The Modular Function λ and its Universal Covering of the Twice Punctured Plane
(2023) In Bachelor's Theses in Mathematical Sciences MATK11 20231Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- In this thesis we introduce the notion of elliptic functions, and in particular Weierstrass' Elliptic Function, in order to define the modular function $\lambda$ on the upper half plane. We establish a group ismorphism between the unimodular transformations and $\Aut{\C\setminus\{0,1\}}$. In particular this gives two functional equations involving $\lambda$, and with these equations we show that $\lambda$ is real along the boundary of its fundamental domain. Integrating along this boundary and using the argument principle we deduce that the right side of its fundamental domain is mapped bijectively onto the upper half plane. $\lambda$ is then shown to be locally injective and surjective. We conclude that $\lambda$ possesses the path... (More)
- In this thesis we introduce the notion of elliptic functions, and in particular Weierstrass' Elliptic Function, in order to define the modular function $\lambda$ on the upper half plane. We establish a group ismorphism between the unimodular transformations and $\Aut{\C\setminus\{0,1\}}$. In particular this gives two functional equations involving $\lambda$, and with these equations we show that $\lambda$ is real along the boundary of its fundamental domain. Integrating along this boundary and using the argument principle we deduce that the right side of its fundamental domain is mapped bijectively onto the upper half plane. $\lambda$ is then shown to be locally injective and surjective. We conclude that $\lambda$ possesses the path lifting property and that $\lambda$ gives a covering of $\C\setminus\{0,1\}$. (Less)
- Popular Abstract
- Functions which are differentiable in the complex sense have nice properties that real differentiable functions in general do not have. One such property is that a function which is differentiable once is differentiable infinitely many times. Another property is Liouville's Theorem. It states that a non-constant complex differentiable function defined on the whole complex plane needs to attain values that are arbitrarily large. A severe strengthening of Liouville's Theorem is Picard's Little Theorem which is a result of this thesis. Instead of only requiring that such a function attains arbitrarily large values the theorem remarkably states that the function must attain every single value with possibly one exception.
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9122506
- author
- Ellmer, Isak LU
- supervisor
-
- Yacin Ameur LU
- organization
- course
- MATK11 20231
- year
- 2023
- type
- M2 - Bachelor Degree
- subject
- keywords
- Complex Analysis, Covering Spaces, Modular Group, Modular Function, Picard's Little Theorem
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFMA-4145-2023
- ISSN
- 1654-6229
- other publication id
- 2023:K15
- language
- English
- id
- 9122506
- date added to LUP
- 2025-06-27 15:54:27
- date last changed
- 2025-06-27 15:54:27
@misc{9122506, abstract = {{In this thesis we introduce the notion of elliptic functions, and in particular Weierstrass' Elliptic Function, in order to define the modular function $\lambda$ on the upper half plane. We establish a group ismorphism between the unimodular transformations and $\Aut{\C\setminus\{0,1\}}$. In particular this gives two functional equations involving $\lambda$, and with these equations we show that $\lambda$ is real along the boundary of its fundamental domain. Integrating along this boundary and using the argument principle we deduce that the right side of its fundamental domain is mapped bijectively onto the upper half plane. $\lambda$ is then shown to be locally injective and surjective. We conclude that $\lambda$ possesses the path lifting property and that $\lambda$ gives a covering of $\C\setminus\{0,1\}$.}}, author = {{Ellmer, Isak}}, issn = {{1654-6229}}, language = {{eng}}, note = {{Student Paper}}, series = {{Bachelor's Theses in Mathematical Sciences}}, title = {{The Modular Function λ and its Universal Covering of the Twice Punctured Plane}}, year = {{2023}}, }