Linear fractional transformations

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Linear fractional transformations - Cornell University

SL(2,C) Proposition The map g 7→T g is a group isomorphism between SL(2,C)/{±Id}, and linear fractional transformations Proof Every every fractional transformation is of the form T g with g satisfying detg 6= 0 But g and 1 detg g give the same transformation, and the latter is in SL(2,C) To see that the map is 1-1, note that if T g 0

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Math 418 Spring 2006Linear fractional transformations

0.1.Definitions.We study a special class of mapsT:C? {∞} -→

C? {∞}.

A linear fractional transformation ofC? {∞}is a map of the form (0.1.1)w=T(z) =az+bcz+d, a,b,c,d?C. We think of the transformation as depending on the 2×2 matrix (0.1.2)g:=?a b c d? and writeTgfor the transformation. We assume thatg?= 0,or else we would be dividing

00for allz.In fact, ifc=d= 0, Twould be undefined

for allz.Note that if we multiply the coefficients by the sameλ?= 0, we get the same transformation. In particular, ifad-bc= 0,then there is cancellation, and the transformation becomesT(z) =constant except for maybe one value ofzwhere the transformation is undefined.

Weassumethatad-bc?=0.

The following properties are verified by direct calculation.

1:Tg=Tg?if and only if there isλ?= 0 such that

a ?=λa, b?=λb, c?=λc, d?=λd.

This can be written as

?a?b? c ?d?? =?λ0

0λ?

·?a b

c d?

2:Tg1·g2=Tg1◦Tg2,where·is usual matrix multiplication, and◦

is composition of transformations/functions.

3:Ifg=?1 0

0 1? ,thenTg(z) =z. In view of this, we define dividing any nonzero number by 0 to be∞.

With this convention, we define

(0.1.3)Tg(∞) =ac, Tg(-dc) =∞. Ifc?= 0,acis a finite number. Ifc= 0,thena?= 0,andac=∞.

Similarly for

dc. 1 2

But we still have to assume that

00and 0· ∞are undefined.

0.2.Relation to elementary transformations.Recall the elemen-

tary transformations:

Translation:Ta(z) =z+a

Dilation:Ta(z) =azfora?= 0.

Inversion:R(z) =1z.

These are linear fractional transformations, so any composition of sim- ple transformations is a linear fractional transformations. Conversely any linear fractional transformation is a composition of simple trans- formations. Ifc= 0,this is clear. Ifc?= 0,we can write (0.2.1) az+bcz+d=ac-ad-bcc(cz+d)

The claim follows from this equation.

0.3.Circles and lines.The equation of a circle is given by

(0.3.1)Azz+Bz+Bz+C= 0, A,C?R, AC <|B|2.

The equation of a line is

(0.3.2)Bz+Bz+C= 0, C?R. It is easy to verify that translations and dilations send circles to circles, and lines to lines. Inversion also sends a circle to a circle, except if it passes through 0. In this case,C= 0.A pointz?= 0 on such a circle (0.3.1) withC=) gives the equation (0.3.3)Bw+Bw+A= 0, w=1z. It makes sense to make the definition that every line contains∞.Then

0 is mapped by inversion to∞,and every circle going through 0 is

mapped to a line not going through 0. Lines going through 0 have equation (0.3.2) withC= 0. Using our conventions, it follows that inversion takes lines through 0, to lines through 0. Thus elementary transformations take circles and lines to circles or lines. The complement of a line or circle in the extended plane is a union of two connected regions. being continuous, a fractional transformation takes a connected region to a connected region. Exercise 1.These conventions are compatible with stereographic pro- jection. Prove this fact. Precisely, show that (1) Any circle through the north pole is mapped to a line, (2) Any circle on the sphere not passign through the north pole is mapped to a circle. 3 Hint:Any circle on the sphere is obtained by intersecting the sphere with a plane.

0.4.Uniqueness.The set of 2×2 matrices with complex entries, and

of determinant 1 forms a group under matrix multiplication. It is called

SL(2,C).

Proposition.The map

g?→Tg is a group isomorphism betweenSL(2,C)/{±Id},and linear fractional transformations. Proof.Every every fractional transformation is of the formTgwithg satisfying detg?= 0.Butgand1detgggive the same transformation, and the latter is inSL(2,C).To see that the map is 1-1, note that if T g?=Tg,impliesTg?◦Tg-1=Id,soTg?·g-1=Id.Thus it is enough to see that if (0.4.1)Tg(z) =z, then g=Id.

This means

(0.4.2) az+bcz+d=z, az+b=cz2+dz. Plugging inz= 0,we getb= 0.Plugging inz=∞,we getc= 0. Then plugging inz= 1,we getad= 1.Since we also assumed detg=ad= 1, we finda=d=±1.? Theorem.Let{z0,z1,z∞}and{z?0,z?1,z?∞}be two sets of triplets of distinct points inC? {∞}.Then there is a unique transformationT such that

T(z0) =z?0, T(z1) =z?1, T(z∞) =z∞.

Proof.We show existence first. Assume that{z?0,z?1,z?∞}={0,1,∞}. If we can prove the assertion for this choice, then the general existence statement follows. This is because for arbitrary{z0,z1,z∞}we can construct (0.4.3)

T(z0) = 0, T(z1) = 1, T(z∞) =∞,

T ?(z?0) = 0, T?(z?1) = 1, T?(z?∞) =∞, and thenT?-1◦Tis the desired transformation.

If none of{z0,z1,z∞}are∞,then

(0.4.4)T(z) =z-z0z1-z∞·z1-z∞z1-z0 4 is the transformationT.If sayz0=∞,then (0.4.5)T(z) =z1-z0z-z∞.

The other cases are left as an exercise.

Now we show uniqueness. Suppose thatTandT?satisfy the claim of the theorem. ThenT?-1◦Tfixes{z0,z1,z∞}.SupposeT0takes

But then we can check easily thatS=Tgmust satisfy

(0.4.6) bd= 0,ac=∞. Thusb=c= 0,and thereoforead= 1 as well. ThusS=Id.It follows thatT?-1◦T=Id,or equivalentlyT?=T.? Exercise 2.Complete the proof of the theorem,i.e., find the trans- formations whenz1=∞and whenz∞=∞.?

0.5.Transformations preserving the real line.

Exercise 3.

(1) Show that any linear fractional transformation that maps the real line to itself can be written asTgwherea,b,c,d?R. (2) The complement of the real line is formed of two connected re- gions, theupper half plane{z?bC:Imz >0},and the lower half plane{z?C:Imz <0}.Show that a trans- formation presrving the real line preserves the two half planes if detg >0,and interchanges them if detg <0.In particu- lar, show that the groupSL(2,R) is precisely the subgroup of

SL(2,C) that preserves the upper half plane.

0.6.The upper half plane.Denote the upper half plane byH.Then

SL(2,R) acts on it by

(0.6.1)Tg(z) =az+bcz+d, g=?a b c d? , a,b,c,d?R, ad-bc= 1.

The stabilizer ofiis the subgroup satisfying

(0.6.2) 5 Thusac=-bd, andc2+d2= 1.Settingc= sinθandd= cosθ,we finda= cosθandb=-sinθ.Thus the stabilizer ofiis the subgroup calledSO(2), (0.6.3)SO(2) =??cosθ-sinθ sinθcosθ? :θ?R?

On the other hand, letz=x+iy? H,i.e.y >0.Then

(0.6.4)?y1/2xy-1/2

0y-1/2?

has the property thatTg(i) =x+iy.Thus for anyz? H,there is g?SL(2,R) such thatTg(i) =z. Exercise 4.Show that anyg?SL(2,R) can be written uniquely as (0.6.5)?a b c d? =?y1/2xy-1/2

0y-1/2?

·?cosθ-sinθ

sinθcosθ? Exercise 5*.Find all fractional linear transformations coming from

SL(2,C) which preserve the region

z?C:|z|<1?

0.7.Modular forms.The groupSL(2,R) contains a discrete sub-

group (0.7.1)SL(2,Z) =? g?SL(2,R) :a,b,c,d?Z? Functions which are analytic (to be defined later in the course) on H

¯satisfying

(0.7.2)f(Tγ(z)) = (cz+d)kf(z)γ?SL(2,Z), are called modular forms of weightk.Their properties are closely con- nected to number theory.

Some examples.

•Eisenstein series:G2k(z) =? (m,n)?=(0,0)1(mz+n)2k. •Theta series:θ(z) =? n?Zein2πz.This is not really a modu- lar form forSL(2,Z),but rather satisfies a more complicated invariance property for a subgroup ofSL(2,Z).You may have seen this function in connection with the heat kernel.quotesdbs_dbs15.pdfusesText_21

[PDF] Linear fractional transformations - Cornell University

0.1.Definitions.We study a special class of mapsT:C? {∞} -→

C? {∞}.

00for allz.In fact, ifc=d= 0, Twould be undefined

Weassumethatad-bc?=0.

1:Tg=Tg?if and only if there isλ?= 0 such that

This can be written as

0λ?

·?a b

2:Tg1·g2=Tg1◦Tg2,where·is usual matrix multiplication, and◦

3:Ifg=?1 0

With this convention, we define

Similarly for

But we still have to assume that

00and 0· ∞are undefined.

0.2.Relation to elementary transformations.Recall the elemen-

Translation:Ta(z) =z+a

Dilation:Ta(z) =azfora?= 0.

Inversion:R(z) =1z.

The claim follows from this equation.

0.3.Circles and lines.The equation of a circle is given by

The equation of a line is

0 is mapped by inversion to∞,and every circle going through 0 is

0.4.Uniqueness.The set of 2×2 matrices with complex entries, and

SL(2,C).

Proposition.The map

This means

T(z0) =z?0, T(z1) =z?1, T(z∞) =z∞.

T(z0) = 0, T(z1) = 1, T(z∞) =∞,

If none of{z0,z1,z∞}are∞,then

The other cases are left as an exercise.

But then we can check easily thatS=Tgmust satisfy

0.5.Transformations preserving the real line.

Exercise 3.

SL(2,C) that preserves the upper half plane.

0.6.The upper half plane.Denote the upper half plane byH.Then

SL(2,R) acts on it by

The stabilizer ofiis the subgroup satisfying

On the other hand, letz=x+iy? H,i.e.y >0.Then

0y-1/2?

0y-1/2?

·?cosθ-sinθ

SL(2,C) which preserve the region

0.7.Modular forms.The groupSL(2,R) contains a discrete sub-

¯satisfying

Some examples.