Mathematics  An Introduction to Riemann Surfaces

1
Mod01 Lec01 The Idea of a Riemann Surface
by nptelhrd 2,615 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
To develop a suitable definition of a structure of a Riemann
Surface on a 2dimensional surface that will allow us to carry out
Complex Analysis (i.e., study of holomorphic (or) analytic functions)
on the given surface.
Keywords:
Complex plane, open set, analytic (or) holomorphic function, CauchyRiemann
equations, complex differentiable, convergent power series, Taylor expansion,
Taylor coefficients, open map, biholomorphic map (or) holomorphic isomorphism,
homeomorphism (or) topological isomorphism, complex coordinate chart,
compatibility of charts, transition functions, Riemann surface structure 
2
Mod01 Lec02 Simple Examples of Riemann Surfaces
by nptelhrd 679 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To see how the real plane can be equipped with two different Riemann Surface structures
 To see how the real 2dimensional sphere can be equipped with a Riemann surface structure
Keywords for Lecture 2:
Complex coordinate chart, compatible charts, transition function, complex atlas, Riemann surface, holomorphic function, unit disc, complex plane, Uniformisation theorem, simply connected, Riemann sphere, Riemann mapping theorem 
3
Mod01 Lec03 Maximal Atlases and Holomorphic Maps of Riemann Surfaces
by nptelhrd 411 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To give a better definition of a Riemann surface using equivalence of atlases
 To define the notion of a holomorphic (or) analytic map from one Riemann surface
into another Riemann surface and in particular an isomorphism of Riemann surfaces
Keywords for Lecture 3:
Complex atlas, equivalent atlases, union of equivalent atlases, maximal atlas, holomorphic (or) analytic mapping between Riemann surfaces, isomorphism of Riemann surfaces 
4
Mod01 Lec04 A Riemann Surface Structure on a Cylinder
by nptelhrd 353 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To interpret a cylinder as a suitable quotient of the complex plane;
 To use the above interpretation to give a Riemann surface structure on a cylinder and
to raise the question as to how many such nonisomorphic structures exist.
Keywords for Lecture 4:
Translation by a complex number, equivalence relation, equivalence class, set modulo an equivalence relation, glueing edges of a strip, inverse image of an equivalence class, quotient topology, quotient map, open map, homeomorphism, Moebius transformation, group action on a set, orbits of an action, set modulo (action of) a group 
5
Mod01 Lec05 A Riemann Surface Structure on a Torus
by nptelhrd 369 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To interpret the torus as a suitable quotient of the complex plane;
 To use the above interpretation to give a Riemann surface structure on a torus and
to raise the question as to how many such nonisomorphic structures exist.
Keywords for Lecture 5:
Translation by a complex number, equivalence relation, equivalence class, set modulo an equivalence relation, glueing edges of a parallelogram, inverse image of an equivalence class, quotient topology, quotient map, open map, homeomorphism, Moebius transformation, group action on a set, orbits of an action, set modulo (action of) a group 
6
Mod02 Lec06 Riemann Surface Structures on Cylinders and Tori via Covering Spaces
by nptelhrd 326 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To look at the set of all possible Riemann surface structures on a cylinder and the
need for a method to distinguish between them
 To explain the motivation for the use of the Theory of Covering Spaces to distinguish
Riemann surface structures
 To motivate the notion of a covering map by examples
 To get introduced to the fact (called General Uniformisation) that any Riemann surface
is the quotient (via a covering map) of a suitable simply connected Riemann surface
 To understand the idea of the Fundamental group and where it fits into our discussion
Keywords for Lecture 6:
Cylinder, punctured plane, punctured unit disc, annulus, Riemann's Theorem on removable singularities, covering map, covering space, pathwise connected, locally pathwise connected, admissible neighbourhood or admissible open set, universal covering space or simply connected covering space, fundamental group, uniformisation of a general Riemann surface 
7
Mod02 Lec07 Moebius Transformations Make up Fundamental Groups of Riemann Surfaces
by nptelhrd 301 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 Every good topological space possesses a unique simply connected covering
space called the Universal covering space
 The fundamental group of the topological space shows up as a subgroup of
automorphisms of its universal covering space
 The universal covering map expresses the target space as the quotient of
the universal covering space of the target, by the fundamental group of the target
 A covering map can be used to transport Riemann surface structures from source
to target and viceversa, thus making it into a holomorphic covering map
 Any Riemann surface is the quotient of the complex plane, or the upper halfplane,
or the Riemann sphere by a suitable group of Moebius transformations isomorphic to
the fundamental group of the Riemann surface
 The study of any Riemann surface boils down to studying suitable subgroups of
Moebius transformations
Keywords:
Covering map, covering space, admissible open set or admissible neighborhood,
simply connected covering or universal covering, local homeomorphism, Riemann
surface structure inherited by a topological covering of a Riemann surface, uniformisation, fundamental group, Moebius transformation 
8
Mod02 Lec08 Homotopy and the First Fundamental Group
by nptelhrd 476 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To understand the notion of homotopy of paths in a topological space
 To understand concatenation of paths in a topological space
 To sketch how the set of fixedendpoint (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First Fundamental Group
 To look at examples of fundamental groups of some common topological spaces
 To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing nonisomorphic topological spaces
 To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixedendpoint (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant 
9
Mod02 Lec09 A First Classification of Riemann Surfaces
by nptelhrd 244 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
 To get an idea of the classification of Riemann surfaces that can be arrived at
based on the fundamental group, using the theory of covering spaces
 To get introduced to the notions of : moduli problem, moduli space, number of
moduli, fine and coarse classification, and to write these down for simple Riemann
surfaces
Keywords:
Biholomorphic map or isomorphism of Riemann surfaces, classification of Riemann
surfaces, universal covering of a Riemann surface, abelian fundamental group,
complex plane, unit disc, upper halfplane, punctured plane, punctured unit disc,
cylinder, complex torus, annulus, Riemann sphere, gtorus, coarse classification,
fine classification, moduli problem, moduli theory, moduli space, number of moduli 
10
Mod03 Lec10 The Importance of the Pathlifting Property
by nptelhrd 164 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 10:
* To explore the reasons for the fundamental group occurring both as the inverse image of any point under the universal covering map as well as a subgroup of automorphisms of the universal covering space
* To understand the notions of lifting property, uniquelifting property and uniquenessoflifting property
* To understand the Covering Homotopy Theorem
* To note that surjective local homeomorphisms have the uniquenessoflifting property
* To note that a surjective local homeomorphism is a covering iff it has the pathlifting property
* To deduce that covering maps have the unique pathlifting property
Keywords for Lecture 10:
Lifting of a map, lifting of a path, lifting property, uniquelifting property, uniquenessoflifting property, Covering Homotopy Theorem, local homeomorphism, unique pathlifting property, existence of lifting, fundamental group, universal covering 
11
Mod03 Lec11 Fundamental groups as Fibres of the Universal covering Space
by nptelhrd 163 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 11:
* To see that the formation of the fundamental group is a covariant functorial operation, from the category whose objects are pointed topological spaces and whose morphisms are basepointpreserving continuous maps, to the category whose objects are groups and whose morphisms are group homomorphisms
* To deduce the pathlifting property for a covering map as a consequence of the Covering Homotopy Theorem
* To deduce from the Covering Homotopy Theorem that the fundamental group of a covering space can be identified naturally with a subgroup of the fundamental group of the space being covered
* To note that the inverse image of a point (fibre over a point) under a covering map may be identified with the space of cosets of the fundamental group (based at a point fixed above) inside the fundamental group at the point below
* To note that the universal covering of a space may be pictured as a fibration consisting of fundamental groups over that space
Keywords for Lecture 11:
Covering Homotopy Theorem, stationary homotopy, lifting of a homotopy, pathlifting property, category, objects of a category, morphisms of a category, covariant functor, functorial operation, fundamental group as a covariant functor, pointed topological space, group action, transitive action, fundamental group, universal covering, subgroup, cosets of a subgroup in a group 
12
Mod03 Lec12 The Monodromy Action
by nptelhrd 168 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goal of Lecture 12:
To understand how the fundamental group based at a point of the target of a covering map acts naturally on the fiber of the covering map over that point, the fiber being thought of as embedded inside the source of the covering map.
Keywords for Lecture 12:
Path, lifting of a path, uniquepathlifting property, Covering Homotopy Theorem, surjective local homeomorphism, universal covering space, injective group homomorphism, fundamental group, simply connected space, trivial group, fiber of a covering map, coset space, group action, orbit, orbit map, stabilizer subgroup, fibration. 
13
Mod03 Lec13 The Universal covering as a Hausdorff Topological Space
by nptelhrd 164 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 13:
* To ask the question as to why the fundamental group of a space occurs as a subgroup of automorphisms of its universal covering
* To define the universal covering space intuitively as a space of paths
* To give a natural topology on the space defined above and to show that this topology is Hausdorff
Keywords for Lecture 13:
Path, Fixedendpoint (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, subbase for a topology 
14
Mod03 Lec14 The Construction of the Universal Covering Map
by nptelhrd 270 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 14:
* In the previous lecture, the universal covering space was constructed for a given space as a Hausdorff topological space along with a natural map into the given space. In this lecture, we show that this natural map is a covering map
* It would follow that if the given space is locally arcwise connected and locally simply connected, then the same properties hold
for the universal covering space as well
Keywords for Lecture 14:
Path, Fixedendpoint (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, subbase for a topology, admissible neighborhood 
15
Mod03 Lec15A Completion of the Construction of the Universal Coveringl
by nptelhrd 141 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 15 Part A:
* In the previous couple of lectures, the universal covering space was constructed for a given space as a Hausdorff topological space along with a natural map into the given space. That natural map was shown to be a covering map. In this lecture, we show that the universal covering space we constructed is indeed simply connected and has a universal property
* We show that the universal covering space we have constructed is also a covering space for any other covering space. We further show that any covering space which is simply connected is homeomorphic to the universal covering space we have constructed. It follows that any two simply connected covering spaces thereby are not only just homeomorphic, but homeomorphic by a map that respects the covering projections, i.e., are isomorphic as covering spaces; in fact, even the isomorphism becomes unique if a point of the source and one of the target are fixed. These results show the universality of a simply connected covering space, which is why such a space is called "the" universal covering space
Keywords for Lecture 15 Part A:
Path, Fixedendpoint (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, subbase for a topology, admissible neighborhood, isomorphism of covering spaces, universal property 
16
Mod03 Lec15B Completion of the Construction of the Universal Covering: The Fundamental Group
by nptelhrd 122 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 15 Part B:
* In the previous couple of lectures, the universal covering space was constructed for a given space as a Hausdorff topological space along with a natural map into the given space. That natural map was shown to be a covering map. In the first part (part A) of this lecture, we showed that the universal covering space we constructed is indeed simply connected and has a universal property. In this lecture (part B), we show that we can naturally identify the fundamental group of the base space with a subgroup of selfisomorphisms of the universal covering space called the Deck Transformation Group
* For any covering space, we may define the socalled Deck Transformation Group. This is the subgroup of selfhomeomorphisms of the covering space that respect the covering projection map. If the covering space is the universal covering space, then the fundamental group of the base space (the space whose coverings we are concerned with) gets naturally identified with the deck transformation group.
Thus the fundamental group of the base acts on the universal covering via the socalled deck transformations. These act along the fibers of the covering projection map. This action is called the Monodromy Action
Keywords for Lecture 15 Part B:
Path, Fixedendpoint (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, subbase for a topology, admissible neighborhood, isomorphism of covering spaces, universal property, deck transformation, deck transformation group, monodromy action 
17
Mod04 Lec16 The Riemann Surface Structure on the Topological Covering of a Riemann Surface
by nptelhrd 500 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To extend the theory of topological coverings to that of holomorphic (complex analytic) coverings
* To show that any Riemann surface structure on the base space of a topological covering induces a Riemann surface structure on the covering space in such a way that the covering projection map is holomorpic. To achieve this using the technique of "pulling back charts from below"
* To see why the Riemann surface structure induced above is essentially unique
* In particular, we get a unique Riemann surface structure on the topological covering of a Riemann surface. The deck transformations therefore become holomorphic automorphisms of this Riemann surface structure
Keywords: Topological covering, holomorphic covering, admissible neighborhood, chart, pulling back charts by local homeomorphisms, locally biholomorphic, pulling back Riemann surface structures, holomorphicity or complex analyticity of continuous liftings, deck transformation 
18
Mod04 Lec17 Riemann Surfaces with Universal Covering the Plane or the Sphere
by nptelhrd 235 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To see how the topological quotient of the universal covering of a space by the deck transformation group (which is isomorphic to the fundamental group of the space) gives back the space
* In particular, the topological universal covering of a Riemann surface (which inherits a unique Riemann surface structure as shown in the previous lecture) modulo (or quotiented by or divided by) the fundamental group gives back the Riemann surface
* To see that nontrivial deck transformations are fixedpoint free
* To see why any Riemann surface with universal covering the Riemann sphere is isomorphic to the Riemann sphere itself
* To get a characterisation of discrete subgroups of the additive group of complex numbers
* To use the above characterisation to deduce that a Riemann surface with universal covering the plane has to be isomorphic to either the plane itself, or to a complex cylinder, or to a complex torus
Keywords: Holomorphic covering, holomorphic universal covering, group action on a topological space, orbit of a group action, equivalence relation defined by a group action, quotient by a group, topological quotient, quotient topology, quotient map, transitive action, deck transformation, open map, Riemann sphere, onepoint compactification, stereographic projection, Moebius transformation, unique lifting property, group of translations, admissible neighborhood, module, submodule, subgroup, discrete submodule, discrete subgroup 
19
Mod04 Lec18 Classifying Complex Cylinders Riemann Surfaces
by nptelhrd 172 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To characterize discrete subgroups of the additive group of complex numbers and to use this characterization to classify Riemann surfaces whose universal covering is the complex plane
* To see how the twisting of the universal covering space by an automorphism (of the universal covering space) leads to the identification of the fundamental group (of the base of the covering) with a conjugate of the deck transformation group of the original covering
* To show that the natural Riemann surface structures, on the quotient of the complex plane by the group of translations by integer multiples of a fixed nonzero complex number does not depend on that complex number; in other words that all such Riemann surfaces are isomorphic
Keywords: Uppertriangular matrix, complex plane, universal covering, deck transformation, abelian fundamental group, additive group of translations, module, submodule, discrete submodule, discrete subgroup 
20
Mod04 Lec19 Characterizing Moebius Transformations with a Single Fixed Point
by nptelhrd 255 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To realize that in order to study Riemann surfaces with abelian fundamental group and having universal covering the upper halfplane, one needs to first classify Moebius transformations in general and in particular study among those that are automorphisms of the upper halfplane
* To motivate how the classification of Moebius transformations can be done using two seemingly unrelated aspects: one of them being the set of fixed points in the extended complex plane and the other being the value of the square of the trace of the transformation. To show that these two aspects, though one of them is geometric while the other numeric, are in fact precisely related to each other
* To characterize Moebius transformations with exactly one fixed point in the extended complex plane as precisely those that are conjugate to a translation; to show that such transformations are also precisely the socalled parabolic transformations, where parabolicity is defined as the square of the trace being equal to four
Keywords: Upper halfplane, unit disc, abelian fundamental group, deck transformation group, Moebius transformation, universal covering, holomorphic automorphism, group isomorphism, linear fractional transformation, bilinear transformation, fixed point of a map, square of the trace of a Moebius transformation, parabolic Moebius transformations, translations, conjugation by a Moebius transformation, special linear group, projective special linear group, uppertriangular matri 
21
Mod04 Lec20 Characterizing Moebius Transformations with Two Fixed Points
by nptelhrd 129 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To analyze Moebius transformations with more than one fixed point in the extended complex plane
* To continue with the classification of Moebius transformations begun in the previous lecture by defining the notions of loxodromic, elliptic and hyperbolic Moebius transformations using the values of the square of the trace of the transformation
* To characterize geometrically the loxodromic, elliptic and hyperbolic Moebius transformations by showing that they can be conjugated by suitable Moebius transformations to multiplication by a complex number
* To show that the elliptic Moebius transformations are precisely those that are conjugate to a rotation about the origin
* To show that the hyperbolic Moebius transformations are precisely those that are conjugate to a real scaling
Keywords: Parabolic, elliptic, hyperbolic and loxodromic Moebius transformations, fixed point of a Moebius transformation, square of the trace of a Moebius transformation, translation, conjugation by a Moebius transformation, special linear group, projective special linear group 
22
Mod04 Lec21 Torsionfreeness of the Fundamental Group of a Riemann Surface
by nptelhrd 236 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To analyze what the conditions of loxodromicity, ellipticity or hyperbolicity imply for an automorphism of the upper halfplane, i.e., to characterize the automorphisms of the upper halfplane. This is required for the classification of Riemann surfaces with universal covering the upper halfplane
* To show that the fundamental group of a Riemann surface is torsion free i.e., that it has no nonidentity elements of finite order
* To show that the Deck transformations of the universal covering of a Riemann surface have to be either hyperbolic or parabolic in nature
* To deduce that the fundamental group of a Riemann surface is torsion free
Keywords: Moebius transformation, special linear group, projective special linear group, parabolic, elliptic, hyperbolic, loxodromic, fixed point, conjugation, translation, Riemann sphere, extended complex plane, upper halfplane, square of the trace (or trace square) of a Moebius transformation, torsionfree group, element of finite order of a group, torsion element of a group, universal covering, fundamental group, Deck transformations 
23
Mod04 Lec22 Characterizing Riemann Surface Structures on Quotients of the Upper Half
by nptelhrd 183 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To show that any Riemann Surface with nonzero abelian fundamental group and universal covering the upper halfplane has fundamental group isomorphic to the additive group of integers i.e., that it is cyclic of infinite order
* To classify the Riemann surface structures naturally inherited by annuli in the complex plane, and to show that there is a family of such distinct (i.e., nonisomorphic) structures parametrized by a real parameter
* To deduce that if a Riemann surface has fundamental group isomorphic to the product of the additive group of integers with itself, then it has to be isomorphic to a complex torus, and hence in particular that it has to necessarily be compact
Keywords: Upper halfplane, unit disc, annulus, torus, simply connected, abelian fundamental group, additive group, translation, deck transformation, Moebius transformation, universal covering, holomorphic automorphism, parabolic, elliptic, hyperbolic, loxodromic, fixed point, commuting Moebius transformations, conjugation, translation, universal covering, discrete subgroup, discrete submodule, generator of a group 
24
Mod04 Lec23 Classifying Annuli up to Holomorphic Isomorphism
by nptelhrd 301 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: To show that the various annuli with inner radii in the real open unit interval and with outer radius unity are all nonisomorphic as Riemann surfaces
Keywords: Upper halfplane, universal covering, fundamental group, deck transformation group, Moebius transformations, real special linear group, real projective (special) linear group, simply connected, biholomorphic map, holomorphic isomorphism, infinite cyclic group, parabolic Moebius transformation, hyperbolic Moebius transformation, fixed point, extended plane, abelian fundamental group, commuting Moebius transformations, commuting deck transformations, punctured unit disc, annulus, unique lifting property 
25
Mod05 Lec24 Orbits of the Integral Unimodular Group in the Upper HalfPlane
by nptelhrd 97 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To ask for a description of the set of holomorphic isomorphism classes of complex tori
* To state the Theorem on the Moduli of Elliptic Curves that not only answers the question above but also shows that the set above has a beautiful Godgiven geometry
* To see how the upper halfplane and the unimodular group (integral projective special linear group) enter into the discussion
* To use the theory of covering spaces to prove a part of the Theorem on the Moduli of Elliptic Curves, namely
that the set of holomorphic isomorphism classes of complex 1dimensional tori is in a natural bijective correspondence with the set of orbits of the unimodular group in the upper halfplane
Keywords: Real torus, complex torus, Moebius transformation, translation, abelian group, holomorphic universal covering, admissible neighborhood, fundamental group, deck transformation group, biholomorphism class (or) holomorphic isomorphism class, locally biholomorphic map, upper halfplane, projective special linear group, unimodular group, orbits of a group action, action of a subgroup, underlying fixed geometric structure, superimposed (or) overlying (or) extra geometric structure, variation of extra structure for a fixed underlying structure (or) moduli problem, quotient by a group, equivalence relation induced by a group action, universal property of the universal covering, unique lifting property, moduli of elliptic curves, forming the fundamental group is functorial 
26
Mod05 Lec25 Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
by nptelhrd 248 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To ask the question as to when the quotient of a space, by a subgroup of automorphisms (selfisomorphisms) of that space, becomes again a space with good properties. For example: when does the quotient of a Riemann surface, by a subgroup of holomorphic automorphisms, again become a Riemann surface?
* To define properly discontinuous (free) actions and note that they are fixedpointfree
* To see that the action of the Deck transformation group on the covering space is properly discontinuous
* To define Galois (or) Regular (or) Normal coverings and characterize them precisely as quotients by properly discontinuous actions
Keywords: upper halfplane, biholomorphism class (or) holomorphic isomorphism class, complex torus, projective special linear group, unimodular group, quotient by a subgroup of automorphisms, quotient by the Deck transformation group, orbits of a group action, quotient topology, properly discontinuous action, action without fixed points, transitive action, admissible neighborhood, Galois covering (or) Normal covering (or) Regular covering, covariant functor, normal subgroup, equivalence relation induced by a group action, open map, unique lifting property, covering homotopy theorem, Riemann sphere, stabilizer (or) isotropy subgroup, ramified (or) branched covering 
27
Mod05 Lec26 Local Actions at the Region of Discontinuity of a Kleinian Subgroup
by nptelhrd 273 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in

28
Mod05 Lec27 Quotients by Kleinian Subgroups give rise to Riemann Surfaces
by nptelhrd 117 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To see how the quotient of the region of discontinuity by a Kleinian subgroup of Moebius transformations is a union of Riemann surfaces
* To see how the quotients above are ramified (or) branched coverings of Riemann surfaces, with ramifications at the points with nontrivial isotropies (stabilizers)
* To see in detail how to get a complex coordinate chart at the image point of a point of ramification
Keywords: Upper halfplane, unimodular group, fixed point, projective special linear group, quotient by a subgroup of Moebius transformations, holomorphic automorphisms, extended plane, properly discontinuous action, stabilizer (or) isotropy subgroup, region of discontinuity of a subgroup of Moebius transformations, limit set of a subgroup of Moebius transformations, elliptic Moebius transformations, isolated point, discrete subset, Kleinian subgroup of Moebius transformations, quotient topology, ramification (or) branch points, ramified (or) branched covering, unramified (or) unbranched covering, branch cut, slit disc 
29
Mod05 Lec28 The Unimodular Group is Kleinian
by nptelhrd 80 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To see that a Kleinian subgroup of Moebius transformations is a discrete subspace of the space of all Moebius transformations and also that such a subgroup is either finite or countable as a set
* To define a subgroup of Moebius transformations to be Fuchsian if it maps a halfplane or a disc onto itself
* To see that a discrete Fuchsian subgroup is Kleinian. For example, the unimodular group is thus Kleinian
* To conclude using the results of the previous lecture that the quotient of the upper halfplane by the unimodular group is a Riemann surface
Keywords: Schwarz's Lemma, Riemann Mapping Theorem, properly discontinuous action, Kleinian subgroup of Moebius transformations, region of discontinuity of a subgroup of Moebius transformations, upper halfplane, unimodular group, projective special linear group, discrete subgroup of Moebius transformations, Fuchsian subgroup of Moebius transformations, holomorphic automorphisms, extended plane, stabilizer (or) isotropy subgroup, orbit map, second countable metric space, space of matrices, space of invertible matrices, space of determinant one matrices 
30
Mod06 Lec29 The Necessity of Elliptic Functions for the Classification of Complex Tori
by nptelhrd 130 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * In the last few lectures, we have shown that the quotient of the upper halfplane by the unimodular group has a natural Riemann surface structure. In order to show that this Riemann surface is isomorphic to the complex plane, we have to realize that we need to look for invariants for complex tori
* To motivate how the search for invariants for complex tori leads us to the study of doublyperiodic meromorphic functions (or) elliptic functions, the stereotype of which is given by the famous Weierstrass phefunction
Keywords: Upper halfplane, unimodular group, projective special linear group, set of orbits, quotient Riemann surface, lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, complex torus associated to a lattice, translation, jinvariant for a complex torus, Felix Klein, groupinvariant function, bounded entire function, Liouville's theorem, singularity of an analytic function, poles, meromorphic function, doublyperiodic meromorphic function (or) elliptic function, Karl Weierstrass, algebraic curve, elliptic curve, Weierstrass phefunction, Residue theorem, double pole 
31
Mod06 Lec30 The Uniqueness Property of the Weierstrass Phefunction
by nptelhrd 181 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To show that the Weierstrass Phefunction is the unique doublyperiodic meromorphic function (i.e., the unique elliptic function)
with residuezero double poles precisely at each point of the lattice and with constant term zero in the Laurent development at the origin
Keywords: Upper halfplane, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, isolated double pole, singular part of the Laurent expansion, deleted neighborhood, even function, entire function, algebraic elliptic cubic curve 
32
Mod06 Lec31 The First Order Degree Two Cubic Ordinary Differential Equation satisfied
by nptelhrd 166 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To show that the Weierstrass phefunction associated to a lattice satisfies a first order degree two cubic ODE
* The ODE mentioned above is the key to studying the geometry of the complex torus associated to the lattice and eventually leads to the classification (moduli) theory of complex tori
Keywords: Upper halfplane, invariants for complex tori, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, isolated double pole, singular part of the Laurent expansion, analytic part of the Laurent expansion, antiderivative for the Weierstrass phefunction, Identity theorem for power series or Laurent series, differentiating termbyterm and integrating termbyterm under uniform convergence, even function, odd function, entire elliptic functions are constants, algebraic elliptic cubic curve 
33
Mod06 Lec32 The Values of the Weierstrass Phe function at the Zeros of its Derivative
by nptelhrd 147 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To find the zeros of the derivative of the Weierstrass phefunction associated to a lattice
* To use the ODE established in the previous lecture to analyze the values of the Weierstrass phefunction at the zeros of its derivative and to show that these values are nonvanishing analytic (holomorphic) functions on the upper halfplane
* To introduce the notion of order for an elliptic function, namely the finite positive integer which is the number of times the function assumes any value in the extended complex plane (Riemann sphere)
Keywords: Upper halfplane, invariants for complex tori, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, simple zero, pole of order three, isolated double pole, Argument Principle, Residue theorem, order of an elliptic function, automorphic function (or) automorphic form, modular function (or) modular form, congruence mod two subgroup of the unimodular group, even function, odd function 
34
Mod07 Lec33 The Construction of a Modular Form of Weight Two on the Upper HalfPlane
by nptelhrd 96 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To construct an analytic function on the upper halfplane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruencemod2 subgroup of the unimodular group
Keywords: Upper halfplane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruencemod2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, universal covering, fundamental group, deck transformation group, holomorphic lifting, conjugation by a Moebius transformation, conjugate subgroup, additive group of translations, isomorphism of lattices, generator of a group, locally invertible map, locally biholomorphic map, conformal map, zeros of the derivative of the Weierstrass phefunction 
35
Mod07 Lec34 The Fundamental Functional Equations satisfied by the Modular Form of Weight
by nptelhrd 120 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * In the previous lecture, we constructed an analytic function on the upper halfplane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruencemod2 subgroup of the unimodular group. We ask what the effect of a general element of the unimodular group is on this weight two modular form
* To see that in order to answer the question above, it is enough to compute the effect under each of five unimodular elements representing preimages of the five nontrivial elements in the quotient by the congruencemod2 subgroup
* To see that the computations above result in five simple and beautiful functional equations satisfied by the weight two modular form
Keywords: Upper halfplane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruencemod2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phefunction 
36
M0d07 Lec35 The Weight Two Modular Form assumes Real Values on the Imaginary Axis
by nptelhrd 166 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * In the last few lectures, we constructed an analytic function on the upper halfplane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruencemod2 subgroup of the unimodular group. We saw that the effect of a general element of the unimodular group on this weight two modular form can be understood by just computing the effect under each of five unimodular elements representing preimages of the five nontrivial elements in the quotient by the congruencemod2 subgroup. We saw that these computations resulted in five simple and beautiful functional equations satisfied by the weight two modular form. Using these functional equations diligently, in the present and the forthcoming lectures, we obtain a suitable region in the upper halfplane on which the mapping properties of the weight two modular form can be studied. In this lecture we show that the weight two modular form assumes only real values on the imaginary axis, which will turn out to be a boundary for such a region
Keywords: Upper halfplane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruencemod2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phefunction 
37
Mod07 Lec36 The Weight Two Modular Form Vanishes at Infinity
by nptelhrd 59 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * In the last few lectures, we constructed an analytic function on the upper halfplane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruencemod2 subgroup of the unimodular group. We saw that the effect of a general element of the unimodular group on this weight two modular form can be understood by just computing the effect under each of five unimodular elements representing preimages of the five nontrivial elements in the quotient by the congruencemod2 subgroup. We saw that these computations resulted in five simple and beautiful functional equations satisfied by the weight two modular form. Using these functional equations, we want to find a suitable region in the upper halfplane on which the mapping properties of this weight two modular form may be easily studied. In the previous lecture we saw that the weight two modular form assumes only real values on the imaginary axis, which will turn out to be a boundary for such a region. In this lecture, we show that the weight two modular form vanishes at infinity
Keywords: Upper halfplane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruencemod2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phefunction, singular part of the Laurent expansion, pole of order two, uniform convergence, Weierstrass Mtest, removable singularity, entire function, periodic function, period of a function, singly periodic function, Liouville's theorem 
38
Mod07 Lec37A The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
by nptelhrd 32 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 37 Part A:
* In the last few lectures, we constructed an analytic function on the upper halfplane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruencemod2 subgroup of the unimodular group. We saw that the effect of a general element of the unimodular group on this weight two modular form can be understood by just computing the effect under each of five unimodular elements representing preimages of the five nontrivial elements in the quotient by the congruencemod2 subgroup. We saw that these computations resulted in five simple and beautiful functional equations satisfied by the weight two modular form. Using these functional equations, we want to find a suitable region in the upper halfplane on which the mapping properties of this weight two modular form may be easily studied.
* In the last couple of lectures we saw that the weight two modular form assumes only real values on the imaginary axis, which will turn out to be a boundary for such a region, and that the weight two modular form vanishes at infinity. In part A of this lecture we estimate that this vanishing at infinity is in fact an exponential decay. This estimation is actually a computation of the Fourier coefficient that matters most in the Fourier development of the weight two modular form which has period two. This estimation is critical for the study of the mapping properties which will be completed in part B of this lecture
Keywords for Lecture 37 Part A:
Upper halfplane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruencemod2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phefunction, singular part of the Laurent expansion, pole of order two, uniform convergence, Weierstrass Mtest, removable singularity, entire function, periodic function, period of a function, singly periodic function, Liouville's theorem, Fourier coefficient, Fourier development 
39
Mod07 Lec37B A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal
by nptelhrd 27 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 37 Part B:
* In the last few lectures, we constructed an analytic function on the upper halfplane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruencemod2 subgroup of the unimodular group. We saw how the effect of a general element of the unimodular group on this weight two modular form can be understood by just computing the effect under each of five unimodular elements representing preimages of the five nontrivial elements in the quotient by the congruencemod2 subgroup. We saw that these computations resulted in five simple and beautiful functional equations satisfied by the weight two modular form. Using these functional equations, we want to find a suitable region in the upper halfplane on which the mapping properties of this weight two modular form may be easily studied
* In the last couple of lectures we saw that the weight two modular form assumes only real values on the imaginary axis, which will turn out to be a boundary for such a region, and further that the weight two modular form vanishes at infinity. In part A of this lecture we estimated, by calculating the Fourier coefficient that mattered most, that this vanishing at infinity is in fact an exponential decay. This estimation is critical for the study of the mapping properties which we complete in part B of this lecture. We show that the weight two modular form assumes every value on the upper halfplane, and that when restricted to a suitable region it actually gives a holomorphic conformal isomorphism onto the upper halfplane with a continuous monotonic conformal extension to the boundary on the Riemann Sphere so that every real value and the point at infinity is also assumed precisely once
Keywords for Lecture 37 Part B:
Upper halfplane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruencemod2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phefunction, monotonic function, contour, winding number, Fourier coefficient, Fourier development 
40
Mod08 Lec38 The JInvariant of a Complex Torus (or) of an Algebraic Elliptic Curve
by nptelhrd 166 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To associate to each complex 1dimensional torus a complex number, called the jinvariant of the complex torus, which depends only on the holomorphic isomorphism class of the torus. This jinvariant will be shown in the forthcoming lectures to completely classify all complex tori
* In the previous unit of lectures, we constructed a weight two modular form on the upper halfplane and studied its mapping properties. In this lecture we use this weight two modular form to define a full modular form, i.e., a holomorphic function on the upper halfplane that is invariant under the action of the full unimodular group. It is this modular form that goes down to give the jinvariant function on the Riemann surface of holomorphic isomorphism classes of complex tori with underlying set consisting of the orbits of the unimodular group in the upper halfplane
Keywords: Upper halfplane, quotient by the unimodular group, orbits of the unimodular group, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, congruencemod2 normal subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower halfplane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, meromorphic functions are holomorphic functions to the Riemann Sphere, jinvariant of a complex torus (or) jinvariant of an algebraic elliptic curve 
41
Mod08 Lec39 A Fundamental Region in the Upper HalfPlane for the Elliptic Modular JInvariant
by nptelhrd 37 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals: * To show that there exists a complex torus with jinvariant any prescribed complex number; in other words, to show that the jinvariant is surjective as a map onto the complex numbers
* To use the functional equations satisfied by the weight two modular form as well as the mapping properties of that form, as studied in the previous unit of lectures, to find a suitable region in the upper halfplane where the mapping properties of the full modular form given by the jinvariant can be clearly studied
Keywords: Upper halfplane, quotient by the unimodular group, orbits of the unimodular group, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruencemod2 normal subgroup of the unimodular group, projective special linear group with mod2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower halfplane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, jinvariant of a complex torus (or) jinvariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group 
42
Mod08 Lec40 The Fundamental Region in the Upper HalfPlane for the Unimodular Group
by nptelhrd 31 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 40:
* To introduce the notion of a fundamental region for a groupinvariant surjective holomorphic map, for example for a holomorphic map that is invariant under the action of a subgroup of holomorphic automorphisms
* To describe a suitable region in the upper halfplane and to show that it is a fundamental region for the unimodular group
Keywords for Lecture 40:
Upper halfplane, quotient by the unimodular group, orbits of the unimodular group, representative of an orbit, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruencemod2 normal subgroup of the unimodular group, projective special linear group with mod2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower halfplane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, jinvariant of a complex torus (or) jinvariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group, ramified (or) branched covering, groupinvariant holomorphic maps, fundamental region for a groupinvariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, fundamental region associated to the quotient map defining a complex torus 
43
Mod08 Lec41 A Region in the Upper HalfPlane Meeting Each Unimodular Orbit Exactly Once
by nptelhrd 36 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 41:
* We prove the fact that a suitable region in the upper halfplane, which was described in the previous lecture and which was shown there to intersect each orbit of the unimodular group, meets each unimodular orbit at precisely one point. All this amounts to showing that the region is indeed a fundamental region for the unimodular group as claimed in the previous lecture
Keywords for Lecture 41:
Upper halfplane, quotient by the unimodular group, orbits of the unimodular group, representative of an orbit, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruencemod2 normal subgroup of the unimodular group, projective special linear group with mod2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower halfplane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, jinvariant of a complex torus (or) jinvariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group, ramified (or) branched covering, groupinvariant holomorphic maps, fundamental region for a groupinvariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, fundamental region associated to the quotient map defining a complex torus 
44
Mod08 Lec42 Moduli of Elliptic Curves
by nptelhrd 119 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 42:
* To complete the proof of the fact that a suitable region in the upper halfplane, described in the previous lecture and shown there to be a fundamental region for the unimodular group, is also a fundamental region for the elliptic modular jinvariant function
* In view of the above, we complete the proof of the theorem on the Moduli of Elliptic Curves: the natural Riemann surface structure, on the set of holomorphic isomorphism classes of complex 1dimensional tori (complex algebraic elliptic curves) identified with the set of orbits of the unimodular group in the upper halfplane, is holomorphically isomorphic via the jinvariant to the complex plane
Keywords for Lecture 42:
Upper halfplane, quotient by the unimodular group, orbits of the unimodular group, representative of an orbit, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruencemod2 normal subgroup of the unimodular group, projective special linear group with mod2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower halfplane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, jinvariant of a complex torus (or) jinvariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group, ramified (or) branched covering, groupinvariant holomorphic maps, fundamental region for a groupinvariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, Galois theory, Galois group, Galois extension of function fields of meromorphic functions on Riemann surfaces, symmetric group, Galois covering 
45
Mod09 Lec43 Punctured Complex Tori are Elliptic Algebraic Affine Plane
by nptelhrd 34 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 43:
* In this and the forthcoming lectures, our aim is to show that complex tori are algebraic, i.e., that they are actually elliptic algebraic projective curves. This is the reason that complex tori exhibit a rich geometry which involves a beautiful interplay between their complex analytic properties and the algebraic geometric and number theoretic properties of the elliptic curves they are associated to. It is a deep and nontrivial theorem that any compact Riemann surface is algebraic, so such Riemann surfaces exhibit a rich geometry as in the case of complex tori
* Towards the above end, in this lecture we begin by identifying any punctured complex torus with a plane curve in complex 2space. This plane curve is called the associated elliptic algebraic affine plane cubic curve. For this identification we make use of the Weierstrass phefunction associated to the complex torus, its derivative, their properties and the first order degree two cubic ordinary differential equation that they satisfy
Keywords for Lecture 43:
Upper halfplane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, triple pole (or) pole of order three, cubic equation, elliptic algebraic cubic curve, zeros of a polynomial equation, bicontinuous map (or) homeomorphism, open map, order of an elliptic function, elliptic integral, Argument principle, even function, odd function, analytic branch of the square root, simply connected, punctured torus, elliptic algebraic affine cubic plane curve, projective plane cubic curve, complex affine twospace, complex projective twospace, onepoint compactification by adding a point at infinity 
46
Mod09 Lec44 The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
by nptelhrd 40 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 44:
* To show that the graph of a holomorphic function is naturally a Riemann surface embedded in complex affine 2space
* To use the Implicit Function Theorem to show that the zero locus of a nonsingular polynomial in two complex variables is naturally a Riemann surface embedded in complex affine 2space
* To show that the elliptic algebraic affine cubic plane curve associated to a punctured complex torus, as described in the previous lecture, has a natural Riemann surface structure which is holomorphically isomorphic to the natural Riemann surface structure on the punctured complex torus (inherited from the complex torus)
Keywords for Lecture 44:
Upper halfplane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, triple pole (or) pole of order three, cubic equation, elliptic algebraic cubic curve, zeros of a polynomial equation, bicontinuous map (or) homeomorphism, open map, order of an elliptic function, elliptic integral, Argument principle, even function, odd function, analytic branch of the square root, simply connected, punctured torus, elliptic algebraic affine cubic plane curve, projective plane cubic curve, complex affine twospace, complex projective twospace, onepoint compactification by adding a point at infinity, Implicit function theorem, graph of a holomorphic function, nonsingular (or) smooth polynomial in two variables, zero locus of a polynomial, solving an implicit equation locally for an explicit function, nonsingular (or) smooth point of the zero locus of a polynomial in two variables, Hausdorff, second countable, connected component, nonsingular cubic polynomial, discriminant of a polynomial, cubic discriminant 
47
Mod09 Lec45A Complex Projective 2Space as a Compact Complex Manifold of Dimension Two
by nptelhrd 63 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 45A:
* In Part A of this lecture, we define complex projective 2space and show how it can be turned into a twodimensional complex manifold. In Part B, we show that any complex torus is holomorphically isomorphic to the natural Riemann surface structure on the associated elliptic algebraic cubic plane projective curve embedded in complex projective 2space
Keywords for Lecture 45A:
Upper halfplane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, triple pole (or) pole of order three, cubic equation, elliptic algebraic cubic curve, zeros of a polynomial equation, bicontinuous map (or) homeomorphism, open map, order of an elliptic function, elliptic integral, Argument principle, even function, odd function, analytic branch of the square root, simply connected, punctured torus, elliptic algebraic affine cubic plane curve, projective plane cubic curve, complex affine twospace, complex projective twospace, onepoint compactification by adding a point at infinity, Implicit function theorem, graph of a holomorphic function, nonsingular (or) smooth polynomial in two variables, zero locus of a polynomial, solving an implicit equation locally for an explicit function, nonsingular (or) smooth point of the zero locus of a polynomial in two variables, Hausdorff, second countable, connected component, nonsingular cubic polynomial, discriminant of a polynomial, cubic discriminant, homogeneous coordinates on projective 2space, punctured complex 3space, quotient topology, open map, complex twodimensional manifold (or) complex surface, complex onedimensional manifold (or) Riemann surface, complex coordinate chart in two complex variables, holomorphic (or) complex analytic function of two complex variables, glueing of Riemann surfaces, glueing of complex planes, zero set of a homogeneous polynomial in projective space, degree of homogeneity, Euler's formula for homogeneous functions, homogenisation, dehomogenisation, a complex curve is a real surface, a complex surface is a real 4manifold 
48
Mod09 Lec45B Complex Tori are the same as Elliptic Algebraic Projective Curves
by nptelhrd 65 views
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of Lecture 45B:
* In Part A of this lecture, we defined complex projective 2space and showed how it can be turned into a twodimensional complex manifold. In Part B, we show that any complex torus is holomorphically isomorphic to the natural Riemann surface structure on the associated elliptic algebraic cubic plane projective curve embedded in complex projective 2space
Keywords for Lecture 45B:
Upper halfplane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doublyperiodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phefunction associated to a lattice, ordinary differential equation satisfied by the Weierstrass phefunction, zeros of the derivative of the Weierstrass phefunction, pole of order two (or) double pole with residue zero, triple pole (or) pole of order three, cubic equation, elliptic algebraic cubic curve, zeros of a polynomial equation, bicontinuous map (or) homeomorphism, open map, order of an elliptic function, elliptic integral, Argument principle, even function, odd function, analytic branch of the square root, simply connected, punctured torus, elliptic algebraic affine cubic plane curve, projective plane cubic curve, complex affine twospace, complex projective twospace, onepoint compactification by adding a point at infinity, Implicit function theorem, graph of a holomorphic function, nonsingular (or) smooth polynomial in two variables, zero locus of a polynomial, solving an implicit equation locally for an explicit function, nonsingular (or) smooth point of the zero locus of a polynomial in two variables, Hausdorff, second countable, connected component, nonsingular cubic polynomial, discriminant of a polynomial, cubic discriminant, homogeneous coordinates on projective 2space, punctured complex 3space, quotient topology, open map, complex twodimensional manifold (or) complex surface, complex onedimensional manifold (or) Riemann surface, complex coordinate chart in two complex variables, holomorphic (or) complex analytic function of two complex variables, glueing of Riemann surfaces, glueing of complex planes, zero set of a homogeneous polynomial in projective space, degree of homogeneity, Euler's formula for homogeneous functions, homogenisation, dehomogenisation, a complex curve is a real surface, a complex surface is a real 4manifold