New class for Selmer groups of number fields

[JohnCremona]

There should be a proper class for Selmer groups of number fields K(S,m), similar to the existing and related S-unit groups and S-class-group classes: it fits into the short exact sequence

1 --> O<sup>*_{K,S}/(O</sup>*_{K,S})^m --> K(S,m) --> Cl_{K,S}[m] --> 1 .

Currently K.selmer_group(S,m) just returns a list of generators; it should be a proper finite abelian group. Angelos Koutsianas already has code which expresses a given element in terms of the generators.

CC: @pjbruin

Component: number fields

Issue created by migration from https://trac.sagemath.org/ticket/16496

[JohnCremona]

comment:2

I have code which implements a new function which on input (K,S,p) with p prime, returns K(S,p) as an abstract vector space together with two maps, one from the abstract space to K^ and one (the harder one) which takes an element of K^ whose image in K^/(K)^p lies in K(S,p) and returns the associated abstract vector space element. This is needed from some applications (in progress).

My implementation is based on code I wrote ~15 years ago in Magma. It does not actually define a new class but provides all the functionality one needs.

[videlec]

comment:3

update milestone 8.3 -> 8.4

[JohnCremona]

comment:4

I created a new ticket #31345 for this in the case m=p. I probably should have reused this ticket instead, but this one can stay in case the improved versions are extended to general m in future.