Moebius transformation



Group Theory

Complex geometry

Elliptic cohomology



Let kk be a field, and let 1(k)\mathbb{P}^1(k) be the projective line over kk. A Möbius transformation (also called a homography, a linear fractional transformation, or a fractional linear transformation which is my own preference – Todd) is a function f: 1(k) 1(k)f: \mathbb{P}^1(k) \to \mathbb{P}^1(k) defined by the rule

f(x)=ax+bcx+df(x) = \frac{a x + b}{c x + d}

where a,b,c,dka, b, c, d \in k and adbck ×a d - bc \in k^\times. Möbius transformations form a group under composition, isomorphic to the projective linear group

PGL 2(k)GL 2(k)/{λI:λk ×}PGL_2(k) \coloneqq GL_2(k)/\{\lambda I: \lambda \in k^\times\}

If, as in the case k=k = \mathbb{C}, each element of k ×k^\times has a square root, then this group is identified with

PSL 2(k)SL 2(k)/±I.PSL_2(k) \coloneqq SL_2(k)/\pm I.

Alternatively, a fractional linear transformation can be considered as synonymous with an automorphism of the field of rational functions k(x)k(x) as a field over kk of transcendence degree 1.

In complex analysis (which is the usual context when one speaks of Möbius transformations; otherwise one usually calls them by some combination of “linear” and “fractional”), Möbius transformations are precisely the biholomorphisms of the Riemann sphere, hence exactly its bijective conformal transformations.

Often, and particularly when kk is the the commutative ring of integers \mathbb{Z}, one considers a modular group where the coefficients a,b,c,da, b, c, d are assumed to lie in an integral domain and adbc=1a d - b c = 1. (The homotopy quotient of the upper half-plane by the group PGL 2()PGL_2(\mathbb{Z}) is the moduli stack of elliptic curves over the complex numbers.)




The action PGL 2(k)× 1(k) 1(k)PGL_2(k) \times \mathbb{P}^1(k) \to \mathbb{P}^1(k) is 3-transitive, i.e., any triplet of distinct points (a,b,c)(a, b, c) may be mapped to any other triplet of distinct points (a,b,c)(a', b', c') by applying a group element.


It suffices to consider a=0,b=1,c=a' = 0, b' = 1, c' = \infty where one applies the transformation x(xa)(bc)(xc)(ba)x \mapsto \frac{(x-a)(b-c)}{(x-c)(b-a)}.

This motivates the following definition: given a 4-tuple (a,b,c,d)(a, b, c, d) of distinct points in 1(k)\mathbb{P}^1(k), its cross-ratio is

γ(a,b,c,d)=(da)(bc)(dc)(ba).\gamma(a, b, c, d) = \frac{(d-a)(b-c)}{(d-c)(b-a)}.

It is not hard to see that the group action preserves the cross-ratio, i.e., γ(ga,gb,gc,gd)=γ(a,b,c,d)\gamma(g \cdot a, g \cdot b, g \cdot c, g \cdot d) = \gamma(a, b, c, d). Moreover, the group action is transitive on each cross-ratio-equivalence class of 4-tuples.

In the case k=k = \mathbb{C} where 1\mathbb{P}^1 is interpreted as the Riemann sphere, it turns out that the cross-ratio of a 4-tuple is a real number if and only if the four points lie on a circle (or a line which is a circle passing through \infty). Hence Möbius = conformal transformations take circles to circles.

Action on hyperbolic space

As explained at Poincare group, the group PSL 2()PSL_2(\mathbb{C}) can be identified with those linear transformations of Minkowski space 4\mathbb{R}^4 that preserve the Minkowski form QQ, are orientation-preserving, and take the forward light cone {v=(x,t):Q(v)=0,t>0}\{v = (\vec{x}, t): Q(v) = 0, t \gt 0\} to itself. It follows that PSL 2()PSL_2(\mathbb{C}) acts on the hyperboloid sheet

H 3={v=(x,t):Q(v)=1,t>0}H^3 = \{v = (\vec{x}, t): Q(v) = 1, t \gt 0\}

which is naturally identified with hyperbolic 3-space. There is a Poincaré disk model for H 3H^3; consider the disk D 3D^3 that is the intersection of the future cone {v=(x,t):Q(v)0,t>0}\{v = (\vec{x}, t): Q(v) \geq 0, t \gt 0\} with the hyperplane t=1t = 1. Its interior is an open 3-disk int(D 3)int(D^3) which can be placed in perspective with H 3H^3 by considering lines through the origin in 4\mathbb{R}^4: each line that passes through a unique point in H 3H^3 passes through a unique point of int(D 3)int(D^3). In this way, D 3D^3 is viewed as a natural compactification of H 3H^3, and the action of PSL 2()PSL_2(\mathbb{C}) on H 3H^3 induces an action of PSL 2()PSL_2(\mathbb{C}) on D 3D^3. The restriction of this action to the boundary S 2=D 3S^2 = \partial D^3 (“the heavenly sphere”) coincides with the action on the Riemann sphere S 2= 1()S^2 = \mathbb{P}^1(\mathbb{C}).

Modular group

The modular group Γ\Gamma is the subgroup PSL 2()PSL 2()PSL_2(\mathbb{Z}) \hookrightarrow PSL_2(\mathbb{C}) consisting of Möbius transformations zaz+bcz+dz \mapsto \frac{a z + b}{c z + d} where a,b,c,da, b, c, d \in \mathbb{Z} and adbc=1a d - b c = 1.

The group PSL 2()PSL_2(\mathbb{R}) acts on the upper half-plane H={z:Im(z)0}H = \{z \in \mathbb{C}: Im(z) \geq 0\} (or rather H{}H \cup \{\infty\} as a subspace of the Riemann sphere), by restriction of the action of PSL 2()PSL_2(\mathbb{C}) on the Riemann sphere. Indeed, the action of PSL 2()PSL_2(\mathbb{R}) takes the real line {}\mathbb{R} \cup \{\infty\} to itself, and any element f(z)=az+bcz+df(z) = \frac{a z + b}{c z + d} takes ii to (b+ai)(dci)/(c 2+d 2)(b + a i)(d - c i)/(c^2 + d^2), whose imaginary part (adbc)/(c 2+d 2)=1/(c 2+d 2)(ad - b c)/(c^2 + d^2) = 1/(c^2 + d^2) is positive. By continuity it follows that the action preserves the sign of the imaginary part, hence takes the upper-half plane HH to itself.

It is illuminating to think of complex numbers τ\tau such that Im(τ)>0Im(\tau) \gt 0 as representing elliptic curves EE. Indeed, the field of meromorphic functions on an elliptic curve (i.e., a complex projective curve EE of genus 11, or a torus equipped with a structure of complex analytic 1-manifold) can be identified with a field of doubly periodic holomorphic functions /L 1()\mathbb{C}/L \to \mathbb{P}^1(\mathbb{C}) where L=ω 1,ω 2L = \langle \omega_1, \omega_2\rangle is a fundamental lattice (a discrete cocompact subgroup of the additive topological group \mathbb{C}) attached to EE. In essence, this field is generated by Weierstrass elliptic functions (z),(z):/L 1()\wp(z), \wp'(z): \mathbb{C}/L \to \mathbb{P}^1(\mathbb{C}) (here \wp' is the derivative of \wp) which satisfy a cubic algebraic relation

() 2=4 3g 2g 3(\wp')^2 = 4\wp^3 - g_2\wp - g_3

where the constants g 2,g 3g_2, g_3 are expressed as certain Eisenstein series in the fundamental periods ω 1,ω 2\omega_1, \omega_2. The \mathbb{Z}-linear basis elements ω 1,ω 2\omega_1, \omega_2 of the lattice may be arranged so that τ=ω 2/ω 1\tau = \omega_2/\omega_1 has positive imaginary part. Of course, if there is a homothety zλzz \mapsto \lambda z that takes a lattice LL to a lattice LL', then the elliptic curves E=/LE = \mathbb{C}/L and E=/LE' = \mathbb{C}/L' are analytically isomorphic, so the map

τ/1,τ\tau \mapsto \mathbb{C}/\langle 1, \tau\rangle

gives a surjection from complex numbers with positive imaginary part to isomorphism classes of smooth elliptic curves. Thus we may restrict attention to lattices of the form L=1,τL = \langle 1, \tau \rangle.

Of course, LL admits more than one such basis (1,τ)(1, \tau), but for any other (1,τ)(1, \tau') there is a linear transformation γΓPSL 2()\gamma \in \Gamma \coloneqq PSL_2(\mathbb{Z}) such that γ(τ)=τ\gamma(\tau) = \tau'. In summary, the orbit space

{τ:Im(τ)>0}/Γ\{\tau \in \mathbb{C}: Im(\tau) \gt 0\}/\Gamma

is a coarse moduli space for elliptic curves. In this context, one often says that elliptic curves are paramatrized by the jj-invariant, a certain modular form j(τ)j(\tau) defined on the upper half-plane such that j(τ)=j(τ)j(\tau) = j(\tau') if and only if τ=γτ\tau' = \gamma \cdot \tau for some γΓ\gamma \in \Gamma.

Of course, in some cases there may be more than one γΓ\gamma \in \Gamma that fixes a given τ\tau; this is notably the case when τ\tau is a fourth root of unity or a sixth root of unity. A more refined approach then is to consider, instead of the coarse orbit space, the (compactified) moduli stack (H{})//Γ(H \cup \{\infty\})//\Gamma for elliptic curves, as a central geometric object of interest.

Abstract structure of modular group

As an abstract group, Γ=PSL 2()\Gamma = PSL_2(\mathbb{Z}) is a free product /(2)*/(3)\mathbb{Z}/(2) \ast \mathbb{Z}/(3); explicitly, we may take the generator of order 22 to be given by the Moebius transformation z1/zz \mapsto -1/z, and the generator of order 33 to be given by z(z1)/zz \mapsto (z-1)/z.

This concrete group and certain of its subgroups, such as congruence subgroups, are fairly ubiquitous; for example the modular group appears in the theory of rational tangles and of trivalent maps, and these groups frequently crop up in the theory of buildings (stuff on hyperbolic buildings to be filled in).

It is also worth pointing out that Γ\Gamma is a quotient of the braid group B 3B_3. Consider the standard Artin presentation of B 3B_3, with two generators σ 1\sigma_1, σ 2\sigma_2 subject to the relation

σ 1σ 2σ 1=σ 2σ 1σ 2.\sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2.

Then z(σ 1σ 2) 3z \coloneqq (\sigma_1 \sigma_2)^3 is a central element of B 3B_3, and there is a central extension

11zB 3qΓ11 \to \mathbb{Z} \stackrel{1 \mapsto z}{\to} B_3 \stackrel{q}{\to} \Gamma \to 1

where qq is the unique homomorphism mapping σ 1σ 2\sigma_1\sigma_2 to λz.(z1)/z\lambda z. (z-1)/z, and σ 1σ 2σ 1\sigma_1\sigma_2\sigma_1 to λz.1z\lambda z. \frac{-1}{z}.


Named after August Möbius.

Last revised on June 11, 2017 at 10:02:40. See the history of this page for a list of all contributions to it.