Unit Shapes Distinguish Non-CM D6 Sextic Fields

A number field has both an arithmetic lattice, its ring of integers, and a geometric object built from the logarithms of its units. The former has a substantial literature on lattice shapes; the latter has received much less systematic treatment. Cruz, Holmes, Jalalvand, Nunez Lon-Wo, Scheidler, and Tran study this second object for fields of unit rank 2, asking where its shape sits in the moduli space of rank-2 lattices and how much field-theoretic information the shape retains.

The paper's central point is that a log-unit lattice is not merely a coarse geometric invariant. In several rank-2 settings, its position in a fundamental domain reflects the Galois group of the Galois closure. For a particular family of imaginary sextic fields, the invariant is still stronger: the unit shape determines the field up to isomorphism, subject to an essential non-CM hypothesis.

Core Contribution

The authors organize rank-2 unit shapes through the standard fundamental domain for shapes of planar lattices. Rather than treating the regulator lattice only through its covolume, they retain its similarity class. This preserves angular information: an orthogonal log-unit lattice, for example, occupies a distinguished geometric position that cannot be recovered from the regulator alone.

Their first main result gives, in many cases, a characterization of where a unit shape lies in that domain in terms of the Galois group of the field's Galois closure. The paper also identifies cases in which the resulting shape is transcendental. This connects a geometric coordinate of a lattice-shape space to arithmetic symmetries of the field, rather than presenting shape as an unstructured numerical by-product of computing units.

For totally imaginary $D_6$ non-CM sextic fields, the authors prove an injectivity statement: equality of unit shapes forces isomorphism of fields. That claim is deliberately family-specific. The corresponding assertion fails in the CM case, so the result does not establish unit shape as a universal complete invariant for sextic fields.

Technical Approach

The construction begins with the log embedding of the unit group. After quotienting by torsion, a field of unit rank 2 supplies a rank-2 lattice in the relevant logarithmic hyperplane. Its shape is then represented in the fundamental domain for the action of $S_2$ on rank-2 lattice shapes. The analysis uses that representation to compare the geometry of a basis of logarithmic units with the arithmetic constraints imposed by the Galois closure.

The $D_6$ analysis separates non-CM and CM imaginary sextics. That separation is structural rather than cosmetic: the non-CM setting permits recovery of the field from the shape, while the CM setting supplies counterexamples to such recovery. The paper further isolates subfamilies of $D_6$ non-CM fields for which a simple sufficient condition makes the log-unit lattice orthogonal. This turns a geometric question about an angle into an arithmetic criterion that can be counted across a family.

Figure 1 displays unit shapes of signature $(0,3)$ $D_6$ sextics inside the $S_2$ fundamental domain. The plot makes the paper's use of shape concrete: the objects being classified are points or loci in a fixed moduli region, not only regulator values.

Results and Analysis

The strongest result is the uniqueness theorem for totally imaginary non-CM $D_6$ sextic fields. Within that stated family, the unit shape carries enough information to identify the isomorphism class. This is a sharper conclusion than locating a shape or distinguishing broad Galois behavior, but its scope matters: the authors explicitly report failure in the CM case.

The orthogonality results give the paper a second, population-level direction. For selected $D_6$ non-CM subfamilies, the authors provide a sufficient condition for orthogonal log-unit lattices and lower bounds on the proportion of fields satisfying it. The supplied material establishes the existence of those bounds but does not provide a numerical proportion here, so the result should be read as a rigorous family-specific counting statement rather than a broad empirical frequency claim.

This is theoretical evidence, not a benchmark study. Its value is the exact link between lattice geometry and field structure. For researchers working with regulators, unit groups, or arithmetic statistics of sextic fields, it offers a more discriminating invariant than covolume alone; for broader classes of number fields, the paper leaves open how often comparable rigidity survives.