Quadratic Fields
Class Numbers from Analytic Class Number Formulas
Using L-functions and the analytic class number formula to compute or verify class numbers and regulators.
Engineering / MathematicsQuadratic Fields2 min readKV-MATH-0605
The analytic class number formula relates class number and regulator to a value of an L-function. It provides an independent route to these invariants and, more importantly, an independent check on results obtained algebraically.
The formula for quadratic fields
For a quadratic field, the class number is expressed in terms of the value at one of the Dirichlet L-function attached to the discriminant character.
Computing the L-value
Evaluating the L-function at one
- Compute the characterKronecker symbols of the discriminant at successive integers — see symbol computation.
- Sum the seriesThe naive Dirichlet series converges far too slowly for direct use.
- AccelerateUse a rapidly convergent expression involving an incomplete gamma factor, or a class number formula expressed as a finite character sum.
- Bound the errorTruncation error must be small enough to identify the integer answer.
Use as verification
| Field type | What the formula gives | How it is used |
|---|---|---|
| Imaginary quadratic | The class number directly | Unconditional verification of an algebraic computation |
| Real quadratic | The product of class number and regulator | Verifies the pair jointly, not individually |
| General number field | The product of class number and regulator, via the Dedekind zeta function | Same joint verification |
Direct computation
For small discriminants the formula is a practical computation method, not merely a check. It becomes expensive as the precision requirement grows with the discriminant, at which point algebraic methods dominate — but the check remains available and remains worth running.
Related L-functions
The same analytic machinery in the elliptic curve setting gives the conjectural formula relating the L-function to rank — see L-functions and BSD.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 5.3.3. 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.