Hilbert-Siegel Modular Forms

Problem 5.1.1   Use quaterion algebras and Brandt modules to find examples of Hilbert-Siegel modular forms.

If possible, also study the Galois representations corresponding to Siegel modular forms.

Skoruppa has done work on tables of Siegel modular forms:

   http://wotan.algebra.math.uni-siegen.de/~modi/

Problem 5.1.2   Make the data available in Skoruppa's tables easily accessible in SAGE. If there are dimension formulas, implement them in SAGE. If there are tables, download them all and put them into SAGE. Be sure to appropriately acknowledge Skorrupa's contribution.

Remark 5.1.3 (From Richard Taylor)   I would be very interested in this.

You can also compute on unitray groups (Lassine has been doing something along these lines and Kevin was planning to too.)

If you are going to work with siegel modular forms of level $>1$ (which you should do!), you have to be very careful what levels you use. I have never understood this properly, but I think it does not suffice to work with what people call $\Gamma_0(N)$ - defined similarly to $\Gamma_0(N)$ for ${\mathrm{GL}}_2$ but using $2\times 2$ blocks.

Remark 5.1.4 (From John Cremona:)  

Why is there not a Chapter similar to this but about imaginary quadratic fields? [By the way, I once mentioned that I would have a new PhD student starting in September who would implement higher weight modular symbols over such fields. But he has decided to stay in Cambridge, with Tom Fisher, so that project remains open.]