Canonical representatives for a Cusps

[edgarcosta]

Our implementation of computing cusps only works if the narrow class representative and level are coprime.

How do we convert the cusp info back to the desired representative?
We only need to convert the representatives of the points in P1(F) from Y_b to Y_b'.

Is there a way to also pick the coordinates of these points in a canonical way?

[edgarcosta]

@jvoight , input would be appreciated, but for now, I'm just going to generate data with the hope that, at some point, we know how to convert it back and canonicalize it.

[jvoight]

Interesting!

Can we choose a coprime representative, compute cusps, then use the appropriate scalar to map them back?

This kind of headache also occurs with Shimura curves. There are situations where you can still make things work even when it's not coprime, and that's usually both very interesting and very hard.

[SamSchiavone]

We were struggling with the "map them back" part, but I now think the cusp conversion map is just (a:c) |-> (lambda^(-1) a:c). See my rambling under the Converting narrow class reps topic here: https://agntc.zulipchat.com/#narrow/stream/306633-Hilbert-modular-forms/topic/Converting.20narrow.20class.20reps/near/316402166