Elliptic Curves

An elliptic curve is a smooth cubic curve whose points carry a natural group law; over the rationals this group is finitely generated, making elliptic curves a uniquely tractable yet deep family of Diophantine equations.

Definition

An elliptic curve over a field is a smooth projective curve of genus one with a chosen base point; equivalently, away from small characteristics, the set of solutions of a Weierstrass cubic together with a point at infinity, forming an abelian group.

Scope

This topic covers Weierstrass equations and the discriminant and j-invariant, the chord-and-tangent group law, elliptic curves over the rationals and the Mordell-Weil theorem, torsion subgroups and Mazur's classification, the rank and methods of descent, reduction modulo primes and the local-global picture, the L-function of an elliptic curve, and the Birch and Swinnerton-Dyer conjecture relating rank to the order of vanishing of that L-function.

Core questions

• How does the chord-and-tangent construction make the points of an elliptic curve into an abelian group?
• Why is the group of rational points finitely generated, and how are its rank and torsion determined?
• How does reduction modulo a prime relate the curve to curves over finite fields and to its L-function?
• What does the Birch and Swinnerton-Dyer conjecture predict about the rank?

Key theories

Group law and Mordell-Weil theorem

Three points on a line on an elliptic curve sum to the identity, giving an abelian group; over the rationals this group is finitely generated, equal to a finite torsion part plus a free part of some rank.

Torsion and Mazur's theorem

The torsion subgroup of a rational elliptic curve is one of fifteen explicit groups (Mazur's theorem), so the only mystery in Mordell-Weil is the rank.

L-functions and Birch-Swinnerton-Dyer

The Hasse-Weil L-function built from point counts modulo primes is conjectured to vanish at the central point to order equal to the rank, a Millennium Prize problem partially proven in low-rank cases.

Clinical relevance

Elliptic curves over finite fields power elliptic-curve cryptography, including key exchange and digital signatures, whose efficiency and security rest on the group law and the hardness of the elliptic-curve discrete logarithm problem; they also underlie isogeny-based post-quantum proposals.

History

Elliptic curves arose from elliptic integrals studied by Abel and Jacobi. Poincare and Mordell established the group law and finite generation over the rationals in the early twentieth century; Weil generalized this to abelian varieties, and the Birch and Swinnerton-Dyer conjecture emerged from numerical experiments in the 1960s.

Key figures

• Louis Mordell
• Andre Weil
• Barry Mazur
• Bryan Birch
• Peter Swinnerton-Dyer

Seminal works

• silverman2009

Frequently asked questions

Are elliptic curves shaped like ellipses?

No. The name comes from elliptic integrals used to compute arc lengths of ellipses; an elliptic curve is a cubic curve and looks nothing like an ellipse.

What is the rank of an elliptic curve?

It is the number of independent rational points of infinite order; computing it is difficult, and the Birch and Swinnerton-Dyer conjecture relates it to the behaviour of the curve's L-function at the central point.