nLab
elliptic curve

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 $\hat{E}$ by completing $D$ along its identity section ${\sigma }_0$

$$E\to \mathrm{Spec}(R)\stackrel{{\sigma }_0}{\to }E,$$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

$${\hat{O}}_{E\times E,(0,0)}\simeq {\hat{O}}_{E,0}{\hat{\otimes }}_k{\hat{O}}_{E,0}\simeq k[[x,y]]$$\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)

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

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

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

$$f(x,y)=\frac{x\sqrt{1-2\delta y^2+ϵy^4}+y\sqrt{1-2\delta x^2+ϵx^4}}{1-ϵx^2{y}^2}$$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 $\Delta :=ϵ({\delta }^2-ϵ{)}^2\ne 0$ 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.

See also

• A Survey of Elliptic Cohomology - formal groups and cohomology

Related concepts

• abelian variety

• elliptic cohomology

• Szpiro's conjecture

References

an introduction to elliptic curves is at

• A Survey of Elliptic Cohomology - elliptic curves