Generalised Elliptic Units in Class Field Theory

• Generalised elliptic units are explicit algebraic units in ray class fields constructed using multivariate elliptic gamma functions, generalizing classical modular units.
• They underpin explicit class field theory by providing abelian extension generators that satisfy precise norm relations and reciprocity laws.
• Their construction integrates analytic, cohomological, and modular techniques, with computational validations in cubic to quintic fields and links to Hilbert’s twelfth problem.

Generalised elliptic units are explicit algebraic units in ray class fields of number fields, constructed via special values of analytic functions generalizing classical modular forms—most notably through the use of multivariate elliptic gamma and related functions. These units serve as higher-degree analogues of classical elliptic or modular units, which originate in the theory of complex multiplication and explicit class field theory for imaginary quadratic fields. The generalised constructions extend to fields with one complex embedding of arbitrary degree and are conjectured to provide explicit generators for abelian extensions in the spirit of Hilbert's twelfth problem.

1. Classical Context and Modular-Analytic Preliminaries

Classical elliptic units arise from the specialization of modular units at complex multiplication (CM) points on modular curves, with explicit constructions via Siegel functions, division polynomials, and q-expansions on modular curves such as $Y^1(N)$. The units so obtained (e.g., Siegel–Ramachandra invariants) generate abelian extensions—specifically, ray class fields of imaginary quadratic fields—and their Galois and norm properties are controlled via modular and reciprocity laws (Streng, 2015). These units are described analytically by products and expansions involving the classical theta function $\theta_0(z;\tau)$ and generalize via $p$-adic and $\ell$-adic cohomological constructions to provide norm-compatible systems, crucial for Iwasawa theory and arithmetic applications (Kings, 2013, Bley et al., 2018).

2. Multivariate Elliptic Gamma Functions and Higher Analytic Theory

The elliptic gamma function, introduced in mathematical physics and studied for its modularity under $\mathrm{SL}_3(\mathbb{Z})$ by Felder and Varchenko, is a holomorphic function of three complex variables and forms the analytic backbone of higher-dimensional constructions of generalised elliptic units (Bergeron et al., 2023).

For $z, \tau, \sigma \in \mathbb{C}$ with $\operatorname{Im}\tau, \operatorname{Im}\sigma > 0$, the function is defined by: $$\Gamma(z;\tau,\sigma) = \prod_{j,k\geq 0} \frac{1 - e^{2\pi i ((j+1)\tau + (k+1)\sigma - z)}}{1 - e^{2\pi i (j\tau + k\sigma + z)}}$$ with meromorphic extensions determined by periodicity and relations involving the Jacobi theta function. Its transformation law under $\mathrm{SL}_3(\mathbb{Z})$ analogizes the modularity of classical theta functions, and this higher modularity property underpins its arithmetic utility in ray class field constructions (Bergeron et al., 2023).

The multiple elliptic gamma functions $G_r(z;\tau_0,\dots,\tau_r)$ for $r\geq 0$, introduced as further generalizations, provide special values whose arithmetic properties are conjecturally tied to abelian extensions of number fields with a single complex place (Morain, 17 Jan 2026). Their functional equations, pseudo-periodicity, and explicit modular-type identities generalize the classical one-variable case and are crucial for defining higher-degree elliptic units.

3. Construction of Generalised Elliptic Units

For a complex cubic field $K$ (or more generally, for a number field $K$ of degree $n$ with a single complex embedding), and ray class field $K^+(\mathfrak{f})$, a conjectural construction proceeds as follows (Bergeron et al., 2023, Morain, 17 Jan 2026):

• Fix auxiliary data (conductor $\mathfrak{f}$, suitable ideals, smoothing primes $\mathfrak{a}$, and embedding $\sigma_C$).
• Define appropriate algebraic parameters governing the division points and fundamental units of $K$.
• Construct analytic invariants via ratios of appropriately parameterized multivariate elliptic gamma functions (for cubic fields, the Ruijsenaars elliptic gamma; for higher degrees, $G_{n-2}$).
• The general formula for the unit $u_{k,\mathfrak{b}}$ involves a product over permutations in the symmetric group $S_{n-2}$: $$u_{k,\mathfrak{b}} = \prod_{\rho\in S_{n-2}} \frac{G_{n-2}\left(\tfrac{k m_\rho}{q}+\delta_\rho ;\, \tau_{1,\rho},\dots,\tau_{n-1,\rho}\right)^{\nu_\rho N}}{G_{n-2}\left(N(\tfrac{k m_\rho}{q}+\delta_\rho) ;\, N\tau_{1,\rho},\dots,N\tau_{n-1,\rho}\right)^{\nu_\rho}}$$ where $q=\operatorname{Norm}_{\mathbb{Q}}(\mathfrak{f})$, $N=\operatorname{Norm}_{\mathbb{Q}}(\mathfrak{a})$, and the parameters arise from cycle decompositions of the totally positive units modulo $\mathfrak{f}$ (Morain, 17 Jan 2026).

The units thus defined conjecturally satisfy:

• They form a full Galois orbit under $\operatorname{Gal}(K^+(\mathfrak{f})/K)$, exhibiting explicit reciprocity (Artin's law):

$$\sigma_{[\mathfrak{c}]}(u_{k,\mathfrak{b}}) = u_{k, \mathfrak{b} \mathfrak{c}}$$

• They satisfy exact norm relations analogous to those of classical elliptic units.
• Their absolute values are governed by Kronecker limit formulae involving partial zeta function derivatives at $s=0$:

$$N\,\zeta_{\mathfrak{f}}'([k\mathfrak{b}],0) - \zeta_{\mathfrak{f}}'([k\mathfrak{a}\mathfrak{b}],0) = \log|u_{k,\mathfrak{b}}|^2$$

(Bergeron et al., 2023, Morain, 17 Jan 2026).

4. Explicit Examples and Computational Evidence

Worked cases in cubic, quartic, and quintic fields provide computational validation for the conjectural properties (Morain, 17 Jan 2026). In these, explicit evaluations of $G_r$-quotients at algebraic parameters yield units whose minimal polynomials over $K$ match those predicted for ray class field generators. The Kronecker limit formula holds numerically to high precision in each example.

For instance, in the cubic case with $q=11$ and smoothing ideal norm $17$, quotient constructions using the Ruijsenaars $\Gamma$ at normalized parameters yield ten Galois-conjugate units whose minimal polynomial defines the narrow ray class field $K^+(\mathfrak{f})$. Analogous results are verified for quartic and quintic examples, with computations matching theoretical predictions on analytic behavior and field generation (Morain, 17 Jan 2026).

5. Cohomological, $\ell$-adic, and Iwasawa-Theoretic Extensions

Generalised elliptic units admit a rich cohomological framework. The $\ell$-adic theory, framed via Iwasawa modules and sheaves on the modular curve, connects Kato's norm-compatible elliptic units, Soulé elements, and Eisenstein classes. The elliptic polylogarithm sheaf realizes explicit norm-compatible classes whose residues and cup-product formulas can be computed via Bernoulli distributions and have implications for $p$-adic and $\ell$-adic $L$-functions (Kings, 2013). In this perspective:

• The units give rise to Euler systems in Iwasawa theory.
• Distribution and norm compatibility is controlled by recurrence and functional relations in cohomology.
• The residues, expansions, and class field-theoretic properties of generalised elliptic units can be interpreted entirely in the polylogarithmic-hereditary framework (Kings, 2013, Bley et al., 2018).

6. Interplay with Hilbert’s Twelfth Problem and Class Field Theory

The analytic construction of generalised elliptic units realizes a major ambition of explicit class field theory: to provide transcendental generating functions whose values, at appropriately chosen algebraic arguments, generate abelian extensions of number fields. While the construction for imaginary quadratic fields via classical theta and modular units is classical, the multivariate elliptic gamma and related functions provide the higher-dimensional analogues conjecturally solving Hilbert's twelfth problem for complex cubic and, more generally, fields of degree $n$ with a single complex place (Bergeron et al., 2023, Morain, 17 Jan 2026). These generalised functions take the place of the Jacobi theta in dimension $n=2$, bringing explicit analytic generators, precise reciprocity laws, and limit formulae into the context of non-CM and higher-degree abelian extensions.

7. Connections to Modular, Siegel, and Soulé Units

Beyond analytic and cohomological frameworks, the theory is deeply linked with the structure of modular units on curves like $Y^1(N)$. Modular units generated by elliptic divisibility sequences or Siegel functions specialize at CM points to the classical elliptic units, with the divisibility and norm relations generalizing naturally to settings of higher rank (Streng, 2015). The construction of generalised elliptic units thus synthesizes approaches from modular forms, arithmetic geometry, Iwasawa theory, and the explicit class field theory, unifying the subject across degrees and forms.