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This paper claims that special values of higher elliptic Gamma functions at carefully chosen points of a number field with exactly one complex place form units in the narrow ray class field, are Galois conjugates with a simple reciprocity l

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2026-08-03 09:54 UTC pith:6OBOXYOK

load-bearing objection A serious conjectural and computational paper extending elliptic Gamma constructions to degrees 4–6, with strong 1000-digit numerical evidence, but the parameter machinery behind it is deferred to an unpublished companion and one check is fitted. the 4 major comments →

arxiv 2601.11961 v2 pith:6OBOXYOK submitted 2026-01-17 math.NT

Computations of higher elliptic units in optimal settings

Pierre L. L. Morain This is my paper
classification math.NT MSC 11R3711R4211G16
keywords higher elliptic unitselliptic Gamma functionsHilbert's 12th problemray class fieldsKronecker limit formulaStark conjecturesunique complex placecomputational algebraic number theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a conjectural solution to Hilbert's 12th problem for number fields with exactly one complex place, by constructing 'higher elliptic units' as special values of multiple elliptic Gamma functions. These are the natural higher-degree analogue of the theta function that produces classical elliptic units over imaginary quadratic fields. The main conjecture states that, under two simplifying assumptions on the ideal f, the numbers u_{k,b} defined by an explicit product of smoothed G_{n-2} functions are algebraic units in the narrow ray class field K^+(f), that they are Galois conjugates over the narrow Hilbert class field, that the Artin action follows the explicit rule σ_{k'b'}(u_{k,b})=u_{kk',bb'}, and that they verify a Kronecker limit formula relating their absolute values to derivatives of partial zeta functions. The paper supports this with high-precision computations in degrees 3, 4, 5 and 6. A sympathetic reader would care because, if true, it gives an analytic, computationally explicit description of abelian extensions for a large class of number fields — a long-sought analogue of complex multiplication.

Core claim

Conjecture 2 states: for a number field K of degree n≥3 with exactly one complex place, if f∩Z=qZ and O^{+,×}_f=O^{+,×}_K, then the numbers u_{k,b}=∏_{ρ,j} [G_{n-2}(k m_ρ/q+δ_{j,ρ}, τ_{1,ρ},...,τ_{n-1,ρ}) / G_{n-2}(N(...))] are units in the narrow ray class field K^+(f), are Galois conjugates over the narrow Hilbert class field, satisfy σ_{k'b'}(u_{k,b})=u_{kk',bb'}, and satisfy N ζ'_f([kb],0)−ζ'_f([kab],0)=log|u_{k,b}|^2. The paper leaves the conjecture open and supplies computational evidence in degrees 3–6.

What carries the argument

Central object: the hierarchy of multiple elliptic Gamma functions G_r, meromorphic functions given by infinite products (G_0 = θ, G_1 = elliptic Gamma). The units are N-smoothed ratios G_{n-2}(...)^N / G_{n-2}(N...). Parameters τ, δ, m come from an (n−2)-cycle built from fundamental units and a rational polyhedral cone decomposition (19); the different ideal D_ρ of a linear form controls the minimal number of cone terms via Proposition 3's t_min = λ̃∏p_j^{v_j}. In optimal cases t_ρ=1, the product has minimal size (n−2)!; algorithms sieve for such fields and compute the units. The machinery converts a cohomological cocycle into explicit evaluation points whose G-values are conjectured units.

Load-bearing premise

The construction stands on the correctness of the explicit parameters τ_{l,ρ}, δ_{j,ρ}, m_ρ produced by the cone decomposition of a companion preprint; this step is only sketched here, with the proof of Proposition 3's t_min formula deferred to an unpublished text — if that decomposition is wrong, the numerical verifications would not test the conjecture as stated.

What would settle it

For any listed optimal field, compute (13) with the published parameters, then run an independent algebraic check (minimal polynomial over K, norm in the ray class field) at 1000-digit precision; the conjecture fails if the values are not units in K^+(f) or if σ_{k'b'}(u_{k,b}) ≠ u_{kk',bb'}. More directly, compare u_{k,(1)} to an independently computed Stark unit for the same modulus — the paper predicts σ(u_{Stark}^{−2}) in the quartic example — and require agreement at full precision rather than after fitting signs.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A proof of Conjecture 2 would give an analytic construction of abelian extensions of any number field with exactly one complex place, extending the role of the theta function for imaginary quadratic fields to all higher degrees.
  • The reciprocity law σ_{k'b'}(u_{k,b})=u_{kk',bb'} would make the Artin map on the narrow ray class group explicit, allowing class field towers to be computed from analytic data.
  • The Kronecker limit formula (14) would give a higher-rank analogue of Kronecker's second limit formula, expressing derivatives of partial zeta functions at s=0 in terms of finite products of special values — a computable source of zeta-derivative information.
  • The optimal-condition sieve (Algorithm 1) plus the computation pipeline (Algorithm 2) yields many fields of degree 3, 4, 5 and 6 where these units can be computed to high precision quickly, so the conjecture is testable beyond the examples printed.
  • In the optimal case the units are essentially powers of the conjectural Stark units (as shown in the quartic example), so a proof would simultaneously resolve the rank-one abelian Stark conjecture for these fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The numerical checks are not fully independent: the signs ν_ρ in (13) are chosen by verifying the Kronecker limit formula (14), so the agreement is partly built in. A genuinely stronger test would derive ν_ρ from a closed formula, e.g., the signed Shintani-cone signs mentioned in Remark 6.
  • If the conjecture holds, the same hierarchy G_0, G_1, G_2, ... suggests a whole ladder of 'higher elliptic unit' constructions for fields of every signature: the smoothing and cone machinery seem to be signature-independent, and the case of totally real fields should reduce to Shintani zeta functions and Gross–Stark units.
  • The rarity of optimal examples in degree 7 (none found despite good heuristics) hints that either the optimal condition is genuinely restrictive, or the computational parameter search has not been exhaustive; this makes degree 7 a natural stress-test for the conjecture.
  • A direct way to strengthen the paper's evidence would be to compare u_{k,b} against an independently constructed Stark unit or a known algebraic unit (as done implicitly in Example 2's relative polynomial), but using a base field with nontrivial class group to test the reciprocity law across all classes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes Conjecture 2, a simplified version of a conjectural construction of higher elliptic units for number fields K of degree n ≥ 3 with exactly one complex place. For an ideal f satisfying f ∩ Z = qZ and O_f^{+,×} = O_K^{+,×}, the conjecture associates to classes [b] and k ∈ (Z/qZ)^× a complex number u_{k,b} defined by a product of smoothed higher elliptic Gamma functions G_{n−2}, and asserts that the u_{k,b} are units in the narrow ray class field K^+(f), are Galois conjugate, satisfy an Artin reciprocity law, and satisfy the Kronecker limit formula (14) involving partial zeta derivatives. The computational sections give Algorithm 1 for sieving 'optimal' fields using Proposition 3's t_min formula, and Algorithm 2 for computing u_{k,b}, with signs ν_ρ chosen by matching (14). Five examples in degrees 3, 4, 5, and mention of degree 6, are provided; each value is matched to 1000 digits to roots of an explicit polynomial defining a subextension of K^+(f), and in some cases zeta derivatives are matched. The construction depends on the author's cone construction [Mor25] and the deferred companion [Mor26b].

Significance. If correct, the paper would give a concrete analytic recipe for units in narrow ray class fields of number fields with one complex place, extending [BCG23] and advancing Hilbert's 12th problem for this class of fields. The numerical evidence is extensive and honestly reported: examples list concrete fields, parameters, and polynomials, and the 1000-digit agreement with class-field polynomials and with the Kronecker limit formula is strong support for the algebraicity of the proposed values. The paper also ships reproducible code and data (footnote 2), which is a genuine strength. The author explicitly identifies the sign ambiguity (Remark 6) and the technical nature of Proposition 3, so the load-bearing assumptions are visible. The main limitations are: (i) the parameters are produced by a construction in unpublished companion papers [Mor25], [Mor26b], with the proof of Proposition 3 deferred; and (ii) the Kronecker-limit check is weakened because the signs ν_ρ are chosen precisely to make that formula hold. The reciprocity law asserted in Conjecture 2 is not numerically verified. These issues reduce the degree to which the examples independently test the full conjecture, but they

major comments (4)
  1. [§3.2, Proposition 3 and (19)] Proposition 3 is the central computational tool: its formula t_min = λ̃ ∏ p_j^{v_j} is used in Algorithm 1 step 5 to select the fields in §4, and the parameters τ_{l,ρ}, δ_{j,ρ}, m_ρ of (13) come from the cone decomposition (19), both attributed to [Mor25]/[Mor26b]. The proof of Proposition 3 is only sketched and explicitly deferred to the unpublished companion [Mor26b]. A defect in that proof or in the existence of the generators α_{l,h} with the asserted minimal t_h would make the numerical examples irrelevant to Conjecture 2. This is not merely an exposition gap: the sieve that produces the examples uses the unverified formula. Please include a complete proof of Proposition 3 and of the parameter-generation statements, or state the conjecture and the numerical evidence as explicitly conditional on those companion results.
  2. [§3.3, Algorithm 2 step 6; Remark 6] In Algorithm 2 step 6 the signs ν_ρ in (13) are determined by checking the expected Kronecker limit formula (14), and Remark 6 acknowledges that the conjecture is 'expressed up to the signs ν_ρ which we determine by checking the Kronecker limit formula (14)'. Consequently, the reported agreement of (14) is not an independent verification of that part of Conjecture 2; it is a fitting procedure. Statements such as 'We may also check formula (14) up to 1000 digits' (end of §2) therefore overstate the evidence. To make the check meaningful, the signs should be determined a priori (Remark 6 suggests a relation to Espinoza's signs) or the signs should be fitted on a subset of classes b and then confirmed on the remaining classes, with the independence explicitly demonstrated.
  3. [§4 and Conjecture 2 (reciprocity law)] Conjecture 2 asserts the Artin reciprocity law σ_{k'b'}(u_{k,b}) = u_{kk',bb'}, but the numerical sections do not verify this action. The examples identify the set {u_{k,b}} as (a subset of) roots of an explicit polynomial in O_K[X], which tests algebraicity and Galois conjugacy over K but not the finer Artin action on individual conjugates. Please add at least one example where the Artin symbol is computed explicitly and the predicted identity σ_{k'b'}(u_{k,b}) = u_{kk',bb'} is checked numerically, for instance by evaluating the action on the relative polynomial. Without this, the numerical support for the class-field-generation part of Conjecture 2 is incomplete.
  4. [§2 and §3.2 (self-containedness of the conjecture)] Conjecture 2 is stated as a mathematical conjecture, but the 'explicitly computable' elements τ_{l,ρ}, δ_{j,ρ}, m_ρ are defined through a cone construction in [Mor25] and [Mor26b], including the helper ideal H in (22). A reader cannot independently evaluate or reproduce the conjecture without access to the companion papers. Please make the conjecture self-contained by including the necessary definitions and existence statements, or by explicitly marking Conjecture 2 as conditional on a construction that is only announced elsewhere.
minor comments (6)
  1. [§2, Remark 1] Remark 1 says Conjecture 2 contains two statements (algebraicity and Kronecker limit formula), but the conjecture also asserts Galois conjugacy and an Artin reciprocity law. Please align the remark with the full statement.
  2. [§3.1, after (18)] The bound O(∏_{k=1}^r |ℑ(z)|/ℑ(τ_k)) for the number of translation steps is stated without proof ('we shall omit this proof'). For a computational paper this is acceptable, but please mark it as a heuristic or provide a reference/proof in an appendix.
  3. [§4.1, formula for u_k] The displayed formula for u_{k,(1)} has negative exponents in both factors. The notation is mathematically clear from (13), but a one-sentence clarification of which exponent is the smoothing exponent and which is ν_ρ would help readers.
  4. [§4.3, Eq. (24)] The convergence discussion for the real-parameter case is plausible, but the assertion that such trigonometric-sum formulas exist for all G_r is given without a precise reference. Please cite the relevant statement in [Nis01] or provide a derivation.
  5. [References] [Mor24], [Mor25], [Mor26a], [Mor26b] are unpublished or in progress. Please mark their status clearly and, for statements used in this paper (especially Proposition 3), indicate which are fully proved there. This is important for assessing the reproducibility of the numerical work.
  6. [§3.1, typo] Minor typo: 'pratical' should be 'practical' in the paragraph before §3.2.

Circularity Check

2 steps flagged

The Kronecker limit formula is calibrated into the construction: Remark 6 and Algorithm 2 fix the signs ν_ρ by imposing (14), so the reported check of (14) is not independent. The algebraicity content remains nontrivial.

specific steps
  1. fitted input called prediction [Section 2, Remark 6 (after Conjecture 2); Section 3.3, Algorithm 2 step 6]
    "This conjecture is expressed up to the signs ν_ρ which we determine by checking the Kronecker limit formula (14). ... For each b and each k∈Z/qZ×, compute u_{k,b} for all possible choices of signs ν_ρ and identify the correct one using the expected Kronecker limit formula (14)."

    The signs ν_ρ are part of the data defining u_{k,b} in (13). Algorithm 2 chooses them by imposing the target identity (14), and Section 4 then reports the same identity as checked ('we checked that the values we obtain ... satisfying the expected Kronecker limit formula (14)'). Thus the numerical agreement with (14) is a calibration, not an independent prediction. The algebraicity/conjugacy checks (via algdep and polynomial identification) retain real content, but the Kronecker-limit part of Conjecture 2 is not independently tested by these examples.

  2. self citation load bearing [Section 3.2, Proposition 3 proof remark and Algorithm 2 step 5]
    "The proof of Proposition 3 is technical and requires a deeper dive in the construction (see [Mor26b] for more details). ... For each b, compute a helper ideal H and the vectors h_ρ (see [Mor26b])."

    The optimality sieve in Algorithm 1 discards fields unless Proposition 3's formula t_min = λ̃ ∏ p_j^{v_j} holds, and Algorithm 2 uses the h_ρ from the same source to produce τ_{l,ρ}, δ_{j,ρ}, m_ρ in (13). The proof is deferred to an unpublished paper by the same author, so the numerical examples are not self-contained checks of Conjecture 2: a defect in [Mor26b] would invalidate the parameter choice. This is a load-bearing same-author citation, though it is a support/correctness gap rather than an equation-level reduction.

full rationale

The clearest circular step is the sign calibration. Conjecture 2 defines u_{k,b} only up to signs ν_ρ, and the paper explicitly says it determines those signs by checking the Kronecker limit formula (14). The same formula is later listed among the facts checked numerically, so that particular numerical agreement is forced by construction rather than earned as a prediction. By contrast, the algebraicity portion of the evidence is not fitted: the computed values are fed to lindep/algdep and matched to polynomials defining K^+(f), which is a nontrivial check that does not reduce to the input. The paper also relies heavily on the author's own [Mor25]/[Mor26b] for the cone decomposition (19) and for Proposition 3, with the full proof deferred to the unpublished [Mor26b]; this makes the computational pipeline non-self-contained, but it is better classified as a load-bearing self-citation/support gap than as an equation-level circularity. Overall, one main component of the conjecture has its numerical support fitted, giving partial circularity rather than total collapse.

Axiom & Free-Parameter Ledger

6 free parameters · 8 axioms · 1 invented entities

Most numerical construction steps (N, units, helper ideal, h, cone generators) are hand-chosen inputs to an algorithm rather than fitted constants; the only quantity actually fitted to the claimed output is the sign vector νρ, which is chosen to satisfy (14). The deeper assumptions are the unproved Proposition 3 and the unpublished cone construction from [Mor25]/[Mor26b].

free parameters (6)
  • Signs νρ for each permutation ρ = ±1; in Example 1: −17 and −1; omitted for other examples
    Chosen in Algorithm 2 step 6 to match the expected Kronecker limit formula (14); therefore the later report of (14) being 'checked' is not an independent test.
  • Smoothing prime N = N(a) = 17, 5, 23, 19, 11 in Examples 1–5
    Picked as the smallest prime satisfying coprimality conditions; affects the value u_{k,b} and the left side of (14), but should not affect algebraicity according to the conjecture.
  • Choice of fundamental units ε1,...,ε_{n−2} of O^{+,×}_K = LLL-reduced units; explicit per example (e.g. ε=z^2+2z−1 in §4.1)
    The optimality conditions (λ̃ρ=1, v_{ρ,j}=0) and the ideals Dρ depend on this choice; Algorithm 1 screens choices by hand.
  • Helper ideal H in (22) = O_K or p/P; in §4.2 helper ideals above 13 and 41 are used
    Chosen so that the fractional ideal for h lies in the required class and h is admissible; the units and parameters depend on it.
  • Admissible vector hρ / integer mρ = mρ=1 in several examples; hρ generates qN/(ab) D'ρ H
    Among admissible hρ attaining t_min, the specific choice changes τ and δ parameters, hence the numerical representatives of u_{k,b}.
  • Generators α_{l,hρ} of the cone C_h = implicit in τ_{l,ρ} values listed in examples
    Choice of minimal generators in (19) is not unique; different minimal choices alter δ_j and τ_l, so it is a hand-fixed input.
axioms (8)
  • standard math Modular transformation property (6) for G_r (Felder–Varchenko; Narukawa)
    Used in Sec. 2 to justify the cocycle construction; accepted from prior literature, not reproved.
  • standard math Nishizawa product formula (16), inversion (17), pseudo-periodicity (18)
    The computational method in §3.1 relies on these prior identities.
  • standard math Dirichlet unit theorem: O^{+,×}_f has rank n−2 and is free
    Used in Sec. 2 to form the cycle Υ (9).
  • standard math Chebotarev density theorem guarantees existence of representatives b_c and smoothing prime a
    Algorithm 2 steps 3–4; used to assert computations terminate.
  • domain assumption Simplifying hypotheses f∩Z=N(f)Z and O^{+,×}_f=O^{+,×}_K
    These restrict the conjecture to the 'optimal settings' of the title; the paper notes only finitely many such f exist (Lemma 1).
  • domain assumption Rank-one abelian Stark conjecture for fields with one complex place
    Motivational frame (Remark 2); not used in the algebraicity numerical checks.
  • ad hoc to paper Proposition 3 formula t_min=λ̃∏ p_j^{v_j} with proof deferred to [Mor26b]
    Load-bearing for screening optimal examples and computing parameters; the proof in §3.2 is only a sketch.
  • ad hoc to paper Author's cone construction [Mor25]/[Mor26b] attaches G_{n−2} to arbitrary lattices/cones and guarantees center-strip parameters
    The parameters τ_{l,ρ}, δ_{j,ρ} in (13) come from this unpublished construction; if flawed, examples do not test the conjecture.
invented entities (1)
  • Higher elliptic units u_{k,b} independent evidence
    purpose: Conjectured algebraic units in the narrow ray class field K^+(f), meant to generate abelian extensions of K
    They are numerically computable and the paper recognizes them as roots of explicit class-field polynomials and checks (14); a falsifiable handle exists outside the paper.

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In this paper we present a simplified form of a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. They are conjectural algebraic numbers which are obtained as special values of higher elliptic Gamma functions. These functions form a collection of multivariate meromorphic functions which were studied in the late 1990s and early 2000s in mathematical physics. Our construction extends the scheme of a recent article by Bergeron, Charollois and Garc\'ia where they constructed conjectural elliptic units above complex cubic fields using the elliptic Gamma function. The higher elliptic units we construct are expected to generate specific abelian extensions of the base field where they are evaluated, thus giving a conjectural solution to Hilbert's 12th problem for the number fields with exactly one complex place. We provide several examples to support our conjecture in optimal settings for number fields of degree 3, 4, 5 and 6.

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