Elliptic Curves
L-Functions and the Birch-Swinnerton-Dyer Conjecture
The L-function of an elliptic curve, the Birch-Swinnerton-Dyer conjecture, and what can and cannot be computed about rank.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0642
The Birch-Swinnerton-Dyer conjecture relates the rank of a curve's group of rational points to the order of vanishing of its L-function at one. It remains unproven in general and is central to computational practice regardless.
The Mordell-Weil group
- Torsion
- Computable and bounded: only fifteen possibilities occur, by Mazur's theorem.
- Rank
- Not known to be computable in general. This is the hard part.
- Generators
- Even when the rank is known, finding generators can be very hard because they may have enormous height.
The conjecture
The L-function of the curve, built from the local point counts, is conjectured to vanish at one to order exactly the rank.
The strong form additionally gives the leading coefficient in terms of the regulator, the order of a conjecturally finite group, the torsion and local factors.
What is known
| Situation | Status |
|---|---|
| Analytic rank zero or one | The conjecture is proved |
| Analytic rank two or more | Open |
| Modularity of the L-function | Proved; the L-function has an analytic continuation |
| Finiteness of the relevant group | Open in general |
Computing rank in practice
Determining the rank
- Compute the analytic rankEvaluate the L-function and its derivatives near one.
- Search for pointsFind independent rational points, giving a lower bound on the rank.
- DescentTwo-descent or higher gives an upper bound.
- CompareWhen the bounds meet, the rank is determined unconditionally.
Height and generators
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.3.2. 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.