Why Elliptic Curves have so many nice properties

Question

As the definition referred from Silverman's book:

An elliptic curve is a pair $(E,O)$, where $E$ is a nonsingular curve of genus one and $O\in E$. (We generally denote the elliptic curve by $E$, the point $O$ being understand.) The elliptic curve $E$ is defined over $K$, written $E/K$, if $E$ is defined over $K$ as a curve and $O\in E(K)$.

We use Riemann-Roch theorem to prove $E$ has a Weierstrass form(This need us go into the deep part of algebraic geometry). There are some other nice properties, like the dual isogeny, elliptic have CM(there are two simple forms of $End(E)\neq \mathbb{Z}$)...

I wonder why it is so beautiful and what may be the most important part that influenced elliptic curve has this properties.

Thank you for sharing your mind.

Answer 1

As Jakob notes in the comments, it's difficult to say "why" to such a "philosophical" question. However, the remarks below may provide a starting point. Briefly, a smooth plane curve admits a "natural" group structure if and only if the curve has degree three.

In more detail, a smooth plane curve $E$ of genus one is a cubic by the genus-degree formula, and $E$ turns out to be isomorphic to its own Jacobian variety under identification of a point $p$ in $E$ with the divisor $p - O$ in $J(E)$, with $O$ denoting the identity element of $E$.

To express these facts more geometrically:

• A smooth plane cubic (over an algebraically closed field) intersects each line exactly three times (counting multiplicity).

• If $p_{1}$, $p_{2}$, $p_{3}$ are $q_{1}$, $q_{2}$, $q_{3}$ are triples of collinear points of $E$, then $\sum p_{i}$ and $\sum q_{i}$ are linearly equivalent: If $P$ and $Q$ are linear forms on $\mathbf{P}^{2}$ (a.k.a., sections of the hyperplane bundle $\mathcal{O}_{\mathbf{P}^{2}}(1)$) such that $P(p_{i}) = 0$ and $Q(q_{i}) = 0$ for each $i$, then $P/Q$ defines a rational function on $E$ such that $(P/Q) = \sum p_{i} - \sum q_{i}$.

• A divisor $p_{1} + p_{2} + p_{3} - 3O$ is principal if and only if the $p_{i}$ are collinear. (The "only if" direction amounts to the claim that $E$ and $J(E)$ are isomorphic, and is not as obvious as the "if" direction above. Over the complex numbers, the "only if" direction amounts to an addition formula for the Weierstrass $\wp$-function.)

• Consequently the group law for $E$ has a (beautiful and well-known) geometric interpretation.

  "Numerologically", it's significant that each line intersects $E$ in three points because addition is a binary operation: A condition $p + q = r$, i.e., $p + q - r = O$, involves triples of points.