Search by squarefree conductor, discriminant and other local behaviour

[jvoight]

This was a request made by a colleague when we were playing around: it would be nice if we could search for abelian varieties like we do for number fields in that we can specify good reduction away from a set of primes.

[JohnCremona]

More suggestions:

When searching for elliptic curves with certain properties it would be helpful if one could further require that the curve

1. Has squarefree conductor, or
2. Has squarefree discriminant, or
3. Has a prime with a particular Kodaira type reduction (or other local behaviour)

[roed314]

We've just added the ability to search for classical modular forms with prime/prime-power/square/squarefree level. To support the same for elliptic curves it suffices to just add some boolean columns to the schema. To support the original request we just need to store a set of bad primes with a GIN index.

One way to implement a search on Kodaira type reduction would to just list the reduction types as a list of strings (which we can again GIN index). This would support looking for curves with split multiplicative reduction at some prime, but not looking for curves with split multiplicative reduction at 5.

[JohnCremona]

Right now local data for elliptic curves is stored as a list (with one entry per bad prime) of dicts (with keys 'p', 'kod', etc). That is fine for storing and displaying but not for searching. I have been thinking about changing this -- in effect, transposing the matrix -- to have separate columns for 'bad_primes', 'kodaira_symbols', etc -- one for each of the current keys in those dictionaries.

Would this be a good idea? If so I would do it for both ec/Q and ec/nf. In any case I was about to do a rewrite of the ec_nfcurves collection to add local root numbers and so this would be an opportunity to add these new columns. (Remark: the bad primes listed are the primes dividing the disciminant of the model stored, which is a global minimal model when such a model exists, but when there is no such model there will be one prime in the list for which the reduction type is 'good'.)

[edgarcosta]

@JohnCremona, having an entry for bad_primes of the type int4[] (i.e. arrays of signed-four byte integers) will allow you to quickly search for bad/good reduction regarding a set of primes

With respect to Kodaira symbols, I suggested text[], this will allow you to search certain types of reduction.

However, we won't be able to support searches like a certain type of reduction at the prime p.

[AndrewVSutherland]

This will be addresses by #2912 (easy to do both at once).

[edgarcosta]

When we closed this issue, we forgot about Vladimir Dokchitser's request to search on:

1. Has squarefree conductor, or
2. Has squarefree discriminant, or
3. Has a prime with a particular Kodaira type reduction (or other local behaviour)

@JohnCremona, @AndrewVSutherland are you still open to adding the columns?

[JohnCremona]

Sure. A true/false flag for the two square-free conditions would be very easy to add. The last one would be much harder with the data stored as it currently is.

As I just wrote on #3238 I have no objection to anyone adding or changing columns as long as they -- at the same time -- make appropriate changes to the data upload scripts.

[edgarcosta]

Great, thanks for the quick reply.

[AndrewVSutherland]

I agree with @JohnCremona, adding booleans for square free conductor/discriminant is trivial and something we should do. Searching on local invariants is a lot more complicated. I'm happy to leave this as an open feature request but I don't think we should try to address this now.