A periodic torus orbit in the space of unimodular lattices is a compact orbit of the positive diagonal subgroup. With row lattices and right multiplication, the ambient space and subgroup are
 |
(1)
|
 |
(2)
|
The stabilizer of a compact orbit is a lattice in the logarithmic coordinates of 
.
Totally real fields give arithmetic families
of these orbits. If 
is a nonzero fractional ideal and 
lists the real embeddings, its normalized lattice is
 |
(3)
|
Taking a complete ideal-class collection together with the appropriate diagonal sign translates gives a finite packet of distinct compact orbits. Its volume-weighted probability measure is
 |
(4)
|
where 
is the invariant probability on 
and each distinct orbit is counted once. The averaging is
over a complete packet, rather than a single orbit.
For a full point lattice 
in a degree-
totally real field 
, the multiplier number
field order is 
,
and its absolute discriminant is ![|Disc(K)|[O_K:O(M)]^2](/images/equations/PeriodicTorusOrbit/Inline10.svg)
.
Primitivity means that there is no proper intermediate field between 
and 
.
