Skip to main content

LUP Student Papers

LUND UNIVERSITY LIBRARIES

The Modular Function λ and its Universal Covering of the Twice Punctured Plane

Ellmer, Isak LU (2023) In Bachelor's Theses in Mathematical Sciences MATK11 20231
Mathematics (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:
author
Ellmer, Isak LU
supervisor
organization
course
MATK11 20231
year
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}},
}