Certifying completeness of search results

[AndrewVSutherland]

One of the great features of the Jones-Roberts number field database is that it will tell you when the set of results is provably complete (i.e. every number field that satisfies the criteria you have specified is included). This is an extremely valuable feature (it allows you to prove theorems, not just find examples).

It would be great if we could provide a similar service to parts of the LMFDB where it is feasible to do so. Local and global number fields would be obvious candidates, as would Dirichlet characters, elliptic curves, CMFs, HMFs, BMFs, and abelian varieties over finite fields (genus 2 curves, on the other hand, would be pretty hopeless).

A possible prerequisite for doing this is storing information in the database that gives a computational explicit description of the scope of the data that is present. For CMFs there is a mf_boxes table that does this.

This issue is potentially vast in scope and likely to never be completely accomplished, but that is no reason not to implement what we can when we can. It would probably make sense to pick a particular section where this might be easier to implement because the search parameters are fairly limited and the completeness of the data is easy to describe; possibly Dirichlet characters or local fields would be good candidates.

[bmatschke]

To add a suggestion: For the user it would be furthermore great to know why the search result is complete. For example if one searches for global number fields, the search result page could state: "This table is provably complete. (more)", and if one clicks on "more", the user could find a list of all complete families of fields that contain the search result, with precise descriptions of their scope and references (which should include proofs and the code that was used in the computation of the table).

[everetthowe]

Question (perhaps naïve) about the original post: Why is it hopeless to ask for a provably complete list of genus-2 curves over finite fields? We know the stacky counts...

[AndrewVSutherland]

Not hopeless at all, we already have provably complete lists of genus 2 curves over Fq (for q up to 211).

The "hopeless" comment was referring to genus 2 curves over Q. It probably isn't hopeless in a sufficiently narrow range, but it requires a lot of luck. The only way I know to do this is enumerate all L-functions of a given conductor, find an abelian surface realizing each of them, and then find all the Jacobians isogenous to that abelian surface. Once you have an isogeny class rep that is a Jacobian you are in business, at least when there are no extra endomorphisms), but I don't know of any systematic way to find an isogeny class rep at all (one just searches strategically), and in general I don't know of any way to tell a priori (just from the L-function) whether the isogeny class contains a Jacobian or not.

Even getting all genus 2 curves over Q with a given discriminant or discriminant supported on a given set of primes is hard (I think to date the only successes are powers of 2 and powers of 7; we still don't know all the genus 2 curves with good reduction away from 3, for example). And even if you could do this, it wouldn't really help you get all curves with a given conductor (which is what we want), since in general the discriminant will have prime factors that do not appear in the conductor.

[roed314]

in general I don't know of any way to tell a priori (just from the L-function) whether the isogeny class contains a Jacobian or not.

If any good Euler factor does not contain Jacobian over Fp (using this criterion for example), then presumably you're also not a Jacobian over Q. I imagine that the converse to this statement is not likely to be true (thus your question about a way to tell in full generality), but it would be interesting to see an example where every good Euler factor was a Jacobian over Fp, but the isogeny class did not contain a Jacobian over Q.

[AndrewVSutherland]

Your first sentence is incorrect, it just forces bad reduction at p (I think this is in some sense the "reason" for primes of almost good reduction).

Indeed this happens for the very "first" genus 2 curve over Q, X_0(22), which has the smallest possible conductor 11^2. Here 2 is a prime of almost good reduction (bad for X_0(22) but good for J_0(22)). The Euler factor is https://www.lmfdb.org/Variety/Abelian/Fq/2/2/e_i and this isogeny class does not contain a Jacobian.

A model for this curve is

y^2 + (x^3 + x^2 + x + 1)*y = x^5 + 2*x^4 + 5*x^3 + 2*x^2 + 4*x - 2

I don't know about your second sentence. I would expect there to be lots of examples, but I'm not sure how you would prove that this in any particular example (you could check lots of Euler factors but how do you prove it for all of them?)

[everetthowe]

Ah, OK, so I was misunderstanding the original comment!

Re: reduction to non-Jacobians, see Remark 2.3 here for an interesting example.