Class Groups, Units and Regulators
The Logarithmic Embedding and the Unit Lattice
Mapping units into a real vector space by logarithms of conjugate absolute values, and the lattice this produces.
Engineering / MathematicsClass Groups, Units and Regulators2 min readKV-MATH-0595
The logarithmic embedding converts the multiplicative unit group into an additive lattice, at which point all the lattice machinery becomes available. It is the technical device that makes unit computation possible.
The map
Where units land
A unit has norm plus or minus one, so the sum of its logarithmic coordinates is zero. The image therefore lies in a hyperplane of dimension one less than the number of embedding classes.
Consequences
| Question about units | Becomes |
|---|---|
| Are these units independent? | Are their images linearly independent? |
| Is this unit a root of unity? | Is its image zero? |
| Do these units generate the group? | Do their images generate the full lattice? |
| Regulator | Covolume of the lattice |
| Finding small units | Finding short lattice vectors |
Precision
A practical rule is that precision must exceed the expected size of the regulator in digits, with a comfortable margin, because the regulator is a determinant of these logarithms and cancellation amplifies error.
Finding units by reduction
Because independence and size become lattice questions, LLL applied to a set of known units produces a smaller and more nearly fundamental system. This is the standard final step of regulator recovery.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.9.2. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.