Shimura curve Gamma₀³⁵(1)

Update July 20, 2005: I am copying pages from hverrill.net to www.math.lsu.edu/~verrill, and find that I never actually added a link to this page previously, though it was written probably about 4 years ago.

This program was written to illustarate some joint work doing with David Kohel, which is now published as "fundamental domains for Shimura curves", D. Kohel and H. Verrill, Journal de Theorie des Nombres de Bordeaux 15 (2003), 205-222, which you can also find here.

The above picture shows a fundamental domain for Gamma₀³⁵(1). Edges of the same colour are identified under the action of this group. I'll write down a list of matrices when I have more time.

The interesting thing about this kind of domain is that there are no cusps and no elliptic points, so the boundary has less restrictions than the boundary of domains for congruence subgroups (i.e., those at )

As you move the vertices, you get fundemental domains with the same edge relations. Note that when edges overlap it's (probably) no longer a domain, so really there should be some restriction on how far the vertices can move.

Just click and drag the mouse on the screen to see different domains started.

Note, the quit button only works for the stand alone applet, which you get from here

In the background, the circle is of radius 1, center 0, and the line is the vertical imaginary axis.

There are many other choices of fundamental domain (well, infinitely many). Some are nicer than others for various reasons.

We worked out these domains, and wrote code to draw the rest of the images on this page in magma.

The following is a choice invariant under a certain choice of generators of the normaliser of Gamma₀³⁵(1) in SL(2,R). It's drawn togther with translates, coloured according to edge identifications.

In this picture, the domain is divided into four parts. Each of the four regions is a domain for the normaliser, and together they give a domain for Gamma₀³⁵(1). Note, there are other ways to divide the domain into domains for the normaliser.

Here is a picture of translates of the above under elements of Gamma₀³⁵(1).