Elliptic units for complex cubic fields

Abstract: We propose a conjecture extending the classical construction of elliptic units to complex cubic number fields $K$. The conjecture concerns special values of the elliptic gamma function, a holomorphic function of three complex variables arising in mathematical physics whose transformation properties under $\mathrm{SL}_3(\mathbf{Z})$ were studied by Felder and Varchenko in the early 2000s. Using this function we construct complex numbers that we conjecture to be units in narrow ray class fields of $K$. We also propose a reciprocity law for the action of the Galois group on these units in the style of Shimura. To support our conjecture we offer numerical evidence and also prove a new type of Kronecker limit formula relating the logarithm of the modulus of these complex numbers to the derivatives at $s = 0$ of partial zeta functions of $K$. Our constructions unveil the role played by the elliptic gamma function in Hilbert's twelfth problem for complex cubic fields.