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

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 $R$.

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

Group law

Given an elliptic curve over $R$, $E\to \mathrm{Spec}R$, we get a formal group $\stackrel{^}{E}$ by completing $D$ along its identity section ${\sigma }_{0}$

$E\to \mathrm{Spec}\left(R\right)\stackrel{{\sigma }_{0}}{\to }E\phantom{\rule{thinmathspace}{0ex}},$E \to Spec(R) \stackrel{\sigma_0}{\to} E \,,

we get a ringed space $\left(\stackrel{^}{E},{\stackrel{^}{O}}_{E,0}\right)$

example if $R$ is a field $k$, then the structure sheaf ${\stackrel{^}{O}}_{E,0}\simeq k\left[\left[z\right]\right]$

then

${\stackrel{^}{O}}_{E×E,\left(0,0\right)}\simeq {\stackrel{^}{O}}_{E,0}{\stackrel{^}{\otimes }}_{k}{\stackrel{^}{O}}_{E,0}\simeq k\left[\left[x,y\right]\right]$\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)

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

defines $E$ over $R=ℤ\left[{Y}_{Z},ϵ,\delta \right]$.

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

$f\left(x,y\right)=\frac{x\sqrt{1-2\delta {y}^{2}+ϵ{y}^{4}}+y\sqrt{1-2\delta {x}^{2}+ϵ{x}^{4}}}{1-ϵ{x}^{2}{y}^{2}}$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 $\Delta :=ϵ\left({\delta }^{2}-ϵ{\right)}^{2}\ne 0$ then this is a non-trivial elliptic curve.

If $\Delta =0$ then $f\left(x,y\right)\simeq {G}_{m},{G}_{a}$ (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.