nLab elliptic curve

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 $k$ or indeed over any commutative ring may be defined as a complete irreducible non-singular algebraic curve of arithmetic 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.

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 $R$ (hence in arithmetic geometry) are the well-behaved 1-dimensional group objects parameterized over the space $Spec(R)$ (the prime spectrum of $R$). (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

• Conceptual definition

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

• Coordinatized as solutions to cubic equations

Conceptual definition

Definition

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

Remark

This implies that an elliptic curve has arithmetic genus $1$ (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 $j$-invariant

Definition

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

$$y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$$

for coefficients $a_1, a_2, a_3, a_4, a_6$. This is called the Weierstrass equation.

Remark

Much of the literature on elliptic curves considers def. for the case that $R$ 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 $R$. Non-singularity is embodied in coordinates as follows.

Definition

Let

$$\begin{aligned} b_2 & \coloneqq a_1^2 + 4 a_2 \\ b_4 &\coloneqq a_1 a_3 + 2a_4 \\ b_6 & \coloneqq a_3^2 + 4 a_6 \\ b_8 & \coloneqq a_1^2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 + 4 a_2 a_6 - a_4^2 \end{aligned}$$

and in terms of these

$$\begin{aligned} c_4 & \coloneqq b_2^2 - 24 b_4 \\ c_6 & \coloneqq - b_2^3 + 36 b_2 b_4 - 216 b_6 \\ \Delta &\coloneqq - b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6 \end{aligned} \,.$$

Here $\Delta$ is called the discriminant.

Finally let

$$j \coloneqq c_4^3/\Delta \,,$$

called the j-invariant.

Remark

In def. the discriminant satisfies the relation

$$1728 \Delta = c_4^3 - c_6^2 \,.$$

Remark

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

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. as fundamental then the following is a proposition.

Definition/Proposition

A solution to the Weierstrass cubic, def. , with modular invariants $c_4$, $c_6$ and discriminant $\Delta$ according to def. is

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

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

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

Remark

Adding the nodal curve to the moduli stack of elliptic curves yields its compactification, and the formal neighbourhood of the nodal curve in that compactification is known as the Tate curve.

Localization at 2 and 3

Proposition

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

$$y^2 = 4 x^3 + b_2 x^2 + 2 b_4 x + b_6$$

with the coefficients identified as in def. .

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

$$y^2 = x^3 - 27 c_4 x - 54 c_6 \,.$$

Over the complex numbers

In terms of algebraic geometry

Remark

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

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

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. , over the complex numbers $\mathbb{C}$ is equivalently

• a Riemann surface $X$ of genus 1 with a base point $P \in X$ (e.g.Hain 08, def. 1.1)

• 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. , together with an ordered basis $(a,b)$ of $H_1(X, \mathbb{Z})$ with $(a \cdot b) = 1$

For $n$ 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 $\mathbb{Z}/n\mathbb{Z}$.

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

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 $p$-adic integers and $p$-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 $\Sigma$ 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

$$(a_1, a_2)\in H_1(\Sigma, \mathbb{Z})$$

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

If here one replaces the integers by a cyclic group $\mathbb{Z}/n\mathbb{Z}$ then one obtains what is called a level-n structure on an elliptic curve. Level-$n$ 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) provides a finite covering of the full moduli stack of elliptic curves.

Group structure

A vital reason for the importance of elliptic curves in number theory, arithmetic geometry, and elsewhere is their group structure. This group structure is present by fiat in Definition , and can be constructed given a Weierstrass equation.

The torsion elements of this group structure when the elliptic curve is defined over a number field have good Galois-theoretic properties. See torsion points of an elliptic curve for more.

Constructions

Formal group law

Given an elliptic curve over $R$, $E \to Spec R$, we get a formal group $\hat E$ by completing $E$ 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

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

if $\Delta \coloneqq \epsilon(\delta^2 - \epsilon)^2 \neq 0$ 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 ordinary cohomology and topological K-theory KU, respectively).

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

• Tate curve

• Szpiro's conjecture

• cubic curve

• moduli stack of elliptic curves

  • modular form, Jacobi form

  • topological modular form

• modularity theorem

• elliptic fibration

• elliptic cohomology, tmf

References

Classical accounts of the general case include

• Pierre Deligne, Michael Rapoport, Les Schémas de Modules de Courbes Elliptiques In: Deligne P., Kuijk W. (eds) Modular Functions of One Variable II, Lecture Notes in Mathematics, vol 349. Springer (1973) (doi:10.1007/978-3-540-37855-6_4)

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

• Nicholas Katz, Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, 1985 (jstor:j.ctt1b9s05p)

Introductory lecture notes for elliptic curves over the complex numbers include

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

• Amnon Neeman, section 1.2 of Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, 2007 (publisher)

and for the general case:

• A quick introduction to elliptic curves (pdf)

• R. Sujatha, Elliptic Curves & Number Theory (pdf)

• Andrew Sutherland, Elliptic curves, 2021 (web)

See also

• Matthew Ando, Michael Hopkins, Neil Strickland, appendix B of: Elliptic spectra, the Witten genus, and the theorem of the cube, Inventiones Mathematicae, 146:595–687, 2001, (pdf, doi:10.1007/s002220100175)

  (review of general elliptic curves in the context of elliptic genera, elliptic spectra and the string orientation of tmf)

Discussion of the formal group law of elliptic curves:

• Antonia W. Bluher, Formal groups, elliptic curves, and some theorems of Couveignes, in: J.P. Buhler (eds.) Algorithmic Number Theory ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 (arXiv:math/9708215, doi:10.1007/BFb0054887)

• Stefan Friedl, An elementary proof of the group law for elliptic curves (arXiv:1710.00214)

For more along these lines see also at arithmetic geometry.

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

• Sagemath Elliptic curves over the rational numbers

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. (pdf)

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

See also

• A Survey of Elliptic Cohomology - elliptic curves