Computer simulation of the modular fibration Ihechukwu Chinyere (
[email protected]) African Institute for Mathematical Sciences (AIMS) Supervised by: Dr. Bruce Bartlett Stellenbosch University, South Africa
17 May 2012 Submitted in partial fulfillment of a postgraduate diploma at AIMS
Abstract Lattices are very important concept in number theory as many problems encountered in the field translate to questions regarding the orbits in the space of lattices. This work aims at identifying the space of all lattices in the plane (up to rescaling) with the 3-dimensional sphere, using Eisenstein series. By adapting the code of Johnson [Nil12] for the Hopf fibration, we will give a computer exploration of this idea by using the j-invariant of an elliptic curve as a map from the 3-sphere to the 2-sphere, j : S 3 −→ S 2 3 such that (z, w) 7−→ 20z 320z . The fibers of this map (representing a class of homothetic lattices) are −49w2 3 generically trefoil knots in S , and j can be thought of as the “modular fibration” over the 2-sphere, in analogy to the Hopf fibration, h : S 3 −→ S 2 such that (z, w) 7−→ wz . The difference is that the fibers of the Hopf fibration are circles (unknots), while those of the j-invariant are trefoils. Keywords. Space of lattices, Hopf fibration, modular forms, modular flow.
Declaration I, the undersigned, hereby declare that the work contained in this essay is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly.
Ihechukwu Chinyere, 17 May 2012 i
Contents Abstract
i
1 Introduction
1
2 Fibration
2
2.1
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Hopf map and Geometric visualization . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3 Modular forms and functions
9
3.1
Properties of modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2
Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3
Non-zero lattices in the plane and the 3-sphere . . . . . . . . . . . . . . . . . . . . . . 18
3.4
Special Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Computer simulation
23
4.1
Parametrization of fibres of the j-map . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2
Visualization of the fibers of the j-map . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3
Implementation in Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4
Transformation of the j-map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Conclusion
31
References
34
ii
1. Introduction The discovery of the Hopf fibration by Heinz Hopf was a landmark achievement in the development of algebraic topology. This map has many guises in Physics and Mathematics with some important applications in quantum information theory, rigid body mechanics and magnetic mosopoles. Mathematicians were particularly glad at this discovery as it was enough to clear some ignorance that was in the air. Since the time of the great Poincar´e, the fundamental group was known as well as degree of maps as defined by Brouwer and its homotopy invariance proved. However, it remained unclear whether the map S m into S n with 1 < n < m is necessarily nullhomotopic [Whi83]. The Hopf map is a continuous surjective map from S 3 to S 2 [Hop31]. The actual Hopf map is the cannonical projection π : C2 −→ CP1 such that each vector (z, w) is sent to its equivalence class [z : w] in the projective space. This is interpreted as a map from S 3 to S 2 by identifying S 3 as a subset of C2 of unit norm, and CP1 with S 2 in two steps: identifying CP1 with C ∪ {∞} and then C ∪ {∞} with S 2 . Thus we can say that S 3 is made up of fibres, which in this case are geometric circles, each corresponding to each distinct point on S 2 . This is an example of fibre bundle which we represent h structurally as S 1 ,→ S 3 −→ S 2 , which is a way of showing that S 1 , which we call the fibre is embedded in S 3 , the total space and the Hopf map h : S 3 −→ S 2 is a projection of S 3 onto S 2 ,the base space. This whole construction called the Hopf fibration behaves locally like a product space just like any other fibre bundle. However, it is not globally a product space. It turns out that this property of the Hopf fibration has some useful implications: it tells us that in general, not all homotopy groups are trivial. Chapter 2 contains an overview of what fibration is all about, at least as much as will be applied in the remaining part of this work. It is subdivided into sections which are concerned with explaining the concept of fibre bundle with few example and explaining the Hopf fibration which involves identification of spaces and visualizing the 3-sphere. The aim is to establish a background on which to visualize the fibres of the j-map described in the next chapter. Chapter 3 is devoted towards explaining some basic concepts of modular forms and their relationship with lattices and the Weierstrass ℘ function. One of the most useful tools for this essay will be the first two Eisenstein series which we will show characterizes the lattice. We describe what a lattice is with examples of the special ones namely: square, hexagonal and degenerate lattices. These three lattices will serve as a check to ensure the result of the simulation in the succeeding chapter is correct. Also, another important modular function called the j-map which gives a bijection between the space of lattices up to rescaling and the complex numbers will be introduced here. Chapter 4 will discuss the computer simulation in details. This will begin with explaining the parametrization of the fibres of the j-map, highlighting the modification in the Sage code by Niles Johnson. The analysis of the result in terms of the description of the these fibres is also included here. Chapter 5 is which is the concluding Chapter in this work is just a summary of what was done in the previous chapters.
1
2. Fibration This chapter will introduce the basic concepts that make up what is known as a fibration. Since our interest is the special type of fibration called the Hopf fibration, we will channel our efforts towards understanding this idea. As a starting point, we will define the basic concepts and give some examples. As we shall see that fibration is just a generalization of the idea of fibre bundle. 2.0.1 Definition. [Cro05] Let E, B and F be topological spaces and π : E −→ B be a continuous surjective map. The collection (E, B, F, π) is called a fibre bundle if for every point x ∈ B, there is an open set U ⊂ B containing x with a homeomorphism ψu : U × F −→ π −1 (U ) such that the composition π ◦ ψu : U × F −→ U is the projection (u, f ) 7−→ u. A fibre bundle makes clear the idea of a topological space being parametrized by another topological space. It has the fundamental property of being locally trivial. This simply means that for every point b ∈ B, there exist an open neighborhood U ⊂ B of b such that the diagram below commutes.
π −1(U ) ψu
U ×F
π
proj.
U
In such case, we can refer to a fibre bundle as a differentiable manifold which is locally trivial. 2.0.2 Definition. Given a map g from a space E onto a space B, we can partition the domain into disjoint sets Gb = g −1 (b), b ∈ B. We say that g is a fibration if all the sets Gb are homeomorphic to one another. In this case Gb is called the fiber over b; the space B is called the base space of the fibration, E is the total space , and g is the projection. Thus a fibration differs from a fibre bundle in the sense that it does not necessary have to be locally trivial. In a situation where these fibers are homeomorphic to a space F , we say that F is model fiber or simply fiber [Shv94]. 2.0.3 Example. The natural map of a M¨ obius strip [Ale12] to a circle .Think of the circle that bisects the Mobius strip along the middle and the projection onto it. In this case, the base space is the circle and the fibres are homeomorphic to an interval (see figure 2.1). 2.0.4 Example. If a hollow cylinder is projected orthogonally onto it’s base, we have a fibration. In this case, the base space is the circle and the fibres are homeomorphic to an interval. In general, every map (projection) from a product space to one on the spaces gives rise to a trivial fibre bundle. Also, every covering map defines a trivial fiber bundle.
2
Section 2.1. Identification
Page 3
Mobuis
I = [0, 1]
S1
base space
Total space
fibre
Figure 2.1: M¨obius strip
2.1
Identification
Since we are working in 4-dimensional space, the question that arises is how can one visualize a 4dimensional object. To do this, we will restrict our study to S 3 which is naturally embedded in C2 (≈ R4 ). Having done that, we can now use stereographic projection on S 3 . The stereographic projection has the conformality property, to project onto 2-sphere. We also identify spaces which are homeomorphic. 2.1.1 Definition. The 1-dimensional complex projective space CP1 is the set of all complex lines on C2 passing through the origin. CP1 = {One-dimensional subspaces of C2 } modulo equivalence relation. It is denoted as: CP1 = {[z : w]},
where [z : w] = {[λz, λw]|λ ∈ C − {0}, w and z do not vanish simultaneously}. w-axis b
z-axis
a
Figure 2.2: Complex projective plane 2.1.2 Definition. The 3-sphere of unit radius denoted by S 3 is the subset of C2 defined by: S 3 ={(z, w) ∈ C2 | |z|2 + |w|2 = 1}. Similarly, the ordinary sphere is defined as: S 2 ={(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1}. In Figure (2.2), the points a, b ∈ C2 (≈ R4 ) actually live on the same circle. This we can see via the map that takes (z, w) to (eiθ z, eiθ w). The complex lines intersect the 3-sphere in a circle. Thus for
Section 2.1. Identification
Page 4
each of these lines, we have a circle. Therefore we can identify C2 with CP1 by the map π1 that sends (z, w) to [z : w]. This is the Hopf fibration. w Next, we identify CP1 with C ∪ {∞} by the map π2 that sends [z : w] to . Again this is a bijection z by sending [0, w] to ∞. We identify S 2 with C ∪ {∞} (the Riemann sphere) via stereographic projection. To determine the corresponding ρ, consider the point N (in this case the north pole) of the set: S 2 = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1} and a projection from that point onto the equatorial plane (see figure 2.3).
Figure 2.3: Stereographic projection. 2.1.3 Claim. The map ρ : S 2 − (0, 0, 1) −→ C such that P 7−→ Q is given by: ρ(x, y, z) =
x + iy 1−z
and ρ−1 (x + iy) =
x2
2 (x + iy, x2 + y 2 − 1). + y2 + 1
Proof. For the first case, let P = (x0 , y0 , z0 ) be a point on S 2 . The line v(t) joining N and P has the parametrization : v(t) = (0, 0, 1) + t(x0 , y0 , z0 − 1), = (tx0 , ty0 , t(z0 − 1) + 1).
The line meets the plane C at the point where t =
1 1−z0 .
In which case, we have:
Q = x + iy =
1 (x0 + iy0 ) 1 − z0
ρ(x0 , y0 , z0 ) =
1 (x0 + iy0 ). 1 − z0
Hence,
For the inverse map, suppose ρ(x0 , y0 , z0 ) = (x + iy). Then, 1 (x0 + iy0 ) =x + iy 1 − z0
and
1 (x0 − iy0 ) = x − iy 1 − z0
Section 2.1. Identification
Page 5
Since (x0 , y0 , z0 ) ∈ S 2 , we have that x20 + y02 + z02 = 1. Also, x2 + y 2 = (x + iy)(x − iy), 1 = (x2 + y02 ), (1 − z0 )2 0 1 − z02 = , (1 − z0 )2 = which implies that
2 − 1, 1 − z0
2 = x2 + y 2 + 1, 1 − z0
and so
1 − z0 =
2 . x2 + y 2 + 1
Hence, 2 , + y2 + 1 x2 + y 2 − 1 = 2 . x + y2 + 1
z0 = 1 −
x2
Therefore, ρ−1 (x + iy) = If we denote z = x + iy, then we have:
1 2 2 2x, 2y, x + y − 1 . x2 + y 2 + 1 1 2z, |z|2 − 1 . ρ−1 (z) = 2 |z| + 1
(2.1.1)
This map ρ gives a bijection between S 2 and C ∪ {∞} by sending the north pole to ∞. Most authors think representing a point on the 2-sphere with three coordinates (x0 , y0 , z0 ) is not the cannonical choice since the 2-sphere is a two-dimensional object. We will use the formula in (2.1.3) in that version for the rest of the work. However, we will describe the more natural (two-dimensional) version of it using the spherical coordinates (φ, θ). If θ ranging from 0 to 2π measures the angle around the z-axis and φ ranging from 0 to π measures the angle subtended by the point P = (x0 , y0 , z0 ) and N at the center of the sphere, then the map ρ : S 2 − (0, 0, 1) −→ C such that P 7−→ Q is given by: φ ρ(φ, θ) = cot( )eiθ . 2
(2.1.2)
To show this, notice that the point Q ∈ C lies on the equatorial plane and so can be written as Q = Reiθ , where R is the distance of Q from the origin. The value of R is calculated as follows : using some basic φ idea in circle geometry, we have ∠P QO = ∠P SO = . 2 φ 1 tan = 2 R
Section 2.2. Hopf map and Geometric visualization
Page 6
Hence, φ R = cot . 2
(2.1.3)
The result follows.
2.2
Hopf map and Geometric visualization
Consider the composition: π
ρ−1
π
1 2 S 3 −→ CP1 −→ C ∪ {∞} −−→ S 2 with h = ρ−1 ◦ π2 ◦ π1 , w |w|2 1 π1 π2 w ρ−1 2 , 2 −1 . −−→ (z, w) −→ [z : w] −→ z z |z| |w|2 + 1 2 |z|
1 |w|2 +1 |z|2
|z|2 w |w|2 w |w|2 2 , 2 −1 2 , 2 −1 = z |z| |w|2 + |z|2 z |z|
|w|2 + |z|2 = 1,
w = 2z z¯ , |w|2 − |z|2 , z = 2¯ z w, |w|2 − |z|2 .
Therefore, we obtain the Hopf map as: h : S 3 −→ S 2
such that
(z, w) 7−→ (2¯ z w, |w|2 − |z|2 ).
(2.2.1)
The geometric picture of this map is via stereographic projection where we identify S 3 ≈ R3 ∪ {∞}.ie We extend the formula in (2.1.1) to higher dimensional version of it. ρ−1 (x, y, z) =
x2
+
y2
1 2x, 2y, 2z, x2 + y 2 + z 2 − 1 2 +z +1
(2.2.2)
We want to understand what the Hopf fibres are. One way to do that is to describe points in S 3 using three angles θ, φ and ψ and re-write the Hopf map in terms of these angles. Suppose (z, w) ∈ S 3 , we can write z and w using phase coordinate as: θ z = cos eiφ1 2 θ w = sin eiφ2 2
(2.2.3) (2.2.4)
If we let φ := φ2 − φ1 and ψ := φ2 + φ1 , we can re-write
(φ−ψ) θ z = cos e−i 2 2 θ (φ+ψ) w = sin ei 2 2
(2.2.5) (2.2.6)
The Hopf map in (2.2.1) can be written as h(θ, φ , ψ) = (sin θeiφ , cos θ). We want to observe what the fibre over a circle is in the Hopf fibration. To do this, we make this observation.
Section 2.2. Hopf map and Geometric visualization
Page 7
As a point moves around the circle (zeiα , weiα ), the modulus of z is constant. From (2.2.4) and (2.2.6), we have the fibre of the Hopf map must satisfy the equation: x2 + y 2 4 2 x + y2 + z2 + 1
2
= cos2
θ 2
(2.2.7)
The set of all points (x, y, z) ∈ R3 satisfying equation (2.2.7) forms a surface of revolution, which can be seen by rotating curve (or surface) in the xz-plane around the z-axis. Putting y = 0 and taking square root gives: θ x = ± cos . 2 2 (2.2.8) x + z2 + 1 2 By re-arrangement we obtain: x2 + z 2 + 1 = ± x2 +
2x , cos 2θ
sin2 2θ 2x 2 + z = , θ θ 2 θ cos cos 2 2 2 !2 2 θ sin 2 1 + z2 = x± , θ cos 2 cos2 2θ
1 cos2
±
θ θ (x ± sec )2 + z 2 = tan2 . 2 2
(2.2.9)
Equation (2.2.9) is the curve describing a torus of revolution as shown in figure 2.4. As θ varies, we obtain nested tori.
z-axis
tan θ2 x-axis sec θ2
Figure 2.4: Diagram of coaxial, nested concentric tori parametrized by 0 < θ < π [Ozo12]. If we let U = {h−1 (c)|c ∈ S 2 } ⊂ R3 , it can be shown that each element of U is just homeomorphic to S 1 , but infact geometric circles [Bar10], any two of these circles are linked . h−1 (∞)
Section 2.2. Hopf map and Geometric visualization
Page 8
corresponds to the circle of infinite radius which is seen as a straight line through the origin and passing through the north pole [Sti08]. This corresponds to when θ = π in (2.2.9). When θ = 0, (2.2.9) becomes (x ± 1)2 + z 2 = 0, which describes a torus that has degenerated to a circle. It is contained inside other tori. Since for every circle we have a torus, the 3-sphere can be thought of as the union of two solid tori glued at their boundary. To see the beauty of the Hopf fibration, see the wonderful animation of by Johnson [Nil12]. The map h is a submersion: a map from higher dimensional space into (in this case onto) a lower dimensional space. These Hopf circles are collapsed to points with the 2−sphere left behind. Generalization: At this point, it is worthwhile to mention that there is a corresponding Hopf fibration for each normed division algebra over R (i.e R, C, H, O). These give rise to the Real, Complex, Quaternic and Octonic fibrations respectively. Let V be any of the four division algebra, the total space of the fibration is the space of pairs (v1 , v2 ) ∈ V such that |v1 |2 + |v2 |2 = 1. That means that the base spaces are S 1 , S 3 , S 7 , and S 15 respectively. The corresponding base spaces of the fibration are S 1 , S 2 , S 4 , and S 8 respectively. The Hopf map is defined in each case as: n −1
η : S2
n−1
−→ S 2 v1 (v1 , v2 ) 7−→ . v2
,
n = 1, 2, 3, 4
With the knowledge of the Hopf fibration, we can now define the map j : S 3 −→ S 2 such that 20z 3 w (z, w) 7−→ as a replacement for the Hopf map h : S 3 −→ S 2 such that (z, w) 7−→ . In 3 2 20z − 49w z this case, we obtain trefoil knots instead of unknots as fibers over points on S 2 . With this construction, we are almost ready to identify the space of lattices (up to rescaling) with S 3 using a particular member of the space of modular forms namely the Eisenstein series. In the next chapter, we shall study modular forms in general with particular emphasis on the first two Eisenstein series.
3. Modular forms and functions Modular forms are certain functions defined on the upper half plane. These functions behave nicely under transformations of the upper half plane by the group SL2 (Z). These functions have series expansion whose coefficient are very useful in number theory. In this section, we will study some of these functions with particular emphasis on the Eisenstein series. 3.0.1 Definition. The upper half plane is defined by: H = {z ∈ C : =(z) > 0}. 3.0.2 Definition. The modular group denoted by Γ is defined by: a b SL2 (Z) = : a, b, c, d ∈ Z, ad − bc = 1 . c d az + b a b 3.0.3 Lemma. Γ acts on H by γz = ∀γ= ∈ Γ and z ∈ H. c d cz + d
(3.0.1)
(3.0.2)
Proof. It iseasy to show that for β, γ ∈ Γ and z ∈ H, =(γz) > 0 and β(γz) = (βγ)z. Also, Iz = z, 1 0 is the identity element in Γ. where I = 0 1 3.0.4 Definition. A fundamental domain of SL2 (Z) is a subset D of H such that every orbit of SL2 (Z) has one element in D, and two elements of D are in the same orbit if and only if they lie on the boundary of D. The set D = {z ∈ H |
