Elliptic curve group and line bundles

김현학

연구발표

2026 / 06 / 04

목차

  1. Elliptic curve over ℂ and complex torus
  2. Line bundles
  3. Theta functions
  4. Projective embedding

Definition of Elliptic Curve

Definition 1. Lattice in ℂ

A lattice in ℂ is a discrete subgroup Λ = ℤω1 + ℤω2 where ω1, ω2 ∈ ℂ are linearly independent over ℝ.

Definition 2. Homothety

Two lattices Λ and Λ' in ℂ are homothetic if there exists α ∈ ℂ× such that Λ' = αΛ.

Definition 3. Elliptic curve

An elliptic curve is ℂ / Λ for some lattice Λ ⊂ ℂ.

두 Elliptic curve E1 = ℂ/Λ1, E2 = ℂ/Λ2가 isomorphic하다는 것은
Λ1과 Λ2가 homothety라는 것.

Lattice To Complex Torus

Choose τ and generate Λτ

τ = 0.50 + 1.30i

Drag the red point in the upper half-plane.

Quotient by the lattice

Λτ = ℤ + ℤτ, ℂ / Λτ

The torus grid is the complex-plane grid pushed down to the quotient.

Elliptic Curve Isomorphism의 의미

Elliptic curve를 E = ℂ / Λ 로 보면, E는 단순한 group이 아니라 complex manifold 구조를 가진 group이다.

  • Group으로만 보면 두 elliptic curve E1, E2는 둘 다 torus 형태의 abelian group이다.
  • 하지만 complex Lie group으로 보면 E1, E2가 항상 같은 것은 아니다.
  • 차이는 group law 자체가 아니라 ℂ에서 내려오는 complex structure와 holomorphic structure에 있다.

따라서 elliptic curve의 isomorphism은 group operation뿐 아니라 complex manifold 구조까지 보존해야 한다.

Complex Structure를 보존하는 Isomorphism

Definition 1. Complex manifold

A complex manifold of dimension n is locally modeled on open subsets of ℂn, with coordinate changes given by holomorphic functions.

Definition 2. Holomorphic function

For open U ⊂ ℂn, a function f: U → ℂ is holomorphic if it is complex differentiable locally in each variable.

Definition 3. Isomorphism of elliptic curves

For Ei = ℂ / Λi, an isomorphism E1 → E2 is a bijective holomorphic group homomorphism whose inverse is also holomorphic.

In terms of lattices, E1 and E2 are isomorphic as complex elliptic curves iff there exists α ∈ ℂ× such that

Λ2 = αΛ1.

Main Question

Elliptic curve E = ℂ / Λ는 universal cover ℂ 위에서 lattice Λ만큼 반복되는 점들을 하나로 identify해서 얻는다.

z, z + λ, z + λ'    (λ, λ' ∈ Λ)

는 모두 같은 점을 나타낸다.

각 point가 정확히 하나의 representation을 갖는 ambient space X가 있을까?

그 공간 안에서 elliptic curve E는 어떤 모습으로 나타날까?

우리가 시각적으로 떠올리는 ℝ3 안의 donut 모양 torus는 topological model일 뿐, complex structure를 보존하는 holomorphic embedding은 아니다.

따라서 목표는 E를 projective space 안에 holomorphically embedding해서 algebraic geometry의 대상으로 보는 것이다.

E ↪ ℙn

Projective Embedding: Real Example

Projective embedding은 여러 함수값을 projective coordinate로 묶어서 만든다.

p ↦ [f0(p):f1(p):⋯:fn(p)]

Real circle 위의 점은 p(t) = (cos t, sin t)로 쓸 수 있다.

f0(t)=1,   f1(t)=cos t,   f2(t)=sin t t ↦ [1:cos t:sin t] ∈ ℝℙ2

첫 좌표가 항상 1이므로 affine chart에서는 원래 circle (cos t, sin t)를 그대로 보는 것과 같다.

Real Circle Example

Choose a point on S1

t = 0.85

cos t = 0.66

sin t = 0.75

Coordinate functions

Projective coordinate

[1 : 0.66 : 0.75]

Why Line Bundles?

Theorem. Liouville

Every bounded entire function f: ℂ → ℂ is constant.

Since an elliptic curve E = ℂ / Λ is compact, a holomorphic function f: E → ℂ pulls back to a bounded entire function on ℂ.

By Liouville, it must be constant. Thus ordinary holomorphic functions on E are not rich enough to construct a projective embedding.

To get more holomorphic objects, replace functions by sections of line bundles.

s ∈ H0(E,L)

Sections behave like holomorphic functions locally, but they can transform nontrivially between charts.

Definition of Line Bundle

Definition. Line bundle

A line bundle over a space X is a map π: L → X such that each fiber Lx = π-1(x) is a one-dimensional vector space.

Locally, it looks like a product U × ℝ or U × ℂ.

The key point is local triviality: locally product-like, globally possibly twisted.

Example: the Möbius strip is a real line bundle over S1.

base angle t = 0.90

Each point of S1 has a line attached to it, but after going once around the circle, the line returns with a twist.

(m, n) = (0, 0)

e-nλ = 1.00

Loading 3D scene...

Drag the mouse to turn your view. Walk with W/A/S/D or arrow keys. Left and right copies keep the same size, while forward copies shrink and backward copies grow by e-nλ.

Complex Line Bundle on a Torus

(z,r) ∼ (z + 1, r) (z,r) ∼ (z + τ, e-πiτ - 2πizr)

Let E = ℂ / (ℤ + ℤτ). A complex line bundle can be described by identifying fibers over lattice-equivalent points.

The scalar e-πiτ - 2πiz is a transition factor: it tells us how the fiber coordinate r changes when moving by the period τ.

On the unfolded parallelogram, the lower edge 0 → 1 has phase 0, while the upper edge τ → 1+τ is colored by this transition factor.

τ = 0.45 + 1.25i

z = 0.00 + 0.00i

arg factor = 0.00

0.45
1.25

Adjust τ to change the top-edge transition. Drag either picture to move the selected point, or shift-drag to rotate.

Theta Function as a Section

For the line bundle from the previous slide, a typical holomorphic section is given by a theta function.

ϑ(z, τ) = ∑n∈ℤ exp(πin2τ + 2πinz)

This is not an ordinary function on the quotient torus. It transforms in a controlled way under lattice translations, matching the transition rule of the line bundle.

Here we ignore the size of ϑ(z,τ) and color the complex plane by its phase.

τ = 0.45 + 1.25i

partial sum: n = -8,...,8

0.45
1.25

Color represents arg ϑ(z,τ). Adjust τ to change the lattice and theta phase.

Ample Line Bundle

Definition 1. Globally generated

A line bundle L on X is globally generated if for every x ∈ X, there exists s ∈ H0(X,L) such that s(x) ≠ 0.

Definition 2. Very ample line bundle

L is very ample if its global sections define an embedding X ↪ ℙ(H0(X,L)∨).

Definition 3. Ample line bundle

L is ample if some positive tensor power L⊗m is very ample.

For an elliptic curve, a line bundle of degree 3 is very ample and gives an embedding into ℙ2.

Very Ample Line Bundle Example

Degree 3 line bundle

τ = 0.45 + 1.25i

z = 0.00 + 0.00i

sk(z)=Σn∈ℤ exp(3πi(n+k/3)2τ + 6πi(n+k/3)z)

k = 0, 1, 2

Three theta section basis

s0(z), s1(z), s2(z)

Projective coordinate

[s0(z) : s1(z) : s2(z)]

Theta Relations Give a Plane Cubic

Projective coordinates

Let [X0:X1:X2] = [s0(z):s1(z):s2(z)]. The degree 3 theta sections define an embedding φL:E ↪ ℙ2. Since deg L = 3, the image is a plane cubic.

Where the relation comes from

The coordinate functions satisfy one homogeneous cubic relation: Fτ(X0,X1,X2) = 0. Products such as sisjsk are degree 9 theta sections, and theta addition formulas give linear relations among them.

Hesse form

With a symmetric theta basis, the cubic relation can be written as X03 + X13 + X23 - 3λ(τ)X0X1X2 = 0. Here λ(τ) is determined by theta constants.

Cubic Equation from Theta Functions

1. Three theta sections

Start with the degree 3 theta basis sk(z)=Σn∈ℤ exp(3πi(n+k/3)2τ + 6πi(n+k/3)z),   k=0,1,2.

2. Make projective coordinates

These sections define [X0:X1:X2] = [s0(z):s1(z):s2(z)]. So a cubic equation for the image is a relation among cubic monomials in the Xi.

3. Count dimensions

There are 10 cubic monomials: X03, X02X1, …, X23. But they are degree 9 theta sections, and dim H0(E,L⊗3) = 9.

4. One relation remains

Hence the 10 expressions satisfy one relation: Σi+j+k=3 aijk(τ)X0iX1jX2k = 0. A symmetric theta basis means the 3-torsion symmetries act by (X0,X1,X2) ↦ (X1,X2,X0) and (X0,X1,X2) ↦ (X0,ζX1,ζ2X2),   ζ3=1. The relation must be invariant under these actions. The diagonal action leaves only X03, X13, X23, X0X1X2, and the cyclic action makes the three pure-cube coefficients equal: A(τ)(X03 + X13 + X23) + B(τ)X0X1X2 = 0. After rescaling, this becomes X03 + X13 + X23 - 3λ(τ)X0X1X2 = 0.

5. Coefficient from theta constants

The coefficient is determined by theta constants. Evaluating at z=0 gives λ(τ)= {s0(0)3 + s1(0)3 + s2(0)3}/{3s0(0)s1(0)s2(0)}.

From Cubic Equation to Weierstrass Form

1. Start from the Hesse cubic

From theta functions we obtained Cλ: X03 + X13 + X23 - 3λX0X1X2 = 0. Here λ=λ(τ).

2. Choose the origin and tangent

Take a flex point as the group origin: O=[1:-1:0]. Its tangent line is X0 + X1 + λX2 = 0. We want this tangent line to become the line at infinity.

3. Move tangent to infinity

Use new coordinates X=X2,   Y=X0,   Z=X0+X1+λX2. Then O=[0:1:0],    {Z=0} is the tangent line at O. Equivalently, X0=Y,   X2=X,   X1=Z-Y-λX.

4. Substitute into the cubic

Substitution gives a cubic in X,Y,Z: Y3 + (Z-Y-λX)3 + X3 - 3λYX(Z-Y-λX)=0. Since O=[0:1:0] is at infinity and Z=0 is tangent there, it can be rearranged as Y2Z + a1XYZ + a3YZ2 = X3 + a2X2Z + a4XZ2 + a6Z3.

5. Complete square and cube

On Z=1, write y2 + a1xy + a3y = x3 + a2x2 + a4x + a6. Complete the square: y′ = y + (a1x+a3)/2. Then shift x to remove the x2 term: x′ = x + (a2+a12/4)/3. This gives (y′)2=(x′)3+A(τ)x′+B(τ). After homogenizing and rescaling Y, Y2Z = 4X3 - g2(τ)XZ2 - g3(τ)Z3. The coefficients come from λ(τ), hence from theta constants.

Addition After Projective Embedding

1. Addition before embedding

On the complex torus, the group operation is just z1 + z2   mod Λ. The embedding sends a point to theta values: φ(z)=[s0(z):s1(z):s2(z)].

2. A line becomes a theta section

A line in ℙ2 has equation aX0+bX1+cX2=0. Pulling it back by φ gives F(z)=as0(z)+bs1(z)+cs2(z). So a projective line is the same as one linear combination of the three theta sections.

3. Passing through two points

Let P=φ(z1) and Q=φ(z2). If the line passes through P,Q, then F(z1)=0,    F(z2)=0. Since F is a section of the degree 3 line bundle, it has one more zero: z1,   z2,   z3.

4. Theta zeros remember addition

Use an auxiliary degree 1 theta factor ϑ, not one of the degree 3 basis sections. It satisfies ϑ(z+τ) = -e-πiτ-2πizϑ(z). If the zeros are z1,z2,z3, then locally F(z) ∼ ϑ(z-z1)ϑ(z-z2)ϑ(z-z3). Under z↦z+τ, this product picks up e-3πiτ-6πiz · e2πi(z1+z2+z3). But the degree 3 basis sections satisfy sk(z+τ)=e-3πiτ-6πizsk(z). Hence every F=a0s0+a1s1+a2s2 has this same factor, so the extra term must be fixed. With the origin normalized, this gives z1+z2+z3=0   in   ℂ/Λ, hence z3=-(z1+z2).

5. Chord rule appears

The point φ(z3) is exactly the third intersection of the line with the cubic. Therefore φ(z1+z2) = inverse of the third intersection. In Weierstrass coordinates, inverse is the visible reflection (x,y) ↦ (x,-y).

Zero Sum of a Theta Section

z1, z2, z3

z1+z2+z3 = 0 mod Lambda

Choose a linear combination of the three degree 3 theta basis sections. Changing the coefficients changes the projective line.

F(z)=a0s0(z)+a1s1(z)+a2s2(z)

ak=rkeiπpk

a0 1.00 0.00
a1 0.82 0.34
a2 0.72 -0.26

The canvas colors the complex plane by arg F(z). Red circles are numerical zeros of this section. The blue diamond marks their raw sum in ℂ/Λ.

Conclusion

  • Complex elliptic curve는 line bundle의 section들을 통해 projective embedding할 수 있다.
    • Embedding된 object는 Weierstrass equation으로 표현된다.
    • Projective space에서 irreducible equation으로 표현되는 점들의 집합을 projective variety라고 부른다.
    • Projective variety 중 group structure를 가지는 것이 Abelian variety이다.
  • Projective embedding 위의 group operation은 section의 zero들이 만족하는 관계로 나타난다.
    • Degree 3 line bundle의 section은 세 개의 zero를 갖는다.
    • 앞서 본 예시에서 세 zero의 합은 ℂ/Λ 안에서 고정된다.

Next Questions

  • Complex field가 아닌 다른 field, 예를 들어 finite field 위에서도 Abelian variety가 있는가?
  • Abelian variety가 주어졌을 때 line bundle은 어떻게 정의되는가? (hint: sheaf)
  • Abelian variety 위의 group operation은 line bundle과 항상 관련되는가? (pairing and polarization)