Improve counting of local solutions for QuadraticForm at p=2

[WvanWoerden]

Problem Description

Given a QuadraticForm Q it is currently infeasible to compute Q.siegel_product(m) when Q.dim() >= 8.
The reason for this is that Q.local_density(p,m) for p=2 needs to compute local (good) solutions modulo 8 which is done with a naive brute-force approach by count_congruence_solutions__good_type(2, 3, m, ..., ...).

Proposed Solution

The solutions mod 8 can be computed efficiently by:

1. put Q in local normal form (which is already done),
2. count all solutions Q_block[v] = k mod 8 per Jordan block for each k=0,...7 (this can be done naively)
3. compute the convolution of the solutions of all blocks
4. return the count at m

Of course this could be implemented in a general way and not just for p=2 or modulo 8.

Alternatives Considered

None.

Additional Information

I will propose a pull request with the feature soon.

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.