The authors study unimodular positive definite Hermitian lattices over the ring of integers in the imaginary quadratic number field where is a squarefree positive integer. Such a lattice is a finitely generated projective -submodule of a -vector space of, say, dimension with basis equipped with the Hermitian form with such that contains a basis of and . being unimodular means that . One has for all , and is called even if for all , odd otherwise.
In this paper, one assumes throughout . When , then there is only one genus of unimodular such lattices that are necessarily odd, namely that of the standard lattice . If , then there is in addition a genus of even unimodular such lattices denoted by . The mass of a genus of lattices is defined to be where ranges over a system of representatives of the isometry classes of lattices within the genus . A formula for has been determined by K. Hashimoto and H. Koseki [Tôhoku Math. J., II. Ser. 41, No. 1, 1–30 (1989; Zbl 0668.10029)].
The authors first derive a formula for in terms of . Using the trace of the extension , a Hermitian -lattice can be made into a quadratic -lattice of rank , where if and if . An -lattice is called a theta lattice if is even unimodular. The authors derive a formula for the mass of the genus of theta lattices. They explicitly compute the values for the masses and the numbers of isometry classes of theta lattices in the cases where has class number one and . The paper finishes with some results on the Hermitian theta series of theta lattices.