Class Groups, Units and Regulators
The Regulator: Definition and Computation
The regulator as the covolume of the unit lattice, its computation, and the precision and verification it demands.
Engineering / MathematicsClass Groups, Units and Regulators2 min readKV-MATH-0596
The regulator is the covolume of the unit lattice under the logarithmic embedding. It measures how large the fundamental units are, and together with the class number it is what the analytic class number formula controls.
Definition
Computation
Computing the regulator
- Find independent unitsAny independent system will do initially.
- EmbedCompute the logarithmic images to high precision.
- Take the determinantDelete one column and compute the determinant.
- Correct for indexThe result is the true regulator times the index of the subgroup generated. Determine that index and divide.
The index problem
| Method for the index | Character |
|---|---|
| Compare against the analytic class number formula | Standard; gives the product of class number and regulator |
| Lattice reduction on the known units | Finds smaller units, reducing the index; may not reach one |
| Search for units of small norm in the lattice | Exhaustive within a bound; expensive |
| Assume GRH bounds | Makes the search finite and practical |
Precision
Verification
The analytic class number formula relates the product of class number and regulator to a value of the Dedekind zeta function. Computing that value independently and comparing is the standard check — see verification and analytic formulas.
Real quadratic fields
For real quadratic fields the regulator is the logarithm of the fundamental unit, computable classically by continued fractions — see the fundamental unit.
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.