ECQ and ECNF pages: make more consistent

[JohnCremona]

As reported by @AndrewVSutherland at #6273:

I notice that the ECQ/ECNF Mordell-Weil Invariants sections use rather different layouts, are there reasons to format them differently? I like that everything is grouped under one heading in the ECNF layout, but I also like the way the structure of the MW group is shown on the ECQ pages (including both the free and torsion parts rather than giving the structure of the torsion subgroup on a separate line). I think it might be worth figuring out the optimal layout (perhaps combining elements of both, but also see (4) below), and then uniformizing them.

I set the milestone to 1.3 rather than 1.2.2 as 1 Dec is only a couple of days away.

[JohnCremona]

See #6276: I am doing both in one PR but will leave this open.

[roed314]

@JohnCremona Now that #6279 has been merged and is on beta, can this and #6276 be closed?

[JohnCremona]

Yes, I think so. I am still working on replacing the ECNF gens with smaller ones but that will take some time ( and will only affect stored data, not code).

[roed314]

Great!

[roed314]

For the record, here are some differences that still exist:

• ECNF is missing integral points
• ECNF is missing Faltings height, stable Faltings height, abc quality, and Szpiro ratio.
• For ECNF the period is labeled "Period" while for ECQ it's labeled "Real period."
• There is no displayed BSD formula for ECNF.
• In the local data for ECNF, the symbols for conductor and discriminant do not match those used above, and $$\mathrm{ord}(j)$$ does not match $$v_p(\mathrm{den}(j))$$ on ECQ. There is also no note when the curve is semistable or count for the number of primes of bad reduction.

@JohnCremona, if any of these are things you think deserve an issue or a PR, feel free to open one!
*

[JohnCremona]

Some of these are easy, some are possible, but some not.

Integral points over number fields: there is no good implementation. The only paper I know about this is full of errors. There is an attempt to implement it in Sage which has been around for years but is based on those papers, I have never been prepared to give it a positive review, and it needs to be completely rewritten. I have an incomplete resolution of that, but we are in no position to compute integral points reliably except over Q. Magma does have an implementation only over totally real fields, but I have no idea how reliable it is given all the errors in the literature. Doing this properly and rigorously is a big project.

I do not know a definition of abc quality or Szpiro ratio except over Q. If there is one, please give a reference. I think that the definition of Faltings height could be made explicit for elliptic curves over number fields and implemented and added. Again, a suitable refernce would be helpful.

Periods: I kept this small difference on purpose. The knowl is sufficiently explit, I think. Over Q there is one infinite place and it is real so the period is (always) called the real period. Over number fields, there's a period for every infinite place and this number is essentially the product of all of them. In the knowl I call it the "global period" as there is no standard name for it in the literature (and the exact definition differs in different sources anyway: this is part of the larger issue about defining all the factots in the BSD formula). We could just call it the "period" over Q, but this would clash with the literature where (for elliptic curves over Q) this is normally called the real period.

Displaying BSD formula: once we have sorted out all the normalisation issues regarding BSD over number fields, for which there is another issue, then someone can do this.

The last few points are easy. The table ec_nfcurves does have columns n_bad_primes and semistable, so we could easily show these in a preamble to the local data table. I will do that and at the same time make the table headings there match.

[roed314]

Thanks for the explanations! I was just going through and recording the differences, not saying that they needed to be fixed.

[JohnCremona]

I have created a feature request issue for adding Szpiro ratio (easy), Faltings height (not too hard) and possibly stable Faltings height (harder) at #6292 . Of these, the Szpiro ratio could be done easily on the fly using what is already in the database.

Adding integral points is a lot more work, so not on the horizon and I have not made an issue even as a feature request.

There's now a PR for the BSD formula #6290 .

I still don't know of a number field version of abc-ratio, so have not made that a feature request issue. If someone knows of a definition, they can!