Introduction to Riemann Surfaces and Algebraic Curves

Video Lectures

Displaying all 48 video lectures.
I. Definitions and Examples of Riemann Surfaces
Lecture 1
The Idea of a Riemann Surface
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The Idea of a Riemann Surface
Goals of the Lecture: To develop a suitable definition of a structure of a Riemann Surface on a 2-dimensional surface that will allow us to carry out Complex Analysis (i.e., study of holomorphic (or) analytic functions) on the given surface. Topics: Complex plane, open set, analytic (or) holomorphic function, Cauchy-Riemann 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
Lecture 2
Simple Examples of Riemann Surfaces
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Simple Examples of Riemann Surfaces
Goals of the Lecture: - To see how the real plane can be equipped with two different Riemann Surface structures - To see how the real 2-dimensional sphere can be equipped with a Riemann surface structure. Topics: 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
Lecture 3
Maximal Atlases and Holomorphic Maps of Riemann Surfaces
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Maximal Atlases and Holomorphic Maps of Riemann Surfaces
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 Topics: Complex atlas, equivalent atlases, union of equivalent atlases, maximal atlas, holomorphic (or) analytic mapping between Riemann surfaces, isomorphism of Riemann surfaces
Lecture 4
Riemann Surface Structure on a Cylinder
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Riemann Surface Structure on a Cylinder
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 non-isomorphic structures exist. Topics: 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, Möbius transformation, group action on a set, orbits of an action, set modulo (action of) a group
Lecture 5
Riemann Surface Structure on a Torus
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Riemann Surface Structure on a Torus
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 non-isomorphic structures exist. Topics: 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, Möbius transformation, group action on a set, orbits of an action, set modulo (action of) a group
II. Classification of Riemann Surfaces
Lecture 6
Riemann Surface Structures on Cylinders and Tori via Covering Spaces
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Riemann Surface Structures on Cylinders and Tori via Covering Spaces
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 discussionKeywords 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
Lecture 7
Möbius Transformations Make up Fundamental Groups of Riemann Surfaces
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Möbius Transformations Make up Fundamental Groups of Riemann Surfaces
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 vice-versa, thus making it into a holomorphic covering map - Any Riemann surface is the quotient of the complex plane, or the upper half-plane, or the Riemann sphere by a suitable group of Möbius transformations isomorphic to the fundamental group of the Riemann surface - The study of any Riemann surface boils down to studying suitable subgroups of Möbius transformations Topics: 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, Möbius transformation
Lecture 8
Homotopy and the First Fundamental Group
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Homotopy and the First Fundamental Group
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 fixed-end-point (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 non-isomorphic 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. Topics: 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, fixed-end-point (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.
Lecture 9
A First Classification of Riemann Surfaces
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A First Classification of Riemann Surfaces
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 Topics: 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 half-plane, punctured plane, punctured unit disc, cylinder, complex torus, annulus, Riemann sphere, g-torus, coarse classification, fine classification, moduli problem, moduli theory, moduli space, number of moduli
III. Universal Covering Space Theory
Lecture 10
The Importance of the Path-lifting Property
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The Importance of the Path-lifting Property
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, unique-lifting property and uniqueness-of-lifting property - To understand the Covering Homotopy Theorem - To note that surjective local homeomorphisms have the uniqueness-of-lifting property - To note that a surjective local homeomorphism is a covering iff it has the path-lifting property - To deduce that covering maps have the unique path-lifting property Topics: Lifting of a map, lifting of a path, lifting property, unique-lifting property, uniqueness-of-lifting property, Covering Homotopy Theorem, local homeomorphism, unique path-lifting property, existence of lifting, fundamental group, universal covering
Lecture 11
Fundamental groups as Fibres of the Universal covering Space
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Fundamental groups as Fibres of the Universal covering Space
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 base-point-preserving continuous maps, to the category whose objects are groups and whose morphisms are group homomorphisms - To deduce the path-lifting 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 Topics: Covering Homotopy Theorem, stationary homotopy, lifting of a homotopy, path-lifting 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
Lecture 12
The Monodromy Action
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The Monodromy Action
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. Topics: Path, lifting of a path, unique-path-lifting 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.
Lecture 13
The Universal covering as a Hausdorff Topological Space
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The Universal covering as a Hausdorff Topological Space
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 Topics: Path, Fixed-end-point (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, sub-base for a topology
Lecture 14
The Construction of the Universal Covering Map
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The Construction of the Universal Covering Map
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 Topics: Path, Fixed-end-point (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, sub-base for a topology, admissible neighborhood
Lecture 15
Universality of the Universal Covering
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Universality of the Universal Covering
Goals of the Lecture: - 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 Topics: Path, Fixed-end-point (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, sub-base for a topology, admissible neighborhood, isomorphism of covering spaces, universal property
Lecture 16
The Fundamental Group of the base as the Deck Transformation Group
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The Fundamental Group of the base as the Deck Transformation Group
Goals of the Lecture: - 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 self-isomorphisms of the universal covering space called the Deck Transformation Group - For any covering space, we may define the so-called Deck Transformation Group. This is the subgroup of self-homeomorphisms 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 so-called deck transformations. These act along the fibers of the covering projection map. This action is called the Monodromy Action Topics: Path, Fixed-end-point (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, sub-base for a topology, admissible neighborhood, isomorphism of covering spaces, universal property, deck transformation, deck transformation group, monodromy action
IV. Classifying Moebius Transformations and Deck Transformations
Lecture 17
The Riemann Surface Structure on the Topological Covering of a Riemann Surface
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The Riemann Surface Structure on the Topological Covering of a Riemann Surface
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 Topics: 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
Lecture 18
Riemann Surfaces with Universal Covering the Plane or the Sphere
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Riemann Surfaces with Universal Covering the Plane or the Sphere
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 fixed-point free - To see why any Riemann surface with universal covering the Riemann sphere is isomorphic to the Riemann sphere itself - To get a characterization 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 Topics: 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, one-point compactification, stereographic projection, Möbius transformation, unique lifting property, group of translations, admissible neighborhood, module, submodule, subgroup, discrete submodule, discrete subgroup
Lecture 19
Classifying Complex Cylinders Riemann Surfaces
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Classifying Complex Cylinders Riemann Surfaces
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 Topics: Upper-triangular matrix, complex plane, universal covering, deck transformation, abelian fundamental group, additive group of translations, module, submodule, discrete submodule, discrete subgroup
Lecture 20
Möbius Transformations with a Single Fixed Point
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Möbius Transformations with a Single Fixed Point
Goals: - To realize that in order to study Riemann surfaces with abelian fundamental group and having universal covering the upper half-plane, one needs to first classify Möbius transformations in general and in particular study among those that are automorphisms of the upper half-plane - To motivate how the classification of Möbius 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 Möbius 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 so-called parabolic transformations, where parabolicity is defined as the square of the trace being equal to four Topics: Upper half-plane, unit disc, abelian fundamental group, deck transformation group, Möbius transformation, universal covering, holomorphic automorphism, group isomorphism, linear fractional transformation, bilinear transformation, fixed point of a map, square of the trace of a Möbius transformation, parabolic Möbius transformations, translations, conjugation by a Möbius transformation, special linear group, projective special linear group, upper-triangular matrix
Lecture 21
Möbius Transformations with Two Fixed Points
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Möbius Transformations with Two Fixed Points
Goals: - To analyze Möbius transformations with more than one fixed point in the extended complex plane - To continue with the classification of Möbius transformations begun in the previous lecture by defining the notions of loxodromic, elliptic and hyperbolic Möbius transformations using the values of the square of the trace of the transformation - To characterize geometrically the loxodromic, elliptic and hyperbolic Möbius transformations by showing that they can be conjugated by suitable Möbius transformations to multiplication by a complex number - To show that the elliptic Möbius transformations are precisely those that are conjugate to a rotation about the origin - To show that the hyperbolic Möbius transformations are precisely those that are conjugate to a real scaling Topics: Parabolic, elliptic, hyperbolic and loxodromic Möbius transformations, fixed point of a Möbius transformation, square of the trace of a Möbius transformation, translation, conjugation by a Möbius transformation, special linear group, projective special linear group
Lecture 22
Torsion-freeness of the Fundamental Group of a Riemann Surface
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Torsion-freeness of the Fundamental Group of a Riemann Surface
Goals: - To analyze what the conditions of loxodromicity, ellipticity or hyperbolicity imply for an automorphism of the upper half-plane, i.e., to characterize the automorphisms of the upper half-plane. This is required for the classification of Riemann surfaces with universal covering the upper half-plane - To show that the fundamental group of a Riemann surface is torsion free i.e., that it has no non-identity 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 Topics: Möbius transformation, special linear group, projective special linear group, parabolic, elliptic, hyperbolic, loxodromic, fixed point, conjugation, translation, Riemann sphere, extended complex plane, upper half-plane, square of the trace (or trace square) of a Möbius transformation, torsion-free group, element of finite order of a group, torsion element of a group, universal covering, fundamental group, Deck transformations
Lecture 23
Characterizing Riemann Surface Structures on Quotients of the Upper Half
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Characterizing Riemann Surface Structures on Quotients of the Upper Half
Goals: - To show that any Riemann Surface with nonzero abelian fundamental group and universal covering the upper half-plane 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., non-isomorphic) 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 Topics: Upper half-plane, unit disc, annulus, torus, simply connected, abelian fundamental group, additive group, translation, deck transformation, Möbius transformation, universal covering, holomorphic automorphism, parabolic, elliptic, hyperbolic, loxodromic, fixed point, commuting Möbius transformations, conjugation, translation, universal covering, discrete subgroup, discrete submodule, generator of a group
Lecture 24
Classifying Annuli up to Holomorphic Isomorphism
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Classifying Annuli up to Holomorphic Isomorphism
Goals: To show that the various annuli with inner radii in the real open unit interval and with outer radius unity are all non-isomorphic as Riemann surfaces Topics: Upper half-plane, universal covering, fundamental group, deck transformation group, Möbius transformations, real special linear group, real projective (special) linear group, simply connected, biholomorphic map, holomorphic isomorphism, infinite cyclic group, parabolic Möbius transformation, hyperbolic Möbius transformation, fixed point, extended plane, abelian fundamental group, commuting Möbius transformations, commuting deck transformations, punctured unit disc, annulus, unique lifting property
V. The Riemann Surface Structure on the Quotient of the Upper Half-Plane by the Unimodular Group
Lecture 25
Orbits of the Integral Unimodular Group in the Upper Half-Plane
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Orbits of the Integral Unimodular Group in the Upper Half-Plane
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 God-given geometry - To see how the upper half-plane 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 1-dimensional tori is in a natural bijective correspondence with the set of orbits of the unimodular group in the upper half-plane Topics: Real torus, complex torus, Möbius transformation, translation, abelian group, holomorphic universal covering, admissible neighborhood, fundamental group, deck transformation group, biholomorphism class (or) holomorphic isomorphism class, locally biholomorphic map, upper half-plane, 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
Lecture 26
Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
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Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
Goals: - To ask the question as to when the quotient of a space, by a subgroup of automorphisms (self-isomorphisms) 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 fixed-point-free - 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 Topics: upper half-plane, 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
Lecture 27
Local Actions at the Region of Discontinuity of a Kleinian Subgroup
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Local Actions at the Region of Discontinuity of a Kleinian Subgroup
Lecture 28
Quotients by Kleinian Subgroups give rise to Riemann Surfaces
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Quotients by Kleinian Subgroups give rise to Riemann Surfaces
Goals: - To see how the quotient of the region of discontinuity by a Kleinian subgroup of Möbius 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 Topics: Upper half-plane, unimodular group, fixed point, projective special linear group, quotient by a subgroup of Möbius transformations, holomorphic automorphisms, extended plane, properly discontinuous action, stabilizer (or) isotropy subgroup, region of discontinuity of a subgroup of Möbius transformations, limit set of a subgroup of Möbius transformations, elliptic Möbius transformations, isolated point, discrete subset, Kleinian subgroup of Möbius transformations, quotient topology, ramification (or) branch points, ramified (or) branched covering, unramified (or) unbranched covering, branch cut, slit disc
Lecture 29
The Unimodular Group is Kleinian
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The Unimodular Group is Kleinian
Goals: - To see that a Kleinian subgroup of Möbius transformations is a discrete subspace of the space of all Möbius transformations and also that such a subgroup is either finite or countable as a set - To define a subgroup of Möbius transformations to be Fuchsian if it maps a half-plane 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 half-plane by the unimodular group is a Riemann surface Topics: Schwarz's Lemma, Riemann Mapping Theorem, properly discontinuous action, Kleinian subgroup of Möbius transformations, region of discontinuity of a subgroup of Möbius transformations, upper half-plane, unimodular group, projective special linear group, discrete subgroup of Möbius transformations, Fuchsian subgroup of Möbius 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
VI. Doubly-Periodic Meromorphic (or) Elliptic Functions
Lecture 30
The Necessity of Elliptic Functions for the Classification of Complex Tori
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The Necessity of Elliptic Functions for the Classification of Complex Tori
Goals: - In the last few lectures, we have shown that the quotient of the upper half-plane 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 doubly-periodic meromorphic functions (or) elliptic functions, the stereotype of which is given by the famous Weierstrass phe-function Topics: Upper half-plane, 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, j-invariant for a complex torus, Felix Klein, group-invariant function, bounded entire function, Liouville's theorem, singularity of an analytic function, poles, meromorphic function, doubly-periodic meromorphic function (or) elliptic function, Karl Weierstrass, algebraic curve, elliptic curve, Weierstrass phe-function, Residue theorem, double pole
Lecture 31
The Uniqueness Property of the Weierstrass Phe-function
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The Uniqueness Property of the Weierstrass Phe-function
Goals: - To show that the Weierstrass Phe-function is the unique doubly-periodic meromorphic function (i.e., the unique elliptic function) with residue-zero double poles precisely at each point of the lattice and with constant term zero in the Laurent development at the origin Topics: Upper half-plane, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, isolated double pole, singular part of the Laurent expansion, deleted neighborhood, even function, entire function, algebraic elliptic cubic curve
Lecture 32
The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function
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The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function
Goals: - To show that the Weierstrass phe-function 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 Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, isolated double pole, singular part of the Laurent expansion, analytic part of the Laurent expansion, antiderivative for the Weierstrass phe-function, Identity theorem for power series or Laurent series, differentiating term-by-term and integrating term-by-term under uniform convergence, even function, odd function, entire elliptic functions are constants, algebraic elliptic cubic curve
Lecture 33
The Values of the Weierstrass Phe function at the Zeros of its Derivative
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The Values of the Weierstrass Phe function at the Zeros of its Derivative
Goals: - To find the zeros of the derivative of the Weierstrass phe-function associated to a lattice - To use the ODE established in the previous lecture to analyze the values of the Weierstrass phe-function at the zeros of its derivative and to show that these values are nonvanishing analytic (holomorphic) functions on the upper half-plane - 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) Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, 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
VII. A Form Modular for the Congruence / Mod 2: Subgroup of the Unimodular Group on the Upper Half-Plane
Lecture 34
The Construction of a Modular Form of Weight Two on the Upper Half-Plane
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The Construction of a Modular Form of Weight Two on the Upper Half-Plane
Goals: - To construct an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 subgroup of the unimodular group Topics: Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 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 Möbius 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 phe-function
Lecture 35
The Fundamental Functional Equations satisfied by the Modular Form of Weight
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The Fundamental Functional Equations satisfied by the Modular Form of Weight
Goals: - In the previous lecture, we constructed an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 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 pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 subgroup - To see that the computations above result in five simple and beautiful functional equations satisfied by the weight two modular formKeywords: Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phe-function
Lecture 36
The Weight Two Modular Form assumes Real Values on the Imaginary Axis
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The Weight Two Modular Form assumes Real Values on the Imaginary Axis
Goals: - In the last few lectures, we constructed an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 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 pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 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 half-plane 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 Topics: Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phe-function
Lecture 37
The Weight Two Modular Form Vanishes at Infinity
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The Weight Two Modular Form Vanishes at Infinity
Goals: - In the last few lectures, we constructed an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 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 pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 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 half-plane 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 Topics: Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phe-function, singular part of the Laurent expansion, pole of order two, uniform convergence, Weierstrass M-test, removable singularity, entire function, periodic function, period of a function, singly periodic function, Liouville's theorem
Lecture 38
The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
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The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
Goals: - In the last few lectures, we constructed an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 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 pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 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 half-plane 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 (Lecture 38) 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 (Lecture 39) of this lecture Topics: Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phe-function, singular part of the Laurent expansion, pole of order two, uniform convergence, Weierstrass M-test, removable singularity, entire function, periodic function, period of a function, singly periodic function, Liouville's theorem, Fourier coefficient, Fourier development
Lecture 39
Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
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Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
Goals: - In the last few lectures, we constructed an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 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 pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 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 half-plane 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 (Lecture 38) 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 (Lecture 39) of this lecture. We show that the weight two modular form assumes every value on the upper half-plane, and that when restricted to a suitable region it actually gives a holomorphic conformal isomorphism onto the upper half-plane 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 Topics: Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phe-function, monotonic function, contour, winding number, Fourier coefficient, Fourier development
VIII. The Elliptic Modular J-invariant and the Moduli of Complex 1-dimensional Tori (or) Elliptic Curves
Lecture 40
The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
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The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
Goals: - To associate to each complex 1-dimensional torus a complex number, called the j-invariant of the complex torus, which depends only on the holomorphic isomorphism class of the torus. This j-invariant 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 half-plane 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 half-plane that is invariant under the action of the full unimodular group. It is this modular form that goes down to give the j-invariant 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 half-plane Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, congruence-mod-2 normal subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, 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, j-invariant of a complex torus (or) j-invariant of an algebraic elliptic curve
Lecture 41
Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
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Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
Goals: - To show that there exists a complex torus with j-invariant any prescribed complex number; in other words, to show that the j-invariant 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 half-plane where the mapping properties of the full modular form given by the j-invariant can be clearly studied Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, 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, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant 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.
Lecture 42
The Fundamental Region in the Upper Half-Plane for the Unimodular Group
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The Fundamental Region in the Upper Half-Plane for the Unimodular Group
Goals: - To introduce the notion of a fundamental region for a group-invariant 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 half-plane and to show that it is a fundamental region for the unimodular group. Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, 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, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant 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, group-invariant holomorphic maps, fundamental region for a group-invariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, fundamental region associated to the quotient map defining a complex torus.
Lecture 43
A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
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A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
Goals of the Lecture: - We prove the fact that a suitable region in the upper half-plane, 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 Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, 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, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant 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, group-invariant holomorphic maps, fundamental region for a group-invariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, fundamental region associated to the quotient map defining a complex torus.
Lecture 44
Moduli of Elliptic Curves
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Moduli of Elliptic Curves
Goals of Lecture 42: - To complete the proof of the fact that a suitable region in the upper half-plane, 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 j-invariant 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 1-dimensional tori (complex algebraic elliptic curves) identified with the set of orbits of the unimodular group in the upper half-plane, is holomorphically isomorphic via the j-invariant to the complex plane Topics: Upper half-plane, 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, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, 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, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant 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, group-invariant holomorphic maps, fundamental region for a group-invariant 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.
IX. Complex 1-dimensional Tori are Projective Algebraic Elliptic Curves
Lecture 45
Punctured Complex Tori are Elliptic Algebraic Affine Plane
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Punctured Complex Tori are Elliptic Algebraic Affine Plane
Goals of the Lecture: - 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 2-space. This plane curve is called the associated elliptic algebraic affine plane cubic curve. For this identification we make use of the Weierstrass phe-function associated to the complex torus, its derivative, their properties and the first order degree two cubic ordinary differential equation that they satisfy Topics: Upper half-plane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, zeros of the derivative of the Weierstrass phe-function, 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 two-space, complex projective two-space, one-point compactification by adding a point at infinity
Lecture 46
The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
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The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
Goals: - To show that the graph of a holomorphic function is naturally a Riemann surface embedded in complex affine 2-space - 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 2-space - 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) Topics: Upper half-plane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, zeros of the derivative of the Weierstrass phe-function, 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 two-space, complex projective two-space, one-point 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.
Lecture 47
Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
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Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
Goals: - In Part A (Lec 47) of this lecture, we define complex projective 2-space and show how it can be turned into a two-dimensional complex manifold. In Part B (Lec 48), 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 2-space Topics: Upper half-plane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, zeros of the derivative of the Weierstrass phe-function, 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 two-space, complex projective two-space, one-point 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 2-space, punctured complex 3-space, quotient topology, open map, complex two-dimensional manifold (or) complex surface, complex one-dimensional 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 4-manifold
Lecture 48
Complex Tori are the same as Elliptic Algebraic Projective Curves
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Complex Tori are the same as Elliptic Algebraic Projective Curves
Goals: - In Part A of this lecture (Lec 47), we defined complex projective 2-space and showed how it can be turned into a two-dimensional complex manifold. In Part B (Lec 48), 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 2-space Topics: Upper half-plane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, zeros of the derivative of the Weierstrass phe-function, 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 two-space, complex projective two-space, one-point 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 2-space, punctured complex 3-space, quotient topology, open map, complex two-dimensional manifold (or) complex surface, complex one-dimensional 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 4-manifold