Elliptic Curves
Curve Reduction and Tate's Algorithm
Reduction of an elliptic curve modulo a prime, the classification of bad reduction types, and Tate's algorithm.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0644
Reducing a curve modulo a prime may or may not give another elliptic curve. Classifying what happens at each bad prime is Tate's algorithm, and its output determines the conductor and the local L-factors.
Reduction types
| Type | Condition | Local L-factor |
|---|---|---|
| Good | Reduction is smooth | Determined by the point count |
| Multiplicative, split | Node with rational tangents | Simple linear factor |
| Multiplicative, non-split | Node with conjugate tangents | Simple linear factor with opposite sign |
| Additive | Cusp | Trivial factor |
Minimal models
Reduction type is a property of the minimal model. A non-minimal model can appear to have bad reduction at a prime where the curve is actually good.
Tate's algorithm
Tate's algorithm
- MinimaliseReduce to a minimal model at the prime.
- Test smoothnessIf the reduction is smooth, the reduction is good and the algorithm stops.
- Classify the singularityNode or cusp.
- Work through the casesA sequence of tests on coefficient valuations determines the Kodaira type.
- OutputKodaira type, conductor exponent, and the number of components.
The conductor
The conductor collects the bad primes with exponents determined by the reduction type. It is the level of the associated modular form and the primary identifier of a curve in tables.
Torsion and components
The number of components of the special fibre feeds into the Birch-Swinnerton-Dyer formula as the local Tamagawa number, and it also constrains the torsion subgroup, which is useful as a cross-check.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.4.2. 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.