Alright so we are discussing Meromorphic functions okay and we were looking at Meromorphic functions
on the extended plane in the last class in the last lecture and we prove that a function
which is Meromorphic on the extended plane is none other than quotient of polynomials
okay namely a rational function okay, so what I need to what I want to tell now is about
the collection of Meromorphic functions on a domain okay, you take a domain in the extended
plane and look at the set of all Meromorphic functions defined on the domain okay then
that it has a has a nice structure in fact algebraic structure it is a field okay and
in fact it is an algebra over the complex numbers okay and so it is a field extension
of the complex numbers and the properties of this field extension, algebraic properties
of this field extension they have captured a lot of topological and geometric properties
of the domain okay.
So this is how there is a link from the complex analysis side to the algebra side okay, so
you know geometry involves an interplay of various ways of mathematics, so studying something
Geometrically will in all studying it from the analysis viewpoint okay and studying it
from the topological viewpoint also studying it from the algebraic viewpoint but when you
looking at a particular nice object okay when yesterday the analytically it will have some
properties okay, some special properties and then when you study it algebraically it will
have some properties, special properties. When you study topologically it will have
some special properties and the fact is that these properties are interrelated.
There is some beautiful relationship, hidden relationship between the analytic, the algebraic
at the topological properties of a nice object and that relationship is what you may call
as geometry okay. So if you want to really understand geometry of an object you have
to analyse it using all the 3 viewpoints algebraic, analytic, topological okay, so in that sense
know how do I do geometry on a domain in the complex plane or in the extended complex plane
okay. What I can do is of course the analysis is there, the analysis will worry about what
kind of functions you can define on the domain, what are the holomorphic functions or analytic
functions on the domain? What are the Meromorphic functions on the domain? and so on that will
be the viewpoint from analysis but then how do you go to algebra.
The point is that you take the set of Meromorphic functions that forms a field okay and that
is the field extension of the complex number and you study the properties of this field
extension okay, so in field theory you have lot of you would have come across in a course
in the algebra in field theory that field extensions are of so many types okay there
are algebraic extensions, there are transcendental extensions and then they were normal extension,
there are splitting fields, there are extension okay there are of course separable and non-separable
extension and we study all these things and of course the most important thing here in
general is the study of the nature of extensions because that connects up group theory which
is it connects up with the so called groups. So you see the moment you look at the field
of Meromorphic functions you get an extension of the complex numbers and then you can do
algebra okay and somehow these things are all connected and I will try to give couple
of examples. So 1st of all let me begin by 1st saying that if I take a domain in the
extended plane then the set of all Meromorphic functions in the domain is actually a field
okay.
So so let me write that down the field of Meromorphic functions, so let D inside C union
infinity be a domain, so you are taking a domain in the extended complex plane C union
infinity, so in particular mind you it is a nonempty open connected set okay and the
advantage of taking a domain in the extended plane is that you can also look at a neighbourhood
of infinity okay that is the advantage, so you are also including the point at infinity
right, so let so here is the notation I will put script m of D be the set of of Meromorphic
functions on D okay.
So what is this script m of D? Script m of D is the collection of all Meromorphic function
on and you know what a Meromorphic function is? You have defined the Meromorphic function
we are function which is analytic at all points but except or points on an isolated sets which
have to be always singularities okay, so it is analytic except for poles and the moment
you say analytic except for poles it means the singularities can be only poles and that
in particular means that the singularities can only be isolated because poles are isolated
singularities by definition okay.
So you take all the Meromorphic functions on the domain okay now the fact is that this
is a field okay, so let us see that let f and g be Meromorphic functions on D okay then
you see then you can notice the following things number 1 is lambda is a complex number
okay then lambda f is also a Meromorphic function okay multiplying a Meromorphic function by
a constant is going to keep it Meromorphic okay of course if the constant is 0 you will
get 0 and this 0 is a constant function. Of course when you say Meromorphic, analytic
is also included okay, so the definition for Meromorphic is that it is analytic except
for poles that does not mean it has to have a pole it can be it can have no poles and
it can be analytic everywhere, so holomorphic functions also included in the set of Meromorphic
functions okay.
So this statement is obvious if I take a Meromorphic function multiply by constant, if the constant
is 0 of course I am going to get the 0 function which is holomorphic which is analytic because
it is a constant function okay but if lambda is not 0, lambda times f will also be Meromorphic
and it will have the same poles, okay by multiplying by lambda you not going to change the poles
and you are not going to change the order of the poles. Essentially you just multiplying
by a constant okay, so this is one obvious thing then the 2nd thing is that if you take
the sum of these 2 Meromorphic functions, this will also be a Meromorphic functions
okay. The sum of f and g will also be Meromorphic why because you see the fact is that f is
Meromorphic so it has some it has a collection of poles okay and an isolated set of points.
Then g is also Meromorphic so it has also poles in another isolated set of points and
then you take the union of these 2 isolated set that is again an isolated set okay and
these are the only points where f plus g will have problems okay so at a point where f does
not have a problem and g does not have a problem, f plus g will not have a problem that is at
a point where f is analytic and g is analytic, f plus g of course will be analytic okay,
so the only problems for the function analyticity of the function f plus g will be at the points
where f and g have problems okay and it is possible that some ofůthere could be some
cancellations okay.
So for example f may be 1 by Z minus Z naught g may be minus 1 by Z minus Z naught, so if
I take f plus g I will get 0 which does not have a pole at Z naught okay, so some poles
can cancel out also and sometimes the order of a pole can come down okay when you add
course when I say add it also includes a subtraction because subtraction is just adding with minus
1 multiplied by the 2nd function okay, so so the moral of the story is that sum of 2
Meromorphic functions is again a Meromorphic function. It could very well be analytic okay
some poles might cancel out all the poles may cancel out for example if you take the
function and you take its negative and add it you will get 0 and that is clearly holomorphic
is a constant function, so some is Meromorphic so you know the moment you look at the first
2 things is will tell you that you know m of D is a vector space or complex numbers
see because it is you see so there is a scalar multiplication.
If you think of complex number a scalars then there is a scalar multiplication and there
is addition, so this becomes aůso m of D is C vector space so you get that immediately
okay. Now let us look at f times g look at f times G, see f multiplied by g will also
be Meromorphic on D this is also very clear because just from the fact that you know what
are the problem points on Mark the problem points are the points where f has problems
and g has problems okay, so if you take out those problem points then f times g will be
analytic, so at a point where f is analytic and g is analytic f times g will be analytic
and the only place where f times g will fail to be analytic it is probably on the of the
set of poles of f and poles of g okay and so you see and of course if you want you can
write out always the principal parts and see that you know if you functions have poles
at the same point they have common pole then if you multiply the product function will
have a pole with higher order in fact it will have order equal to sum of the orders that
is obvious if you write out the principal parts okay, so in the Laurent expansion alright.
So I mean the point is that you know all these algebraic operations of adding, subtracting
multiplying by a constant and just multiplying and of course we are going to see division
all these things they do not change the Meromorphic nature okay, so by adding or subtracting or
dividing or multiplying or multiplying by a constant you cannot change a Meromorphic
function into a non-Meromorphic function. If you are only working with Meromorphic functions
you will get back again Meromorphic function okay, so fine so you have f and g are the
product f time g is also Meromorphic of course by product 1 means point wise product okay,
so f g is a function which at each point Z is defined by f of Z times g of Z alright
then of course I can say the same of f by g this is also a Meromorphic function why
did g is not identically 0 okay of course I should not divided by 0, so the fact is
that see when I take f by g okay what other problems points?
The problem points will be poles of F, poles of g and now you will have extra problem points
at 0 so g because they are zeros of the denominator they become the zeros of g will become poles
of f by G, they are likely to become poles of f by g and of course you know it might
happen some zeros of g may cancel out with some zeros of f because the zeros of f on
the numerator the zeros of g are on the denominator some zeros might cancel but a set of problem
points are just the poles of F, the poles of g and the zeros of g and you know zeros
of an analytic function also isolated, you know that, there is a theorem okay in fact
that is another version of identity theorem if you have seen it in the 1st course in complex
analysis, so therefore the set of points where f by g will have problems is still an isolated
set of points okay and at each of those points you can only get poles you cannot get anything
worse, so therefore f by g is also Meromorphic in particular I could have taken 1 by g I
can put f equal to 1 I will get 1 by g is also Meromorphic and that means so what will
I get I will get 1 by g is Meromorphic if g is not identically 0.
So that means every nonzero Meromorphic function, every Meromorphic function which is not identically
0 as an inverse okay and that is what you require for a field okay. A field should be
basically a group under multiplication if you throughout the 0 element commonly set
okay, so well if you if you look at all these things, these things will tell you that m
of D is a field and you know you put it together with this fact that we saw earlier we have
seen just above that m of the D is also a complex factor space of field extension with
the field which is also a vector space is an algebra okay, so basically you can very
will see that m of D contains complex numbers because the complex numbers sit as constant
channels okay you take any complex number lambda you think of it as a constant function
lambda. Constant function lambda is analytic is defined everywhere.
So it is analytic on every domain and it is Meromorphic because mind you when I say Meromorphic
I am allowing also analytic or holomorphic. Meromorphic means that it can either be analytic
and if it is not analytic that is if it has singularities, the singularities must be only
poles that is what it says. So Meromorphic does not say that it should not be analytic,
so in particular m of D contains the complex numbers as a subfield you know the complex
numbers of those form a field and therefore m of D is a field extension of the field of
complex number, so m of D is aů
So let me write that m of D is a field extension of the field C of complex and the beautiful
thing is that the geometry on the domain D is done by a lot of topology of the domain
D is connected toů and a lot of analysis on the domain D namely the behaviour of the
existence and behaviour of Meromorphic functions on D is connected with the algebraic properties
of this field extension okay that is the geometric content okay. So if this goes back to the
work of classical giants like Riemann and Clifford and Weierstrass and Able and Yakobi
you know all these people who developed theory of Riemann surfaces okay of course principally
from Riemann, so let me write that down.
So let me write it out as a diagram so you have m of D and this is over C, so I am using
this field theory notation you write a field, a bigger field on top and you put a smaller
field below and then you put a vertical line thing that the thing that comes above is a
field extension and things that come below are some fields okay, so
the algebraic properties of this field extension they are connected so there is a, so the analysis
and D which is existence of a special existence and properties
of Meromorphic functions on D that is analysis on D okay and this part is algebra only, the
algebra on the domain is actually studying you might think of it as studying this field
extension and then there is the topology of D, topology of the domain D.
So I am not trying to be very particular or trying to go in detail but topology and the
minimum for example D is of course connected but one of the simplest thing that you can
look at is whether D is simply connected or not okay and then or if it is not simply connected
you can see if it is multiply connected and how many holes it has and so on and so forth
okay after all D could be something like an amoeba with some holes okay after all an open
set can look like that and then the topology worries about whether if it is simply connected,
if it is not simply connected how many holes are there and so on and so forth okay, so
all this topological information for example is encoded in the fundamental group of D and
so on and so forth and in fact more precisely I should say that you have to study the theory
of covering spaces of the domain D that is what is the topology of the means.
So let me write this here the theory of covering spaces of D related which is actually related
over the fundamental group, in fact subgroups of the fundamental
group of D, of course you know the so you know at this point let me tell you that if
you have done decent in topology see there is something called covering space theory
which takes the topological space with decent properties or example something that is house
of locally house of connected locally path connected, locally simply connected and then
you study what are called as coverings of topological covering and there is a Galva
theorem of covering which says that you know there is a Galva correspondence between the
coverings and subgroups of the fundamental group of your topological space and in fact
under this Galva correspondence so this Galva correspondence is an analog of the Galva
correspondence that you have in field theory.
See the Galva correspondence in field theory is you correspondence between field extensions
of a given field and subgroup of the Galva group okay and there is an analog, so the
Galva correspondence in field a really is a correspondence on the one side between field
extensions and on the other side between subgroups of a group and in this case is the Galva group
okay, so it is a connection between field theory and group theory okay and it is very
useful because a lot of field theory problems can be translated to group theory problems
and lot of group theory problems can be translated to field theory problems. In the same way
space theory is very very similar, what it does is it translates topological coverings
that is topological data into subgroups of fundamental group, so it also connects to
topological side, the topological side to group theory side okay so that you can use
some algebra in your topology okay.
So that is where usually this is a part of usually 1st course in algebraic topology okay,
so of course all this is very uninteresting if D is simply connected because if it D is
simply connected then the fundamental group is okay but then it is still not so easy in
fact there is I will explain why. Whatever I have written here, the algebra, the analysis
and the topology of D I have written it for a domain D in the extended plane okay but
what the philosophy is that this holds for any Riemann surface okay. Now there is something
called Riemann surface, the Riemann surfaces something at locally looks like a plane okay
but globally it may be different surface, so for example it may be a cylinder in 3 space
okay it may look like a torus alright or it may look like n torus, so it might look like
several tori which are stuck together by removing disk and open disk and pasting the boundaries
of the open disk okay.
So these are called Riemann surfaces and these were studied by Riemann and Riemann was fascinated
to know that on these Riemann surfaces you can put many complex structures there you
can put non-isomorphic complex structures and you must think of a complex structure
as a structure which allows you to decide whether a function on that surfaces is holomorphic
on not okay, so Riemann found that you know, see Riemann try to do what we do in complex
analysis on the plane. On the plane what we do, we take a domain and ask when a function
is analytic at the point okay and if it is analytic then of course you know if it is
not analytic then you see whether that point is an isolated singularity and so on that
is how you do the analysis.
Now what Riemann wanted to do was he wanted to do it on the surface, so he wanted to say
that suppose I have now a function on a torus okay or say even an open subset of the torus
alright, when I say open subset you take the induced topology from R3 in which the torus
sits okay and then suppose I have function which is complex valued defined on an open
subset of the torus, when can I say it is holomorphic, when can I say it is analytic?
So you are trying to study when a function defined on an open subset of a surface is
analytic, the answer to this is that we should define what is called Riemann surface okay
and there are different Riemann surfaces structures you can put and Riemann found thatůhe was
fascinated by these different Riemann surfaces structures and the most beautiful theorem
in Modley theory is that you actually take the set of all these Riemann surface structures
that itself become a nice object.
It becomes an analog at least on an open set, it becomes an analog higher dimensional analog
of Riemann surface which is called a complex manifold and of course it could have boundary
which could have some singular points but it is a very beautiful object okay, so the
moral story is that I am trying to say that whatever I am writing here for D, D at domain
in the extended plane it also works for a Riemann surface okay and so for example in
that context it is really amazing that you get a lot ofů so you know let me ask you
a fundamental question, the fundamental question suppose if you have a simply connected Riemann
surface okay, so the moment I say simply connected the topology seems to be very trivial because
in the sense that the fundamental group is trivial so you do not expect anything special
but then you can ask how many simply connected Riemann surfaces are there which are not isomorphic
to each other okay.
Now you more or less know the answer partially because the Riemann mapping theorem tells
you if we have seen it in the 1st course in complex analysis which you should have done
at you know any simply connected open subset of the complex plane which is not the whole
plane has to be holomorphically isomorphic that is by holomorphic to the unit test okay,
so if you take domains in the complex plane okay, simply connected domain is in the complex
plane there are only 2 types up to holomorphic isomorphism, one is the whole plane the other
one is the unit disk okay, so now it is an amazing fact that even before that let me
say look at the Riemann sphere okay which is you know we use that study the point at
infinity as the stereographic projection, the Riemann sphere is also a nice surface
of course and is compact okay and you can actually make it into a compact Riemann surface
okay. Now the fact is that there is also simply connected, Spear is simply connected so that
is also another simply connected Riemann surface.
Now it is a very deep theorem that you take any simply connected Riemann surface it has
to be isomorphic, holomorphically isomorphic to one of these 3. Any simply connected Riemann
surface has to be either it should either look like the whole plane okay or it should
look like the unit disk or it should look like a Riemann sphere there are no other possibilities.
It is a very deep theorem there is called uniformization, so even the simply connected
case you get a very deep theorem and the theorem is very hard to prove because you have to
use a lot of techniques from analysis to prove it okay, so it involve a lot of analysis,
it involves study of harmonic functions, Meromorphic functions, et cetera and it involves a reasonable
amount of function analysis and measured theory you have to do all this to get that theorem
okay. So anyway so the fact I want to say was at now given all these 3 aspects of putting
them together is what geometry is all about, okay. So let me wright here so geometry of
D is the interplay between these 3.
The geometry of a domain is actually the interplay between the analysis on the domain, algebra
on the domain and the topology on the domain and I have given you a rough idea, the analysis
on the domain is the complex analysis part okay. The algebra on the domain is to really
study the field of Meromorphic functions, the algebraic properties of the field extension
given by the field of Meromorphic function the domain and the topological part is to
study the curving space theory of the domain okay and it so happens I mean as the great
classical giants like Riemann and Clifford and Able and Yakobi and Weierstrass have found
and Clifford for example at you know all these properties, all these various points of view,
they are all interrelated okay so it is an amazing fact and discovering that is what
doing geometry is all about okay.
So you should not think that high school level the geometry is just about drawing triangle
encircles and you know measuring of angles and arcs and things like that but it is really
higher geometry in the higher sense is actually looking at the interplay of all these things
okay. So I well now you see I want to give you a couple of examples, so here is the 1st
example so here is an example.
You take the domain to be the extended plane itself okay see after all we are studying
domains in the extended plane so take the whole extended plane that is also domain.
In fact it is simply connected because you know there is homeomorphic the Riemann sphere
and the Riemann sphere is simply connected, so this simply connected, it is compact okay
is a very very nice thing. Now what are the what are the field of Meromorphic functions
on D okay, what are the fields of Meromorphic functions on the extended plane, so you know
this is an extension of the complex numbers as we have seen this is an extended of complex
numbers but you know what is it that we proved last time.
A function which is Meromorphic on the extended plane is none other than a quotient of polynomials,
it is a rational function okay and therefore this is exactly equal to the, this is exactly
equal to C round bracket Z this is the algebraic notation C round bracket Z and the C round
bracket Z is actually the field of fractions or quotient field of C of square brackets
and C of square bracket Z is standard notation is the ring of polynomials in the variable
Z with complex coefficients okay and C of Z is the field of fractions which is quotient
of such polynomials, so you take quotient of polynomials but of course you do not put
in the denominator 0 anything other than 0 you put okay, so so the moral of the story
is that you have very nice description of this field extension in the case of the C
union infinity which is the extended plane and usually you know extended plane is thought
of as Riemann sphere you know they are isomorphic but you can make them also isomorphic in a
holomorphic sense by giving the Riemann sphere a Riemann surface structure okay.
So often people do not use if you see the literature you will see that people often
use C union infinity instead of C union infinity they keep saying Riemann sphere all the time.
So now you can see that what are the properties of this field extensions you see this field
extension is actually is purely transcendental and has transcendence degree 1 okay. It is
purely transcendental and has transcendence degree 1. Well the transcendence degree is
actually the number of algebraically independent variable is that it generate the bigger extension
okay, so the bigger extension C of Z is generated by a single variable Z and that is the only
algebraically independent variable okay that one variable is enough, so the transcendence
degree is actually one okay and it is purely transcendental because there is no element
in C Z which is algebraic there is no elements in C Z which is not in C and which is algebraic
and that is you know why that is because Comcast numbers are algebraically closed they are
all algebraically closed.
So field theoretically so this is what is called as a function this is the simplest
example of what is called a function field in one variable okay and the beautiful thing
is that now if you take any compact Riemann surface okay then if you take the field of
Meromorphic functions on that compact Riemann surface what you will get is a function field
in one variable but the only thing is that it may not be purely transcendental about
the there may be an algebraic part okay so it will be 1st of transcendental extension,
purely transcendental extension of degree 1 just like this and then about that you will
have an algebraic extension and which will be a finite extension okay. So that is how
it looks in general okay and well I will give you another example for that, so let me write
this here this is a purely transcendental extension
of transcendence degree 1 okay. So the picture that is associated with this is the Riemann
sphere okay, so this is the picture that is associated with this and for all practical
purposes you think of the extended plane as the Riemann sphere okay.
Now let me tell you more generally what is it that happens with
somethingů so let me give you an example of a more complicated case is the case of
so-called elliptic functions or doubly periodic function okay, so here is what I am going
to do? What am going to do is I am going to take, so I am going to define what doubly
periodic function is? A doubly periodic function so this is the topic of what are
known as elliptic functions this is also an example okay w periodic function
is a function f of Z with f of Z plus Omega 1 is equal to f of Z and f of Z plus Omega
2 is equal to f of Z where Omega 1 by Omega 2 is not real and of course omega 1 omega
2 are nonzero okay so I am just defining what a doubly periodic function is?
So see the definition is very very simple for example sign beta will you know is periodic
with 2 pi because sin of theta plus 2 pi is the same as sin theta, so the idea is that
to the variable of the function or the argument of the function you add the period the function
value should not change okay, so what the 1st equation says is that f of Z plus w 1
is equal to f of Z actually tells you that w 1 is a period and the 2nd equation f of...
The 2nd requirement f of Z plus w 2 equal to f of Z tells you that w 2 is also a period,
so w 1 and w 2 are periods and the fact is that we want these periods to be linearly
independent over R in other words what you want is that if you take these 2 complex numbers
w 1 and w 2 then you join the origin to them okay namely you take the vectors that they
represent in the plane then these should be different the vector should be linearly independent
they should be in different directions you know they will be linearly dependent if and
only if the quotient w 2 by w 1 or w 1 by w 2 is a real number okay.
So this condition that w 1 by w 2 is not real it is just to tell you that these 2 vectors
are 2 different vectors. They will form therefore a basis of C over R, the complex numbers over
R is a 2 dimensional vector space and will from a basis so it is equivalent to saying
that w 1 w 2 form the basis for okay and you are putting this condition in order to make
sure that essentially have 2 distinct periods which are in 2 so it is periodic into directions
okay. The fact that f of Z plus w 1 is equal to f of Z tells you that you know if you translate
along the direction of w 1 by integer multiples of w 1 the function value does not change
okay, so you must remember that when I say f of Z plus w 1 is f of Z it follows that
f of Z plus n 1 w 1 is also f of Z for all integers n 1 okay because I can just use induction
f of Z plus w 1 is equal to f of Z, so f of Z plus 2 w 1 is f of Z plus w 1 plus w 1 which
is f of Z plus w 1 which is f of Z and so on and so forth okay.
So what it tells you is that the moment something is a period then all its integer multiples
are also periods okay and similarly you also have for the other but what is adding
W1, see addition of a complex number is just translation along the direction along the
vector that is represented by a complex number okay, so you know basically if I have a point
Z, what is Z plus W1? It is actually this vector, so this will be Z plus W1. I am just
a translating Z by the vector w 1 okay and then similarly what is Z plus w 2 I am just
translating Z by the vector w 2 alright and that is if I add w 1 but if I add minus w
1 you know I am translating in the other direction. If I add minus 2 w 1 I am translating in the
direction opposite to w 1 2 times and so on and so forth okay.
So the moral of the story is that you know basically the function value do not change
if you translate along 2 different directions okay that is why this called w periodic, it
is periodic and the period there are 2 different periods okay and such functions are called
actually now you have to put some more condition on these functions, the condition you put
on these functions is that you know to make them very interesting these points w 1 these
points which are given by integer multiples of w 1 added to integer multiples of w 2 okay
that they will form a lattice, a grid in the plane okay and the function becomes very interesting
if the function is Meromorphic exactly at those points okay and such functions are called
elliptic functions and believe it or not they are exactly the functions which are the function
is Meromorphic on a torus at a single point okay and this is the beginning of the so-called
Weierstrass phi theory there is something called as Weierstrass phi function which is
fundamental model of this kind of function and the beautiful thing is that every torus
the complex structure on any torus can be controlled by prescribing such a function
okay and so the Weierstrass phi functions completely give you.
So if you want to study the various complex structures you can put on a torus what you
will have to do is you have to study various doubly periodic Meromorphic functions which
are otherwise called elliptic functions, the reason they are called elliptic is because
this is beautiful the moment you put a complex structure on the torus it becomes believe
it or not it becomes a cubic curve, it becomes a cubic curve and therefore it becomes an
algebraic geometric object okay, so algebraic geometry also comes in, geometry also comes
in, algebra comes in in a beautiful way okay and this is also a part of a very deep theorem
which says that you know you take any compact Riemann surface it is algebraic it is just
given by a common 0 set of a bunch of polynomials okay and that is an amazing theorem okay.
So what I want to tell you is that I have given an NPTEL video course on Riemann surfaces
and all these things are explained in detail throughout the course when you find time you
can have a look at that and the other thing that I want to tell you is that there is this
book that I have written and it reads ôAn introduction to families deformation and moduli.
This book is basically available as a freely downloadable copy in the form of a navigable
PDF file and it contains a lot about the geometry of Riemann surfaces, so at least the 1st chapter
so that is also something that can be advanced reading material for people who are interested
in pursuing this. So let me continue so I have also f of Z plus n 2 w 2 is equal to
f of Z or all n 2 in Z, so in totality what I will get is I will get f of Z plus n 1 w
1 plus n 2 w 2 is equal to f of Z for all n 1, n 2 in Z if I put both these together
and what are these points n 1 w 1 plus n 2 w 2? They are the vertices of a grid of parallelogram
okay in fact if you draw this if I draw a diagram it is going to look like this.
So I have this so this is my complex plane and you see I have w 1 here I have w 2 here
okay and you know then if I draw this parallelogram then you know pretty well that this is w 1
is w 2 okay by the parallelogram law of additional vectors if you want and then you know if I
extend this parallelogram below then I am going to get this point is going to be you
know it is going to be w 1 minus w 2 okay and this point is going to be minus w 2 and
if I extend it like this, this point is going to be w 2 minus w 1 and this is going to be
minus w 1 minus w 2 and this is going to be minus w 1 okay and more generally if I drawů
If I look at all these points that go on that I get as the vertices of the parallelograms
that I get by simply starting with this parallelogram and simply displacing it by either plus or
minus w 1 or plus or minus w 2 okay that is by translating it with plus minus w 1 of plus
or minus w 2 I will get so manyůthe whole plane is covered by this parallelograms okay
and the vertices of the parallelogram are precisely the points which are of the form
n 1 w 1 plus n 2 w 2 okay and that is called the lattice okay and the fact is that you
seeůjust to give you an idea of what is going on where is the topology coming in India,
so the fact is that what you do is you divide by the equivalence relation Z 1 is equal in
to Z 2 if and only if there exist n 1 n 2 such that Z 1 is equal to Z 2 plus n 1 w 1
plus n 2 w 2 okay.
So see this is the plane this is the complex plane and I am defining an equivalence relation
on the plane, the equivalence relation is 2 points or equivalent one of them is a translate
of the other by one of these grid points okay and what is the advantage of this? The advantage
of this is that if 2 points were related like that then the w periodic function will have
the same value at both points because f of Z 1 will be equal to f of Z 2 plus n 1 w 1
plus n 2 w 2 but that is also equal to f of Z 2, so f of Z 1 will be equal to f of Z 2
okay because when I apply f to this equation okay on the right side I will get f of Z 2
because of periodic city of f so I will get f of Z 1 equal to f of Z okay. What it means
is that the value of the function is not change if you change the point by a translate by
a vector which belongs to one of the grid points okay.
So if you divide by this equivalence relation what you will get is you will get the torus,
you will get a beautiful torus and you can see at very very easily you just take this
fundamental parallelogram okay, this fundamental parallelogram if you take the interior every
point in the interior will be a unique representative in its equivalence class but for points but
then you will have to only identify the boundaries, see if you identify the top boundary with
the bottom boundary you will get a cylinder okay and then you will have to identify these
2 which will in the cylinder look like circles, so if you identify them you will get a torus
okay.
So the moral of the story is you take the plane divided by a lattice like this you will
get the torus okay and all the points in the grade including the origin, the origin is
here they all will go to a particular special point on the torus okay and the beautiful
thing is that the function at you defined on the complex plane will go down to a function
on the torus okay and if the function is Meromorphic exactly at all these grid points it will go
down to a Meromorphic function on the torus and you know you may wonder why should I worry
about why should I not consider mophet functions in the torus and you very well know the answer
will not be any non-constant holomorphic function on the torus because torus is compact. Since
the torus is compact if you define a holomorphic function on the torus okay you are going to
get something that is holomorphic you can use Liouvilleĺs theorem if you have a holomorphic
function on the torus if you composite with this map that is that goes from the complex
plane to the torus you will get a holomorphic function on the plane but since it is defined
on the torus which is compact its images compact therefore the image is bounded.
So I get an entire function which is bounded and that is going to be constant by Liouvilleĺs
theorem okay and this picture also explains why the only functions on the torus are exactly
the functions on the plane which are doubly periodic with respect to the periods w 1 and
w 2 and if you take this beginning point p which is the image of the grid the Meromorphiců
since you know holomorphic function is not available they have constants, the only things
that are available are the Meromorphic functions and then if you look at Meromorphic functions
on the torus at the point p they will be the same as doubly periodic functions okay and
therefore the moral of the story is that you know if you look at the field of Meromorphic
functions on the torus okay we are the same as the collection of Meromorphic functions
on the collection of doubly purity functions with these 2 periods okay and that is the
field of course. Mind you that what is the domain now?
The domain is the whole plane okay and I am looking at functions which are Meromorphic
with poles at points of the grid okay possibly at points of the grid alright and then what
I get is I get field and what is that field? That will is nothing but the field of Meromorphic
functions on the torus which are Meromorphic at a given point okay and so let me call this
torus as T okay mind you this torus depends on the choice of w 1 and w 2 okay and it is
a different story that there is a lot of geometry there but what I want to tell you is that
well I want to tell you the following things.
Simplest Meromorphic function on C that is doubly periodic
with respect to w 1 and w 2 is the Weierstrass phi function
and this is the phi function so there is pretty symbol for that very special symbol so phi
of Z is so there is a formula for that basically it is a formula that will tell you that it
is a Meromorphic function which has a double pole with the residue 0 at each of those points
of the grid okay, so you know what you are going to get let me write that down here you
can find this in any standard book example book on complex analysis which is a classic
so it is 1 by Z Square summation over n 1, n 2 belonging to Z minus 0, 0 in fact,
so I should write it carefully n 1, n 2 ordered pair belonging to Z across Z minus 0, 0 it
is 1 by Z minus n 1 w 1 minus n 2 w 2 the whole square minus 1 by n 1 w 1 plus n 2 w
2 the whole square.
So this is the expression for the Weierstrass phi function which was discovered by Weierstrass
and of course if you go through in detail the lectures of my video course you will see
that how this comes about but you can see something immediately you see that this 1
by Z square is the principal part at the origin and that will tell you that you know origin
is a double pole and residue is 0 because there is no 1 by Z term and then you look
at each of these other terms 1 by Z minus n 1 w 1 plus n 2 w the whole square tells
you that n 1 w 1 plus n 2 w 2 is a point of the grid is a general point of the grid okay
and if you when I write 1 by Z minus that point the whole square actually I am looking
at a pole of order 2 at that point and again the residue there is 0 alright.
So as a result this already gives you were you know it gives you a Meromorphic channel
which is having a double pole at each of these grid points okay and this extra term that
is added here is for convergence because you know I have added infinitely many poles okay
I have added poles at every point of the grid I have made every point of the grid into a
double pole and I am getting a huge series I want it to converge and it is only for this
convergence that this extra constant term is being added okay and therefore I get this
phi function and here comes the amazing theorem.
The amazing theorem is the following that if you take the complex numbers and you take
the Meromorphic functions on D on the complex plane respect to w 1 and w 2 okay you look
at the Meromorphic functions which are doubly periodic periods at w 1 and w 2 okay and
mind you this is the same as the Meromorphic functions on the torus okay which are Meromorphic
at that unique point which I will call at star which is the image of the grid the whole
grid, the whole grid goes to a single point on the torus because all the points on the
grid are equivalent to each other okay and they all define a single equivalence class
so it is a single point on the torus, so mind you the torus is set of equivalence classes
okay topologically and you give it the quotient topology alright.
Now on the beautiful thing is that what is this set of Meromorphic functions? You know
it is a field, what is that field? You know what that field is, that field is just the
field of fractions of phi of Z and its derivatives it is beautiful and how does this extension
breakup, it breaks up as the 1st phi of Z this is again a transcendental extension,
it is purely transcendental extension of transcendence degree 1 and then from here to here this is
an algebraic extension. This is an algebraic extension because then the derivatives of
the phi functions free prime satisfies the polynomial relation with respect to phi and
that is expresses a differential equation.
It is a very famous differential equation and that differential equation interestingly
it comes from analysis what it tells you that the torus is algebraic, it tells you that
the torus is nothing but a cubic curve okay which is an amazing illustration or the fact
that in general a compact Riemannĺs surface is given is algebraic it is given by algebraic
equations okay. So all these details you can have a look at it in more detail in my video
lecture course but this is to tell you that a lot of geometry is involved by looking at
the field extension given by the field of Meromorphic functions, okay. So I will stop
here.