Modular Forms, Fundamental Domain Question

[binbagsss]

Homework Statement

My question is below it makes more sense there, after I have gave my interpretation of the definition of the fundamental domain to confirm my understanding

Homework Equations

The fundamental domain is defined as:

1) Every point in H is equivalent to a point in F.

2) Points in the interior of F are SL2(Z)-inequivalent: Only identifications are via • For ##|Re(τ )| = \pm 1/ 2 ## via ## τ → τ ± 1 ##• On the arc ##|τ | = 1 ##via ## τ → −1/τ ## .And is given by :

## -½ \leq u \geq ½ ##

## |t| \leq 1 ##

The Attempt at a Solution

- Criteria doesn’t specify one-to-one, some elements may map once, more than once, as long as there is at least one from H to F, and, each point in F does not necessarily need to be mapped to- does it happen that each is by any chance ?

- Definition of ‘boundary’ with respect to the criteria, i.e. It is clear all of ##u=-1/2## and ##u=1/2## can be mapped to each other via T and the arc for negative and positive ##u## can be mapped to one another via ##S##
-On this, I assume the definition of the 'boundary' relative to the fundamental domain is not that there must exists some point in the exterior that maps to it (or vice versa), since the boundary itself covers that, however, it is not disallowed and so a point not in F may map to the boundary, and then, by the transitive property of an equivalence relationship, it is not only equivalent to that point, but to one more point- its reflection in the y-axis (either via T or S).

for eg I have this question:

And the answer is to just translate negatively five times, yielding (2i+1)/5 clearly leaving a magnitude of ##1## and so on the arc of F. However the question asks ‘what is the equivalent point in F’ (implying only one such point) but via S we are still on the boundary, a reflection in y and so there are actually two equivalent points in F (2i+1)/5 and (2i-1)/5Many thanks

 

[mighty2000]

for your question and interpretation of the definition of the fundamental domain. It seems like you have a good understanding of the criteria for the fundamental domain and how points are mapped between H and F. To answer your question, yes, each point in F does not necessarily need to be mapped to, as long as there is at least one point in F that is mapped from H. This means that some points in F may have multiple equivalent points in H, but as long as there is at least one mapping, it satisfies the criteria for the fundamental domain.

As for the definition of the boundary, you are correct in that it is not necessary for a point in the exterior to map to the boundary, but it is not disallowed either. This means that there may be points in the exterior that map to the boundary, and they would be considered equivalent to points on the boundary. This is where the transitive property of an equivalence relationship comes into play.

In regards to your example question, it is true that there are actually two equivalent points in F for (2i+1)/5 and (2i-1)/5, as they are reflections of each other in the y-axis. However, for the purposes of the question, it is asking for the equivalent point in F, so either one of those points would be a correct answer. It is important to note that the fundamental domain is not necessarily unique, as there may be multiple equivalent points for a given point in H.

Overall, it seems like you have a good understanding of the fundamental domain and its criteria. Keep up the good work!