Mordell's Theorem and Elliptic Curves

Elliptic curves are among the most beautiful and important objects in modern mathematics. They appear in diverse areas: from the proof of Fermat’s Last Theorem to modern cryptography. In this post, I’ll summarize the main ideas from my thesis on Mordell’s theorem, which states that the group of rational points on an elliptic curve is finitely generated.

Download the full thesis (PDF)

What are Elliptic Curves?

An elliptic curve over the rationals $\mathbb{Q}$ is a non-singular curve defined by an equation of the form:

$y^2 = x^3 + b_0x + b_1$

where $b_0, b_1 \in \mathbb{Q}$ and the discriminant $\Delta = 27b_1^2 + 4b_0^3 \neq 0$ (ensuring non-singularity).

This is called the Weierstrass normal form. Despite being called “elliptic curves,” these are not ellipses — the name comes from their historical connection to elliptic integrals used in computing arc lengths of ellipses.

Why Projective Geometry?

To properly study elliptic curves, we need to work in the projective plane $\mathbb{P}^2$ rather than the affine plane $\mathbb{A}^2$. The projective plane adds “points at infinity” where parallel lines meet.

In projective coordinates $(x : y : z)$, two points $(x, y, z)$ and $(\lambda x, \lambda y, \lambda z)$ represent the same point for any $\lambda \neq 0$. This allows us to capture behavior “at infinity” — for instance, the curves $y^2 = \alpha x^2 + 1$ and $y = x$ don’t intersect in the affine plane, but they do intersect at the point $(1 : 1 : 0)$ in the projective plane.

Bezout’s Theorem

A cornerstone result in algebraic geometry is Bezout’s theorem, which counts intersection points of algebraic curves.

Theorem (Bezout): Let $C$ and $D$ be two projective curves of degrees $d_1$ and $d_2$ with no common factors over an algebraically closed field $k$. Then:

$\sum_{P \in C \cap D} I(C \cap D, P) = d_1 \cdot d_2$

where $I(C \cap D, P)$ is the intersection multiplicity at point $P$.

The intersection multiplicity captures not just whether curves intersect, but how they intersect — for instance, if they’re tangent at a point, the multiplicity is higher. The proof involves constructing the local ring $\mathcal{O}_P$ at each intersection point and showing that the intersection multiplicity can be computed as:

$I(C_1 \cap C_2, P) = \dim\left(\frac{\mathcal{O}_P}{(f_1, f_2)_P}\right)$

This theorem is fundamental because it gives us precise control over intersection behavior, which we’ll use repeatedly in understanding the group structure on elliptic curves.

The Group Law on Elliptic Curves

One of the most remarkable features of elliptic curves is that the set of rational points naturally forms an abelian group. The group operation is defined geometrically:

The Composition Law: Given two rational points $P, Q$ on an elliptic curve $E$:

1. Draw the line through $P$ and $Q$ (or the tangent line if $P = Q$)
2. This line intersects $E$ at a third point $R$
3. Define $P + Q$ to be the reflection of $R$ across the $x$-axis

The point at infinity $\mathcal{O}$ serves as the identity element.

Proposition: The composition law defines an abelian group structure on $E(\mathbb{Q})$, where the point at infinity is the identity.

The proof of associativity is geometric and uses the fact that if two cubic curves intersect at 8 points, they must intersect at a 9th point. This can be proven using Bezout’s theorem and properties of linear systems of cubics.

The Duplication Formula

When adding a point to itself (computing $2P$), we need the tangent line. For a point $P = (x, y)$ on the curve $y^2 = x^3 + ax^2 + bx + c$, the slope of the tangent is:

$\lambda = \frac{dy}{dx} = \frac{3x^2 + 2ax + b}{2y}$

The $x$-coordinate of $2P$ can then be computed as:

$x(2P) = \lambda^2 - 2x - a = \frac{x^4 - 2bx^2 - 8cx + b^2 - 4ac}{4y^2}$

This formula is crucial for the descent argument, where we’ll need to track how the “size” of points changes under duplication.

The Descent Theorem: Height Functions

The key to proving Mordell’s theorem is the concept of a height function, which measures the arithmetic complexity of rational points.

Definition (Height): For $x = m/n$ written in reduced form with $\gcd(m,n) = 1$, the height is:

$H(x) = \max\{|m|, |n|\}$

For a point $P = (x, y)$ on an elliptic curve, we define $H(P) = \max\{|m|, |n|\}$ where $x = m/e^2, y = n/e^3$ is the reduced form.

We work with the logarithmic height $h(P) = \log H(P)$ because it converts products into sums, making estimates easier to manage.

Key Lemmas of the Descent

The descent proof consists of five technical lemmas that work together:

Lemma 1: If $(x, y)$ is a rational point on $y^2 = x^3 + ax^2 + bx + c$, then there exist integers $m, n, e$ such that $(x, y) = (m/e^2, n/e^3)$ in reduced form.

Lemma 2: For any $M \in \mathbb{R}$, the set $\{P \in E(\mathbb{Q}) \mid h(P) \leq M\}$ is finite.

This is because there are only finitely many integers in any bounded interval, so only finitely many possible numerators and denominators.

Lemma 3: For any point $P_0 \in E(\mathbb{Q})$, there exists $\kappa_0 \in \mathbb{R}$ such that:

$H(P + P_0) \leq 2H(P) + \kappa_0$

or equivalently in logarithmic form:

$h(P + P_0) \leq 2h(P) + \kappa_0$

Lemma 4: Let $\phi(X), \psi(X)$ be relatively prime polynomials with coefficients in $\mathbb{Z}$. If $d = \max\{\deg(\phi), \deg(\psi)\}$, then for all $m/n \in \mathbb{Q}$ with $\psi(m/n) \neq 0$:

$\gcd\left(n^d\phi\left(\frac{m}{n}\right), n^d\psi\left(\frac{m}{n}\right)\right) \mid R$

where $R$ depends only on $\phi$ and $\psi$. This controls the growth of height under rational functions.

Lemma 5: For all $m/n \in \mathbb{Q}$ with $\psi(m/n) \neq 0$:

$dh(m/n) - \kappa_1 \leq h\left(\frac{\phi(m/n)}{\psi(m/n)}\right)$

where $\kappa_1$ depends on $\phi$ and $\psi$.

These lemmas together show that height grows in a controlled way under the group operation, but grows rapidly under duplication. This imbalance is the key to the descent argument.

Mordell’s Theorem

With the descent machinery in place, we can state and prove the main theorem:

Theorem (Mordell, 1922): Let $E$ be an elliptic curve over $\mathbb{Q}$. Then the group $E(\mathbb{Q})$ of rational points is finitely generated. That is,

$E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}$

where $r \geq 0$ is called the rank of the elliptic curve, and $E(\mathbb{Q})_{\text{tors}}$ is a finite abelian group (the torsion subgroup).

Proof Sketch

The proof uses the weak Mordell-Weil theorem (which shows that $E(\mathbb{Q})/2E(\mathbb{Q})$ is finite) combined with the descent argument:

1. Torsion is finite: First, show that the torsion subgroup $E(\mathbb{Q})_{\text{tors}}$ is finite. This follows from Lemma 2: torsion points have bounded height, and there are only finitely many such points.

2. Descent: For any finitely generated subgroup $G \subset E(\mathbb{Q})$, consider the quotient $E(\mathbb{Q})/G$. The height function allows us to show that we can find a finite set of points that, together with $G$, generate a strictly larger subgroup.

3. Weak Mordell-Weil: The key step is showing $E(\mathbb{Q})/2E(\mathbb{Q})$ is finite. This uses the duplication formula and properties of height functions to show that there are only finitely many “essentially different” points modulo duplication.

4. Induction: By considering the chain $E(\mathbb{Q}) \supset 2E(\mathbb{Q}) \supset 4E(\mathbb{Q}) \supset \cdots$ and using the finiteness of each quotient, we can show that $E(\mathbb{Q})$ is finitely generated.

The full details involve careful tracking of height bounds and intersection theory, but the geometric intuition is that the duplication map “stretches” points in a way that only finitely many can fit in any bounded region.

Rank and Examples

The rank $r$ of an elliptic curve is one of the most mysterious invariants in number theory. The Birch and Swinnerton-Dyer conjecture (one of the Millennium Prize Problems) predicts that the rank can be read off from the behavior of the L-function of the elliptic curve.

Some examples:

• The curve $y^2 = x^3 - x$ has rank 0
• The curve $y^2 = x^3 - 2$ has rank 1
• The current record for a curve of known rank is around rank 28 (as of recent computations)

Unlike the torsion subgroup (which is bounded by Mazur’s theorem), the rank can be arbitrarily large — but we don’t know if ranks are unbounded in general!

Conclusion

Mordell’s theorem is a beautiful blend of algebraic geometry, number theory, and group theory. The proof technique — using height functions to perform an infinite descent — has been generalized far beyond elliptic curves and is a cornerstone of modern arithmetic geometry.

The machinery developed here (Bezout’s theorem, the group law, height functions, descent) forms the foundation for deeper results like the Mordell-Weil theorem for abelian varieties and Faltings’s theorem (the former Mordell conjecture).

For the complete details, proofs, and technical lemmas, see the full thesis (PDF).

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This post summarizes my undergraduate thesis on Mordell’s theorem and elliptic curves, completed in April 2026.