Sub-exponential Class Group Computation
Verifying Class Group and Regulator Results
Confirming class group and regulator results against the analytic class number formula, and what such confirmation does and does not establish.
Engineering / MathematicsSub-exponential Class Group Computation2 min readKV-MATH-0631
A class group computed by relation collection is conditional and may be wrong in two specific ways. Verification against the analytic class number formula addresses both, and can make the result unconditional.
The two failure modes
| Failure | Effect on the answer |
|---|---|
| Insufficient relations | Computed class number is a multiple of the truth |
| Factor base does not generate | Computed class group is a quotient of the truth |
The analytic formula
The residue of the Dedekind zeta function at one is expressed in terms of the class number, the regulator, the discriminant, the signature and the roots of unity.
Verification against the analytic formula
- Compute the analytic valueBy an Euler product over prime ideals, truncated with a rigorous error bound.
- Compute the algebraic valueClass number times regulator from the relation computation.
- CompareAgreement within the error bound confirms the result.
- Interpret a mismatchA ratio that is a small integer indicates a missing factor — either relations or units.
What it establishes
What it does not establish
Additional checks
- The unit rank must equal r_1 + r_2 - 1 from the signature.
- Every relation must be verified by recomputing the ideal product and confirming it is principal with the recorded generator.
- The class group order must be consistent with any known genus theory constraints.
- For quadratic fields, small cases can be checked against form enumeration.
Error bounds on the analytic side
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.5.4. 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.