nLab
elliptic curve

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

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→SpecR, we get a formal group E^ by completing D along its identity section σ 0

E→Spec(R)→σ 0E,E \to Spec(R) \stackrel{\sigma_0}{\to} E \,,

we get a ringed space (E^,O^ E,0)

example if R is a field k, then the structure sheaf O^ E,0≃k[[z]]

then

O^ E×E,(0,0)≃O^ E,0⊗^ kO^ E,0≃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=1−2δx 2+ϵx 4y^2 = 1- 2 \delta x^2 + \epsilon x^4

defines E over R=ℤ[Y Z,ϵ,δ].

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

f(x,y)=x1−2δy 2+ϵy 4+y1−2δx 2+ϵx 41−ϵx 2y 2f(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 Δ:=ϵ(δ 2−ϵ) 2≠0 then this is a non-trivial elliptic curve.

If Δ=0 then f(x,y)≃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

References

an introduction to elliptic curves is at

Revised on September 19, 2012 01:31:59 by Urs Schreiber (82.169.65.155)
Edit | Back in time (4 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: tmf, 2009 September changes, algebraic group, A Survey of Elliptic Cohomology, elliptic cohomology, elliptic curve, A Survey of Elliptic Cohomology - formal groups and cohomology, A Survey of Elliptic Cohomology - E-infinity rings and derived schemes, Gromov-Witten invariants, moduli space, basic ideas of moduli stacks of curves and Gromov-Witten theory, A Survey of Elliptic Cohomology - elliptic curves, A Survey of Elliptic Cohomology - A-equivariant cohomology, motive, A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves, derived elliptic curve, A Survey of Elliptic Cohomology - towards a proof, period, A Survey of Elliptic Cohomology - descent ss and coefficients, Gaudin integrable model, heterotic string theory, curve, height of a variety, abelian variety, F-theory, algebraic curve, topological automorphic form