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elliptic curve

Contents

Idea

Classically in complex geometry, an elliptic curve is a connected Riemann surface (a connected compact 1-dimensional complex manifold) of genus 1, hence it is a torus equipped with the structure of a complex manifold, or equivalently with conformal structure.

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 and in the generality of algebraic geometry, an elliptic curve over a field kk or indeed over any commutative ring may be defined as a complete irreducible non-singular algebraic curve of arithmetic genus-1 over kk, 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.

The moduli stack of elliptic curves equipped with its canonical map to the moduli stack of formal group laws plays a central role in chromatic homotopy theory at chromatic level 2, where it serves to parameterize elliptic cohomology theories.

Elliptic curves over the complex numbers are also interpreted as those worldsheets in string theory whose correlators are the superstring’s partition function, which is the Witten genus. Via the string orientation of tmf this connects to to the role of elliptic curves in elliptic cohomology theory.

Definition

Over a general ring

Elliptic curves over a general commutative ring RR are the well-behaved 1-dimensional group objects parameterized over the space Spec(R)Spec(R) (the prime spectrum of RR). (Notice the count of dimension: over the complex numbers a torus is complex 1-dimensional and in this sense one is looking at 1-dimensional group schemes here.) This we discuss below in

More concretely there are various explicit and standard coordinazations of elliptic curves as affine schemes, hence as solution spaces to polynomial equations. This we discuss below in

Conceptual definition

Definition

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

Remark

This implies that an elliptic curve has arithmetic genus 11 (by a direct argument concerning the Chern class of the tangent bundle.)

Definition

An elliptic curve over a field of positive characteristic whose formal group law has height equal to 2 is called a supersingular elliptic curve. Otherwise the height equals 1 and the elliptic curve is called ordinary.

Coordinatized as solutions to cubic Weierstrass equations

Elliptic curves are examples of solutions to Diophantine equations? of degree 3. We start by giving the equation valued over general rings, which is fairly complicated compared to the special case that it reduces to in the classical case over the complex numbers. The more elements in the ground ring are invertible, the more the equation may be simplified.

(See Silverman 09, III.1 for a textbook account and for instance (QuickIntro) for a quick survey.)

Weierstrass cubic, discriminant and jj-invariant
Definition

Let RR be a commutative ring, then Zariski locally over Spec(R)Spec(R) a cubic curve is a solution in the corresponding projective space to an equation of the form

y 2+a 1xy+a 3y=x 3+a 2x 2+a 4x+a 6 y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6

for coefficients a 1,a 2,a 3,a 4,a 6a_1, a_2, a_3, a_4, a_6. This is called the Weierstrass equation.

Remark

Much of the literature on elliptic curves considers def. 3 for the case that RR is an algebraically closed field, in which case there is no need to pass to a cover. But for the true global discussion necessary for the moduli stack of elliptic curves one needs the full generality.

The non-singular such solutions are the elliptic curves over RR. Non-singularity is embodied in coordinates as follows.

Definition

Let

b 2 a 1 2+4a 2 b 4 a 1a 3+2a 4 b 6 a 3 2+4a 6 b 8 a 1 2a 6a 1a 3a 4+a 2a 3 2+4a 2a 6a 4 2 \begin{aligned} b_2 & \coloneqq a_1^2 + 4 a_2 \\ b_4 &\coloneqq a_1 a_3 + 2a_4 \\ b_6 & \coloneqq a_3^2 + 4 a_6 \\ b_8 & \coloneqq a_1^2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 + 4 a_2 a_6 - a_4^2 \end{aligned}

and in terms of these

c 4 b 2 224b 4 c 6 b 2 3+36b 2b 4216b 6 Δ b 2 2b 88b 4 327g 6 2+9b 2b 4b 6. \begin{aligned} c_4 & \coloneqq b_2^2 - 24 b_4 \\ c_6 & \coloneqq - b_2^3 + 36 b_2 b_4 - 216 b_6 \\ \Delta &\coloneqq - b_2^2 b_8 - 8 b_4^3 - 27 g_6^2 + 9 b_2 b_4 b_6 \end{aligned} \,.

Here Δ\Delta is called the discriminant.

Finally let

jc 4 3/Δ, j \coloneqq c_4^3/\Delta \,,

called the j-invariant.

Remark

In def. 4 the discriminant satisfies the relation

1728Δ=c 4 3c 6 2. 1728 \Delta = c_4^3 - c_6^2 \,.
Remark

Over R=R = \mathbb{C} the complex numbers the quantities c 4c_4 and c 6c_6 in def. 4 are proportional to the modular forms called the Eisenstein series (see there) G 4G_4 and G 6G_6.

Elliptic curves, Nodal curves, Cuspidal curves

The following is a definition if one takes the coordinate-description as fundamental. If one takes the more abstract characterization of def. 1 as fundamental then the following is a proposition.

Definition/Proposition

A solution to the Weierstrass cubic, def. 3, with modular invariants c 4c_4, c 6c_6 and discriminant Δ\Delta according to def. 4 is

  1. an elliptic curve iff Δ0\Delta \neq 0;

  2. a nodal curve or cubic curve with nodal singularity iff Δ=0\Delta = 0 and c 40c_4 \neq 0;

  3. a cuspidal curve or cubic curve with cusp singularity iff Δ=0\Delta = 0 and c 4=0c_4 = 0 (which by remark 3 is equivalent to c 4=0c_4 = 0 and c 6=0c_6 = 0)

Localization at 2 and 3
Proposition

If 22 is invertible in RR (is a unit ), and hence generally over the localization R[12]R[\frac{1}{2}] of RR at 2, the general Weierstrass equation, def. 3, is equivalent, to the equation

y 2=4x 3+b 2x 2+2b 4x+b 6 y^2 = 4 x^3 + b_2 x^2 + 2 b_4 x + b_6

with the coefficients identified as in def. 4.

If moreover 33 is also invertible in RR, hence generally over R[12,13]R[\frac{1}{2}, \frac{1}{3}] then this equation is equivalent to just

y 2=x 327c 4x54c 6. y^2 = x^3 - 27 c_4 x - 54 c_6 \,.

Over the complex numbers

In terms of algebraic geometry

Remark

If the ring R=R= \mathbb{C} is the complex numbers, then complex tori are indeed the solutions to the Weierstrass equation as in prop. 1, parameterized by a torus z/Λz \in \mathbb{C}/\Lambda (as discussed in the section in terms of complex geometry) via the Weierstrass elliptic function \wp as (x=(z),y=(z),)(x = \wp(z), y = \wp'(z), ) in the form

(z) 2+4(z) 3g 2p(z)g 3. \wp'(z)^2 + 4 \wp(z)^3 - g_2 p(z) - g_3 \,.

See e.g. (Hain 08, section 5) on how complex elliptic curves are expressed in this algebraic geometric fashion.

In terms of complex geometry

Proposition

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

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)(X,P) in the sense of the first item in prop. 2, together with an ordered basis (a,b)(a,b) of H 1(X,)H_1(X, \mathbb{Z}) with (ab)=1(a \cdot b) = 1

For nn a natural number, a level n-structure on an elliptic curve over the complex numbers is similar data but with coefficients only in the cyclic group /n\mathbb{Z}/n\mathbb{Z}.

A framed lattice in \mathbb{C} is a lattice Λ\Lambda together with an ordered basis (λ 1,λ 2)(\lambda_1, \lambda_2) of Λ\Lambda such that Im(λ 2/λ 1)>0Im(\lambda_2/\lambda_1) \gt 0.

Hence a framed elliptic curve is the quotient of the complex plane by a lattice together with the information on how this quotient was obtained. This is useful for describing the moduli stack of elliptic curves over the complex numbers.

Over the rational numbers

Over the rational numbers: Sagemath: Elliptic curves over the rational numbers

Over the pp-adic integers and pp-adic numbers

Over the p-adic integers, see (Conrad 07).

Over the p-adic numbers. see (Winter 11).

Level structures

In the case over the complex numbers an elliptic curve Σ\Sigma is equivalently the quotient of the complex plane by a framed lattice. If here one remembers the structure given by that framed lattice, this means equivalently to remember an ordered basis

(a 1,a 2)H 1(Σ,) (a_1, a_2)\in H_1(\Sigma, \mathbb{Z})

of the ordinary homology group of Σ\Sigma with coefficients in the integers.

If here one replaces the integers by a cyclic group /n\mathbb{Z}/n\mathbb{Z} then one obtains what is called a level-n structure on an elliptic curve. Level-nn structures on elliptic curves may also be defined over general rings.

These structures are useful in that the moduli stack of elliptic curves with level-n structure (a modular curve in the case over the complex numbers) orivides a finite covering of the full moduli stack of elliptic curves.

Properties

Group law

Given an elliptic curve over RR, ESpecRE \to Spec R, we get a formal group E^\hat E by completing DD along its identity section σ 0\sigma_0

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

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

Example

If RR is a field kk, then the structure sheaf O^ E,0k[[z]]\hat O_{E,0} \simeq k[ [z] ]

then

O^ E×E,(0,0)O^ E,0^ kO^ E,0k[[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=12δx 2+ϵx 4 y^2 = 1- 2 \delta x^2 + \epsilon x^4

defines EE over R=[Y Z,ϵ,δ]R = \mathbb{Z}[Y_Z,\epsilon, \delta].

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

f(x,y)=x12δy 2+ϵy 4+y12δx 2+ϵx 41ϵx 2y 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 Δ:=ϵ(δ 2ϵ) 20\Delta := \epsilon(\delta^2 - \epsilon)^2 \neq 0 then this is a non-trivial elliptic curve.

If Δ=0\Delta = 0 then f(x,y)G m,G af(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

References

Classical accounts of the general case include

  • Pierre Deligne, Courbes Elliptiques: Formulaire (d’apres J. Tate) (web)

  • Nicholas M. Katz, Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR MR772569 (86i:11024)

Introductory lecture notes for elliptic curves over the complex numbers include

  • Richard Hain, Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)

and for the general case

  • A quick introduction to elliptic curves (pdf)

In the context of elliptic fibrations:

  • Rick Miranda, The basic theory of elliptic surfaces, lecture notes 1988 (pdf)

A general textbook account is

  • Joseph Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)

Discussion over the rational numbers includes

Discussion of elliptic curves over the p-adic numbers includes

  • Brian Conrad, Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu 6 (2007), no. 2, 209-278.

  • Rosa Winter, Elliptic curves over p\mathbb{Q}_p, 2011 (pdf)

See also

Revised on April 11, 2014 02:17:08 by Urs Schreiber (185.37.147.12)