An elliptic curve is a curve defined by an equation of the form:

$$y^2 = x^3 + Ax + B$$

for particular values of $A$ and $B$, with the condition that $4A^3 + 27B^2 \neq 0$ (which ensures the curve is smooth — no cusps or self-intersections).

Despite the name, elliptic curves have nothing to do with ellipses. The name comes from their historical connection to elliptic integrals (used to compute the arc length of an ellipse), but the curves themselves are cubic, not conic.

What they look like

Over the real numbers, an elliptic curve is a smooth curve symmetric about the $x$-axis (because of the $y^2$ term). Depending on $A$ and $B$, it has either one or two connected components.

The discriminant $\Delta = -16(4A^3 + 27B^2)$ determines the shape: - $\Delta < 0$: one connected component (a single smooth loop) - $\Delta > 0$: two components (a loop plus a separate egg-shaped piece) - $\Delta = 0$: singular — a cusp ($A = B = 0$) or a self-intersection (not an elliptic curve)

Explore: elliptic curves and point addition

Adjust $A$ and $B$ to change the curve. Click on the curve to place point $P$, then click again for $Q$. The interactive computes $P + Q$ using the geometric construction — draw the secant, find the third intersection, reflect.

The group law

The most remarkable property of elliptic curves is that you can add points. Given two points $P$ and $Q$ on the curve, there’s a natural way to define a third point $P + Q$ that’s also on the curve.

Geometric construction: Draw the line through $P$ and $Q$. A cubic curve intersected by a line gives three points (counting multiplicity). The line hits the curve at $P$, $Q$, and a third point $R$. Reflect $R$ across the $x$-axis to get $P + Q$.

Special cases: - If $P = Q$: use the tangent line at $P$ - The “point at infinity” $\mathcal{O}$ serves as the identity element: $P + \mathcal{O} = P$

This addition law makes the points on an elliptic curve into an abelian group — a set with an addition operation that’s associative, commutative, has an identity, and every element has an inverse. This is extraordinary: a geometric object carrying an algebraic structure.

Points over different fields

The same equation $y^2 = x^3 + Ax + B$ makes sense over different number systems:

Over $\mathbb{R}$ (real numbers): You get the smooth curves described above. Infinitely many points.

Over $\mathbb{Q}$ (rationals): Only some points have rational coordinates. The Mordell-Weil theorem says the rational points form a finitely generated group — they can all be built from a finite set of generators by repeated addition. Finding those generators is hard.

Over $\mathbb{F}_p$ (finite fields): Reduce everything modulo a prime $p$. Now there are only finitely many points. The count $|E(\mathbb{F}_p)|$ varies with $p$, and this variation is the key to the connection with L-functions.

Counting points: the bridge to analysis

For an elliptic curve $E$ over $\mathbb{Q}$ and a prime $p$, define:

$$a_p = p + 1 - |E(\mathbb{F}_p)|$$

This measures how the point count deviates from the “expected” $p + 1$. Hasse’s theorem guarantees $|a_p| \leq 2\sqrt{p}$ — the deviation is bounded.

The sequence $a_2, a_3, a_5, a_7, a_{11}, \ldots$ (one number per prime) encodes the arithmetic of $E$. These coefficients define the L-function of the curve:

$$L(E, s) = \prod_{p} \frac{1}{1 - a_p p^{-s} + p^{1-2s}}$$

This is an Euler product — one factor per prime, but now each factor is quadratic in $p^{-s}$, richer than the linear factors of the Riemann zeta function. The analytic properties of $L(E, s)$ encode deep information about the curve.

Some examples

$$y^2 = x^3 - 2x$$

Integer points: $(0, 0)$, $(-1, \pm 1)$, $(2, \pm 2)$, $(338, \pm 6214)$

The jump from $(2, 2)$ to $(338, 6214)$ is typical — rational and integer points on elliptic curves can be far apart and hard to find.

$$y^2 = x^3 + 17$$

Integer points: $(-2, \pm 3)$, $(-1, \pm 4)$, $(2, \pm 5)$, $(4, \pm 9)$, $(8, \pm 23)$, $(43, \pm 282)$, $(52, \pm 375)$, $(5234, \pm 378661)$

Why elliptic curves matter

Elliptic curves sit at the intersection of geometry, algebra, and number theory:

• Cryptography: Elliptic curve cryptography (ECC) is the backbone of modern secure communication. The group law provides a “one-way function” — easy to compute $nP$ from $n$ and $P$, hard to recover $n$ from $nP$.
• The Birch and Swinnerton-Dyer conjecture: One of the Millennium Prize Problems. It predicts that the rank of the group of rational points (how many independent generators there are) is determined by the order of vanishing of $L(E, s)$ at $s = 1$.
• Fermat’s Last Theorem: The proof hinges on showing that certain elliptic curves (Frey curves, constructed from hypothetical solutions) cannot exist — because they would violate the modularity theorem.