Parameterized families of number fields with a given Galois group

[AndrewVSutherland]

On the Galois group pages, in addition to providing a link to number fields with this Galois group, in cases where a (not necessarily universal) parameterization is known, it would be nice to list known parameterizations on the Galois group page for the benefit of those who would like to be able to generate many examples.

E.g, for 21T2, there are 11 number fields in the LMFDB, but thereisa parameterized family that can be found in the "Generic Polynomials" book by Jensen-Ledet-Yui(https://books.google.com/books/about/Generic_Polynomials.html).

[jwj61]

Wouldn't we also want examples of generic polynomials as well in cases where they are known?

As far as other parameterizations, many of these will come from Belyi maps. Do we list these as well?

In all cases, there may be restrictions on the base field which would need to be noted (e.g. C_3 Belyi maps in our database don't give C_3 fields over Q, and some polynomials in that book work for say fields of characteristic different from 2).

If these are given for a relatively small percentage of the Galois groups, should it be regular database entries or done through bottom knowls?

[AndrewVSutherland]

I think we definitely want generic polynomials when those are known, but I don't think we should require this (the cases where the parameterizations might be most valuable are probably ones where we don't have a generic parameterization).

I'm not sure bottom knowls are the best solution. If the numbers are really small maybe this makes sense, but I worry that people won't notice them, and it's also not necessarily a good reason to have them included in the "interesting number fields" list (which will happen automatically if we add a bottom knowl).

I think it would be reasonable to restrict to parameterizations that work over Q, at least for an initial implementation. In cases were we have a generic polynomial that realizes the Galois group over Q(t) maybe it would be worth noting this.

BTW, this request came from folks at the Algebraic Points on Curves conference that is happening this week at ICERM.

[roed314]

See also the lecture notes from Tim Dokchitser's 2021 PCMI course, which includes a magma package for producing a bunch of regular families.