Modular Forms-Fundamental Domain

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In summary, the fundamental domain for modular forms is not unique, as any image of one fundamental domain by any element of the modular group is another fundamental domain. A cusp is a point where two curves meet, and for modular forms, the cusps are all those points along the X axis. The Wikipedia link provided further explains the concept of cusps in mathematics.
  • #1
binbagsss
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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
 
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  • #2
Here is a link:
https://en.wikipedia.org/wiki/Cusp_(singularity)

For modular forms, the cusps would be all those points along the X axis.
320px-ModularGroup-FundamentalDomain.svg.png
 

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  • #3
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.
 
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1. What are modular forms?

Modular forms are complex-valued functions that are defined on the upper half-plane of complex numbers, and satisfy certain transformation properties under a subgroup of the modular group.

2. What is the fundamental domain of modular forms?

The fundamental domain of modular forms is the smallest region in the upper half-plane that contains all the points that are mapped to each other under the action of the modular group. It is used to study the behavior of modular forms and to calculate their properties.

3. How are modular forms related to number theory?

Modular forms play a crucial role in number theory, particularly in the study of elliptic curves and the proof of Fermat's Last Theorem. They are also used in the study of quadratic forms, modular equations, and other topics in number theory.

4. What are some applications of modular forms?

Modular forms have applications in various areas of mathematics, including algebraic geometry, algebraic topology, and representation theory. They are also used in physics, specifically in string theory and conformal field theory.

5. What is the significance of the concept of "modularity" in mathematics?

The concept of modularity, which is closely related to the theory of modular forms, has been an important tool in solving long-standing mathematical problems, such as Fermat's Last Theorem. It also has connections to other areas of mathematics, including geometry, topology, and algebraic number theory.

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