Today I have a proof I wrote recently, of a relatively fundamental fact: principal congruence subgroups of the modular group contain no elliptic elements.

The proof is available as a PDF or below the fold.

Theorem. The congruence subgroup contains elliptic elements if and only if , the modular group.

Proof. In three parts.

1. It is easy to see that contains elliptic elements. For instance, the following matrix is elliptic, with fixed points :
   

   

2. Assume that . Let:
   

   

   The characteristic polynomial of is . Thus, using the fact that , the discriminant is:
   

   

   To show that is elliptic, we aim to establish that its fixed points are two distinct complex conjugate points, ie, that its discriminant is negative. For this to be true it is necessary and sufficient for the trace of to be between -2 and 2. Since all variables here are integers, we make use of the stronger statement:
   

   

   This implies that or . The only possibilities are that and .

3. Now we suppose that , and return to:
   

   

   This implies that , a contradiction.

Therefore, the only principal congruence subgroup that contains elliptic elements is the modular group itself. QED

The implications of this fact have yet to be fully realized. The question is still open as to whether it is possible to generate all the congruence subgroups of a given level, but the notion of decomposing subgroup generators into constituent generators, then manipulating those one by one is an interesting prospect.

Here are my badges — so far:

Seriously, though, Venn diagrams are a concise and satisfyingly geeky way to convey complex information. Instead of writing a paragraph about how some of the people in my life are geeks, and some of the people in my life are artsy, and explaining that those groups overlap , and further that some subset of the overlapping region will find my webcomic project viscerally satisfying, I can draw three circles in a square and you know exactly what I’m talking about.

And Venn diagrams are versatile! You don’t even have to mouth off to anyone any more if you can just scribble some pithy circles on a bar napkin. Draw three interlocking circles titled “smart people”, “sexy people” and “talented people”, then enclose them in a circle labeled “people who aren’t you” within a larger rectangle labeled “everybody”. See? It’s that easy!

Yes, the Venn diagram could be the key to an new, utopic, entirely non-verbal society. Keep your eye on it!