Elliptic Curves¶
Summary
Elliptic curves are algebraic curves with a natural group structure. They stand at the centre of Wiles' proof – the Frey curve, the Taniyama–Shimura conjecture, and the Galois representations all involve elliptic curves.
Prerequisites¶
1. What Is an Elliptic Curve?¶
An elliptic curve over a field \(K\) is a smooth, projective curve of genus \(1\) with a distinguished point. In practice, one works with the Weierstraß form:
For the curve to be "smooth" (no cusps or self-intersections), the discriminant must be non-zero:
Geometrically, an elliptic curve over \(\mathbb{R}\) is a smooth curve in the plane consisting either of one component (when \(x^3 + ax + b\) has one real root) or of two components.
Why 'elliptic'?
The name has nothing to do with ellipses. It originates historically from elliptic integrals – integrals of the form \(\int \frac{dx}{\sqrt{x^3 + ax + b}}\), which arise in computing the circumference of an ellipse.
Examples¶
| Curve | \(a\) | \(b\) | \(\Delta\) |
|---|---|---|---|
| \(y^2 = x^3 - x\) | \(-1\) | \(0\) | \(64\) |
| \(y^2 = x^3 + 1\) | \(0\) | \(1\) | \(-432\) |
| \(y^2 = x^3 - x + 1\) | \(-1\) | \(1\) | \(-368\) |
The Point at Infinity¶
Technically, an elliptic curve lives in projective space \(\mathbb{P}^2\). Besides the affine points \((x, y)\) with \(y^2 = x^3 + ax + b\), there is an additional point at infinity \(\mathcal{O}\), which serves as the identity element of the group structure.
2. The Group Operation¶
The most remarkable feature of elliptic curves: their points form an abelian group. The operation is defined geometrically by the secant-tangent method:
Addition of two points \(P + Q\): 1. Draw the line through \(P\) and \(Q\) 2. This line intersects the curve in exactly one third point \(R'\) 3. Reflect \(R'\) across the \(x\)-axis: the result is \(P + Q\)
Doubling \(2P = P + P\): 1. Draw the tangent to the curve at \(P\) 2. This tangent intersects the curve in a second point \(R'\) 3. Reflect: the result is \(2P\)
Identity element: The point \(\mathcal{O}\) at infinity. We have \(P + \mathcal{O} = P\) for all \(P\).
Inverse: The inverse of \(P = (x, y)\) is \(-P = (x, -y)\) (reflection across the \(x\)-axis).
Algebraic Formulas
For \(P = (x_1, y_1)\) and \(Q = (x_2, y_2)\) with \(P \neq \pm Q\):
For \(P = Q\) (doubling):
These formulas work over any field – including \(\mathbb{F}_p\) or \(\mathbb{Q}_p\).
3. Rational Points and Mordell's Theorem¶
The set of rational points \(E(\mathbb{Q}) = \{(x, y) \in \mathbb{Q}^2 \mid y^2 = x^3 + ax + b\} \cup \{\mathcal{O}\}\) is a subgroup of \(E\).
Theorem (Mordell, 1922). \(E(\mathbb{Q})\) is a finitely generated abelian group.
By the structure theorem for finitely generated abelian groups:
where: - \(r = \text{rank}(E)\) is the rank of the curve (the number of independent points of infinite order) - \(E(\mathbb{Q})_{\text{tors}}\) is the finite torsion subgroup (points of finite order)
Theorem (Mazur, 1977). The torsion subgroup \(E(\mathbb{Q})_{\text{tors}}\) is isomorphic to one of the following groups:
The rank \(r\), by contrast, is hard to compute, and it remains unknown to this day whether elliptic curves of arbitrarily large rank exist.
4. Reduction Modulo \(p\)¶
An elliptic curve \(E: y^2 = x^3 + ax + b\) with \(a, b \in \mathbb{Z}\) can be reduced modulo a prime \(p\):
If \(p \nmid \Delta\) (the discriminant), then \(\tilde{E}\) is a smooth curve over \(\mathbb{F}_p\) – one says \(E\) has good reduction at \(p\). Otherwise, \(E\) has bad reduction.
The \(a_p\)-Coefficients¶
For primes \(p\) of good reduction, we define:
where \(\#\tilde{E}(\mathbb{F}_p)\) is the number of points of \(\tilde{E}\) over \(\mathbb{F}_p\) (including \(\mathcal{O}\)).
Theorem (Hasse, 1933). We have \(|a_p| \leq 2\sqrt{p}\).
The \(a_p\)-coefficients encode what the curve looks like "prime by prime". They are the building blocks of the \(L\)-series.
Example: For \(E: y^2 = x^3 - x\) and \(p = 5\):
| \(x\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|
| \(x^3 - x \bmod 5\) | \(0\) | \(0\) | \(1\) | \(4\) | \(0\) |
| \(y^2 \equiv \ldots\)? | \(y = 0\) | \(y = 0\) | \(y = \pm 1\) | \(y = \pm 2\) | \(y = 0\) |
This gives \(8\) affine points plus \(\mathcal{O}\), so \(\#\tilde{E}(\mathbb{F}_5) = 9\) and \(a_5 = 5 + 1 - 9 = -3\).
5. The \(L\)-Series of an Elliptic Curve¶
The \(a_p\)-coefficients are assembled into an analytic function – the \(L\)-series (Hasse–Weil):
This \(L\)-series converges for \(\text{Re}(s) > 3/2\) and encodes the entire arithmetic information of the curve.
The BSD conjecture (Birch and Swinnerton-Dyer, one of the Millennium Problems): The rank of \(E(\mathbb{Q})\) equals the order of the zero of \(L(E, s)\) at \(s = 1\).
The Connection to Modular Forms¶
The central question: is the \(L\)-series \(L(E, s)\) equal to the \(L\)-series of a modular form \(f\)?
If so, \(E\) is called modular. The Taniyama–Shimura conjecture (now a theorem) states: every elliptic curve over \(\mathbb{Q}\) is modular. This theorem – more precisely: the semistable case, proved by Wiles – implies Fermat's Last Theorem.
6. Torsion Points and Galois Representations¶
For a prime \(\ell\), the \(\ell\)-division points are the points \(P \in E(\overline{\mathbb{Q}})\) with \(\ell P = \mathcal{O}\):
We have \(E[\ell] \cong (\mathbb{Z}/\ell\mathbb{Z})^2\) – a two-dimensional group over \(\mathbb{F}_\ell\).
The absolute Galois group \(G_{\mathbb{Q}}\) acts on \(E[\ell]\) by permuting the coordinates. This defines a Galois representation:
For the \(\ell\)-adic Tate modules (the projective limit over all \(\ell^n\)-division points), one obtains an \(\ell\)-adic representation:
These representations are the link between elliptic curves and Galois theory – and the central object of Wiles' proof.
7. Elliptic Curves and Cryptography¶
A brief excursion into applications: elliptic curves over finite fields \(\mathbb{F}_p\) form the foundation of modern cryptographic methods (ECC – Elliptic Curve Cryptography).
The security rests on the discrete logarithm problem: given \(P\) and \(Q = nP\) on \(E(\mathbb{F}_p)\), it is computationally extremely difficult to find \(n\) – even though computing \(nP\) from \(n\) and \(P\) is efficient (via repeated doubling and addition).
ECC offers the same security level as RSA, but with significantly shorter keys:
| Security level | RSA key | ECC key |
|---|---|---|
| 128 bit | 3072 bit | 256 bit |
| 256 bit | 15360 bit | 512 bit |
8. Outlook: Modularity¶
This article has presented elliptic curves as self-contained algebraic objects. Their true power unfolds in the interplay with modular forms – the subject of the next tool article.
The chain of connections:
The Taniyama–Shimura conjecture asserts that the arrow in the middle always exists – that every elliptic curve has its "twin soul" in the realm of modular forms. Wiles' proof of this conjecture (for semistable curves) is the key to Fermat's Last Theorem.
Further Reading¶
- Nigel Boston: The Proof of Fermat's Last Theorem, Ch. 6
- Joseph Silverman: The Arithmetic of Elliptic Curves – the standard reference
- Joseph Silverman, John Tate: Rational Points on Elliptic Curves – accessible introduction
- Andrew Wiles: Modular elliptic curves and Fermat's Last Theorem, §1