SchemeHomset_points_abelian_variety_field should be an actual group

[sagetrac-sbesnier]

As discussed in #11474 , SchemeHomset_points_abelian_variety_field is not really a group, but it should be !

However, note that the class of the points of an elliptic curve inherits from AdditiveGroupElement.

CC: @defeo @JohnCremona @pjbruin

Component: elliptic curves

Author: Sébastien Besnier

Issue created by migration from https://trac.sagemath.org/ticket/16184

[defeo]

comment:2

Please set the component field of your tickets.

[pjbruin]

comment:3

Note that there are two distinct but related ways in which objects of type SchemeHomset_points_abelian_variety_field "are not groups":

• the class SchemeHomset_points_abelian_variety_field itself does not inherit from sage.groups.abelian_gps.abelian_group.AbelianGroup_class;
• if A is an instance of SchemeHomset_points_abelian_variety_field, the list of categories returned by A.categories() does not include "Category of commutative groups" (i.e. the category obtained as Groups().Commutative()), let alone "Category of groups" (i.e. Groups()).
  Both points are probably easy to fix, but it would also be good to have an example of (a) concrete new feature(s) that this would introduce.