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 ? ? ?