N}(\mathfrak{a})\zeta_{\mathfrak{f}}'([\mathfrak{b}],0)-\zeta_{\mathfrak{f}}'([\mathfrak{a}\mathfrak{b}],0)=\frac{1}{\#\mathcal{Z}_{\mathfrak{f}}^{1}}\log\left|\prod_{(n,\varepsilon)\in\mathcal{Z}_{\mathfrak{f}}^{1}}I_{r,\mathfrak{f},\mathfrak{b},\mathfrak{a}}\right|^{2} relating it to the s=0s=0 derivatives of partial zeta functions of K\mathbb{K};

(ii) is independent of the auxiliary choices up to a root of unity of order m(r,N)m(r,N) (m(r,2)=4m(r,2)=4, m(r,N)=Nm(r,N)=N for 3≤N≤r+33\leq N\leq r+3, m(r,N)=1m(r,N)=1 for N>r+3N>r+3), and after raising to the power m(r,N)m(r,N) is an algebraic unit of the ray class field K+(f)\mathbb{K}^{+}(\mathfrak{f}) lying on the unit circle at every embedding above a real place of K\mathbb{K};

(iii) obeys the explicit reciprocity law uL,aσc=uLc−1,au_{L,\mathfrak{a}}^{\sigma_{\mathfrak{c}}}=u_{L\mathfrak{c}^{-1},\mathfrak{a}} for c\mathfrak{c} in the ray class group (equivalently σc(uc′)=ucc′\sigma_{\mathfrak{c}}(u_{\mathfrak{c}'})=u_{\mathfrak{c}\mathfrak{c}'} under the Artin map in the successor formulation).

Resolve (prove or refute) each of (i)–(iii), under the optimal-settings hypotheses and, as the natural strengthening asked for in the sources, under the general regime without (∗)(*),(∗∗)(**)-type restrictions (content λ~=1\tilde\lambda=1 removed, general moduli, real evaluation parameters in even-degree fields). Proving the exact denominators and root-of-unity factors m(r,N)m(r,N), including optimality of the von Staudt–Clausen bound r+3r+3, is part of the arithmetic content of (ii).

References

  1. P. L. L. Morain, 'Elliptic Units Above Fields With Exactly One Complex Place', 2024, arXiv:2406.06094, https://arxiv.org/abs/2406.06094
  2. N. Bergeron, P. Charollois, L. E. García, 'Elliptic units for complex cubic fields', 2023, arXiv:2311.04110, https://arxiv.org/abs/2311.04110
  3. P. L. L. Morain, 'Geometric families of multiple elliptic Gamma functions and arithmetic applications, I', 2025, arXiv:2510.16515, https://arxiv.org/abs/2510.16515
  4. P. L. L. Morain, 'Geometric families of multiple elliptic Gamma functions and arithmetic applications, II', 2026, arXiv:2602.06561, https://arxiv.org/abs/2602.06561
  5. P. L. L. Morain, 'Computations of higher elliptic units in optimal settings', 2026, arXiv:2601.11961, https://arxiv.org/abs/2601.11961
  6. P. L. L. Morain, 'Higher elliptic Gamma functions, cohomology of linear groups and higher elliptic units', PhD thesis, Sorbonne Université, 2026, HAL:tel-05674839, https://hal.science/tel-05674839
  7. B. Yalkinoglu, 'A note on Shintani's invariants', 2024, arXiv:2408.07309, https://arxiv.org/abs/2408.07309