Elliptic Curves — Mathematics Knowledge System

Overview

An elliptic curve is a smooth projective cubic curve with a distinguished rational point, equipped with a group law that makes it an abelian group. Elliptic curves over finite fields underpin modern cryptography; over the rationals they are central to the Birch–Swinnerton-Dyer conjecture and Wiles's proof of Fermat's Last Theorem.

Intuition

Draw a cubic curve $y^2 = x^3 + ax + b$ in the plane. Given two points $P$ and $Q$ on the curve, draw the line through them; it meets the curve in a third point $R'$. Reflect $R'$ over the $x$-axis to get $P + Q = R$. This geometric recipe defines an abelian group whose identity is the 'point at infinity'.

Formal Definition

Definition

Over a field $K$ with $\text{char}(K) \neq 2, 3$, an elliptic curve in short Weierstrass form is:

E: y^2 = x^3 + ax + b, \quad a, b \in K, \quad \Delta = -16(4a^3 + 27b^2) \neq 0

Short Weierstrass equation; non-vanishing discriminant ensures smoothness

weierstrass

E(K) = \{ (x, y) \in K^2 : y^2 = x^3 + ax + b \} \cup \{\mathcal{O}\}

Set of K-rational points including the point at infinity O

rational-points

P + Q + R = \mathcal{O} \iff P, Q, R \text{ are collinear on } E

Group law: three collinear points sum to zero

group-law

Notation

Notation	Meaning
$E(K)$	Group of K-rational points on elliptic curve E
$\mathcal{O}$	Identity element (point at infinity)
$j(E)$	j-invariant: j = 1728 · 4a³/(4a³+27b²)
$\#E(\mathbb{F}_p)$	Number of points over the finite field F_p
$\text{rank}(E)$	Rank of the free part of E(Q)

Theorems

Theorem 1: Mordell–Weil Theorem

For an elliptic curve E over \mathbb{Q}, the group E(\mathbb{Q}) is finitely generated: E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}, where r \geq 0 is the rank.

Theorem 2: Hasse's Theorem

For an elliptic curve E over \mathbb{F}_p, |\#E(\mathbb{F}_p) - (p+1)| \leq 2\sqrt{p}.

Theorem 3: Mazur's Torsion Theorem

The torsion subgroup E(\mathbb{Q})_{\text{tors}} is isomorphic to one of the 15 groups: \mathbb{Z}/n\mathbb{Z} (1 \leq n \leq 10, n = 12) or \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2n\mathbb{Z} (1 \leq n \leq 4).

Applications

Elliptic curve Diffie–Hellman (ECDH) and ECDSA digital signatures use the group law over finite fields for public-key cryptography with smaller key sizes than RSA.

Worked Examples

First check: $P = (0,0)$ satisfies $0 = 0 - 0$. For the doubling formula, compute the slope of the tangent at $P$.
$\lambda = \frac{3x_1^2 + a}{2y_1} = \frac{3(0)^2 + (-1)}{2(0)}$
The denominator is 0 at $(0,0)$ since $y_1 = 0$, meaning the tangent is vertical. A vertical tangent at $P$ means $2P = \mathcal{O}$ — the point is a 2-torsion element.

Answer: $2P = \mathcal{O}$; the point $(0,0)$ has order 2.

Practice Problems

Difficulty 7/10

Determine which of the following is smooth: (a) $y^2 = x^3 - 3x + 2$, (b) $y^2 = x^3$. Compute the discriminant.

Difficulty 8/10

On $E: y^2 = x^3 - x + 1$ over $\mathbb{Q}$, compute $P + Q$ for $P = (0,1)$ and $Q = (1,1)$.

Difficulty 9/10

Verify that $(0,1), (0,-1) \in E(\mathbb{Q})$ for $E: y^2 = x^3 + 1$ are inverses, and use Mazur's theorem to constrain the torsion subgroup.

Historical Background

Elliptic integrals arising in arc-length calculations of ellipses were studied by Abel and Jacobi in the 19th century. The modern algebraic theory was developed by Weierstrass, Poincaré, and later Mordell (1922), who proved the rational points form a finitely generated group. Hasse proved the key bound on the number of points over finite fields in 1936. Wiles used elliptic curves decisively in his 1995 proof of Fermat's Last Theorem.

1830s
Abel and Jacobi study elliptic integrals and functions
Niels Abel, Carl Jacobi
1922
Mordell proves the rational points form a finitely generated group
Louis Mordell
1936
Hasse proves |#E(F_p) - (p+1)| ≤ 2√p
Helmut Hasse
1985
Koblitz and Miller independently propose elliptic curve cryptography
Neal Koblitz, Victor Miller
1995
Wiles proves Fermat's Last Theorem via modular elliptic curves
Andrew Wiles

Summary

An elliptic curve $y^2 = x^3 + ax + b$ ($\Delta \neq 0$) carries a natural abelian group law defined geometrically.
Mordell–Weil: $E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}$ with $r$ the rank.
Hasse's theorem bounds $\#E(\mathbb{F}_p)$ within $2\sqrt{p}$ of $p+1$.
Mazur's theorem classifies exactly which torsion groups can occur over $\mathbb{Q}$.
Elliptic curves underlie modern cryptography (ECDH, ECDSA) and Wiles's proof of FLT.

References

BookSilverman, J. The Arithmetic of Elliptic Curves. Springer, 2009.
BookWashington, L. Elliptic Curves: Number Theory and Cryptography. CRC Press, 2008.