Contents

Idea

Classically, an elliptic curve is a connected Riemann surface (a connected compact 1-dimensional complex manifold) of genus 1. The curious term “elliptic” is a remnant from the 19th century, a back-formation which refers to elliptic functions (generalizing circular functions, i.e., the classical trigonometric functions) and their natural domains as Riemann surfaces.

In more modern frameworks, an elliptic curve over a field $k$ may be defined as a complete irreducible non-singular algebraic curve of genus 1 over $k$, or even as a certain type of algebraic group scheme. Elliptic curves have many remarkable properties, and their deeper arithmetic study is one of the most profound subjects in present-day mathematics.

History

Probably should pass through Riemann and Weierstrass, to explain “elliptic”.

Definition

Over a general ring

Definition

An elliptic curve over a commutative ring $R$ is a group object in the category of schemes over $R$ that is a relative 1-dimensional, , smooth curve, proper curve over $R$.

Remark

This implies that an elliptic curva has genus 1. (by a direct argument concerning the Chern class of the tangent bundle.)

Over the complex numbers

Proposition

An elliptic curve, def. 1, over the complex numbers $\mathbb{C}$ is equivalently

• a Riemann surface $X$ of genus 1 with a fixed point $P \in X$

• a quotient $\mathbb{C}/\Lambda$ where $\Lambda$ is a lattice in $\mathbb{C}$;

• a compact complex Lie group of dimension 1.

• a smooth algebraic curve of degree 3 in $\mathcal{P}$.

Remark

From the second definition it follows that to study the moduli space of elliptic curves it suffices to study the moduli space of lattices in $\mathbb{C}$.

Definition

A framed elliptic curve? is an elliptic curve $(X,P)$ in the sense of the first item in prop. 1, together with an ordered basis $(a,b)$ of $H_1(X, \mathbb{Z})$ with $(a \cdot b) = 1$

A framed lattice in $\mathbb{C}$ is a lattice $\Lambda$ together with an ordered basis $(\lambda_1, \lambda_2)$ of $\Lambda$ such that $Im(\lambda_2/\lambda_1) \gt 0$.

Properties

Group law

Given an elliptic curve over $R$, $E \to Spec R$, we get a formal group $\hat E$ by completing $D$ along its identity section $\sigma_0$

$E \to Spec(R) \stackrel{\sigma_0}{\to} E \,,$

we get a ringed space $(\hat E, \hat O_{E,0})$

Example

If $R$ is a field $k$, then the structure sheaf $\hat O_{E,0} \simeq k[ [z] ]$

then

$\hat O_{E \times E, (0,0)} \simeq \hat O_{E,0} \hat \otimes_k \hat O_{E,0} \simeq k[[x,y]]$
Example

(Jacobi quartics)

$y^2 = 1- 2 \delta x^2 + \epsilon x^4$

defines $E$ over $R = \mathbb{Z}[Y_Z,\epsilon, \delta]$.

The corresponding formal group law is Euler’s formal group law

$f(x,y) = \frac{x\sqrt{1- 2 \delta y^2 + \epsilon y^4} + y \sqrt{1- 2 \delta x^2 + \epsilon x^4}} {1- \epsilon x^2 y^2}$

if $\Delta := \epsilon(\delta^2 - \epsilon)^2 \neq 0$ then this is a non-trivial elliptic curve.

If $\Delta = 0$ then $f(x,y) \simeq G_m, G_a$ (additive or multiplicative formal group law corresponding to integral cohomology and K-theory, respectively).

Relation to elliptic cohomology

Elliptic curves, via their formal group laws, give the name to elliptic cohomology theories.