Modular Forms-Fundamental Domain

  • I
  • Thread starter binbagsss
  • Start date
  • #1
1,206
9

Main Question or Discussion Point

Apologies if this is a stupid question, but is the fundamental domain unique?

And what exactly is the definition of a cusp- a quick google tells me it is 'where two curves meet', so looking at the fundamental domain,I would say ##\omega=\exp^{\frac{2\pi i}{ 3}} ## and ##\omega*## are?

thanks
 

Answers and Replies

  • #3
mathwonk
Science Advisor
Homework Helper
10,822
989
no it is not unique. any image of one fundamental domain by any element of the modular group is another fundmental domain. in the illustration posted above in post #2, every curvilinear polygon shown is a fundamental domain. the infinite shaded one is the usual choice simply because it is more symmetric, but the other infinite ones on either side of it are also fundamental domains obtained from it by translation. the bounded ones nearer the x - axis look a different shape but that is because they are images of the standard one by group elements that change shape, i.e. elements like z--> -1/z.
 
Last edited:

Related Threads on Modular Forms-Fundamental Domain

Replies
7
Views
820
  • Last Post
Replies
6
Views
521
  • Last Post
Replies
8
Views
5K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
5
Views
359
Replies
2
Views
615
  • Last Post
Replies
1
Views
1K
Top