Algorithmic problem presence in: MAGMA SAGE PARI/GP (scripts, essentially Villegas)
* Algebraic number theory:
basics of q.a. over a number field F F general F=Q F=Q (need F general)
computing the discriminant X X special case (need)
computing a maximal order/Eichler orders X X special case (need)
computing an algebra with given discriminant X X special case (need)
explicit isomorphism A isom M2(F) X ? (Denis Simon's qf for F=Q)
explicit isomorphism A isom B ?
embeddings of quadratic subfields/suborders X (slow)
norm equations X (slow) special case (need)
Cl(O) set of isomorphism classes of left ideals X X X
principal ideal problem X (slow) ? ?
unit group/norm 1 group special cases special case (A. Rahm: Bianchi) (A. Page: project)
* Automorphic forms:
Brandt matrices X X X
cohomology of unit group and Hecke operators special cases (A. Rahm ?) (A. Page: project)
* Algebraic geometry:
equations for Shimura curves/varieties ? ? ?
CM points ~
modular abelian varieties ~ ~
End of an abelian variety ? ? ?