# Computation of low weight Siegel modular forms

We have these huge tables of elliptic curves, which were generated by computing modular forms of weight $2$ and level $\Gamma_0(N)$ as N increased.

For abelian surfaces over $\mathbb{Q}$ we have very little as far as I know. The Langlands philosophy suggests that every abelian surface should be attached to a Siegel modular form of weight $(2,2)$ on $GSp_4$, but the problem is that this weight is not cohomological, which has the concrete consequence that it's going to be tough to compute such things using group cohomology. In particular one of the reasons that modular symbols work for computing elliptic curves, fails in this situation.

I guess though that one might be able to somehow use the trace formula to compute the trace of various Hecke operators on Siegel modular forms of weight $(2,2)$ and various levels, because presumably the trace formula translates the problem into some sort of "class group" (in some general sense) computation, plus some combinatorics.

[EDIT: from FC's comment, it seems that my guess is wrong.]

Has anyone ever implemented this and tabulated the results?

[NB I know that people have done computations for low level and high weight, for example there's a lovely paper of Skoruppa that outlines how to compute in level 1; my question is specifically about the weights that are tough to access]

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Wouldn't any computational efforts with N > 3 be very difficult simply because the moduli space is general type ? – David Lehavi Apr 29 '10 at 13:58
I don't buy this. Isn't that like saying "wouldn't computational efforts with elliptic curves of conductor 100 or more be hard because the moduli space has genus at least 2", isn't it? The point is that you don't compute the moduli space---that's the last thing you want to do! You use the trace formula applied to a carefully-chosen function which will give you essentially the trace of Frobenius on the cohomology in terms of some much more algebraic/combinatorial/number-theoretic data and compute that instead. Well, that's my suggestion, but a lot of thought needs to go in to making it work. – Kevin Buzzard Apr 29 '10 at 14:52
@Kevin: Curves of genera up 6 have a very pleasant description, and you can comfortably (depends on your standard of comfort of course) live with the description of curves up to genus 15; I'd buy a curve of genus up to 15 as a moduli space any day. I have yet to see a general type three-fold with a nice description (unless you cooked it for this purpose). Disclaimer: I don't know the method involved, you may well be right, it's just my vague intuition speaking here. – David Lehavi Apr 29 '10 at 17:06
@FC: I am pretty sure one can't use the trace formula to compute forms of weight 1 but I have so little understanding of the trace formula that I don't know why this is the case. Seems to me that you're saying that the same obstruction stops me computing low weight Siegel forms :-( – Kevin Buzzard Apr 29 '10 at 19:13
@David: I think that trying to describe the moduli space is not what I am after. I am after an algorithm to compute traces of Hecke operators on spaces of cusp forms that takes a level and a prime (or some slightly more precise data; a diagonal matrix with powers of p on the diagonal) as an input and gives a number as an output and doesn't care less about whether a certain moduli space is general type or not. The same way as modular symbols would do the job for classical modular forms. But from FC's comment it seems that the trace formula won't do it either :-/ – Kevin Buzzard Apr 29 '10 at 19:15