From 93ddb13ad040a8a02c1d78e6dac460670b844159 Mon Sep 17 00:00:00 2001
From: Stevell Muller <78619134+StevellM@users.noreply.github.com>
Date: Wed, 20 Sep 2023 14:31:15 +0200
Subject: [PATCH] `QuadFormWithIsom` Patch 2: towards better time in CI (#2825)

---
 .../QuadFormAndIsom/src/embeddings.jl         | 344 +++++++++---------
 .../QuadFormAndIsom/src/enumeration.jl        |  74 ++--
 .../src/hermitian_miranda_morrison.jl         |  51 +--
 .../src/lattices_with_isometry.jl             |  68 +++-
 src/GAP/gap_to_oscar.jl                       |   4 +-
 src/Groups/abelian_aut.jl                     |   2 +-
 src/Groups/gsets.jl                           |  12 +-
 src/Groups/matrices/MatGrp.jl                 |   2 +-
 8 files changed, 290 insertions(+), 267 deletions(-)

diff --git a/experimental/QuadFormAndIsom/src/embeddings.jl b/experimental/QuadFormAndIsom/src/embeddings.jl
index bb85f6518515..c60691cb1091 100644
--- a/experimental/QuadFormAndIsom/src/embeddings.jl
+++ b/experimental/QuadFormAndIsom/src/embeddings.jl
@@ -13,9 +13,9 @@
 # together with the embeddings O(A) \to O(D) and O(B) \to O(D)
 function _sum_with_embeddings_orthogonal_groups(A::TorQuadModule, B::TorQuadModule)
   D = A+B
-  AinD = hom(A, D, TorQuadModuleElem[D(lift(a)) for a in gens(A)])
-  BinD = hom(B, D, TorQuadModuleElem[D(lift(b)) for b in gens(B)])
-  @assert all(AinD(a)*BinD(b) == 0 for a in gens(A), b in gens(B))
+  AinD = hom(A, D, elem_type(D)[D(lift(a)) for a in gens(A)])
+  BinD = hom(B, D, elem_type(D)[D(lift(b)) for b in gens(B)])
+  @hassert :ZZLatWithIsom 1 all(AinD(a)*BinD(b) == 0 for a in gens(A), b in gens(B))
   OD = orthogonal_group(D)
   OA = orthogonal_group(A)
   OB = orthogonal_group(B)
@@ -25,7 +25,7 @@ function _sum_with_embeddings_orthogonal_groups(A::TorQuadModule, B::TorQuadModu
   for f in gens(OA)
     imgf = data.(union!(AinD.(f.(gens(A))), BinD.(gens(B))))
     fab = hom(gene, imgf)
-    fD = OD(hom(D, D, fab.map))
+    fD = OD(hom(D, D, fab.map); check = false)
     push!(geneOAinOD, fD)
   end
 
@@ -33,7 +33,7 @@ function _sum_with_embeddings_orthogonal_groups(A::TorQuadModule, B::TorQuadModu
   for f in gens(OB)
     imgf = data.(union!(AinD.(gens(A)), BinD.(f.(gens(B)))))
     fab = hom(gene, imgf)
-    fD = OD(hom(D, D, fab.map))
+    fD = OD(hom(D, D, fab.map); check = false)
     push!(geneOBinOD, fD)
   end
   OAtoOD = hom(OA, OD, geneOAinOD; check = false)
@@ -85,11 +85,11 @@ end
 # This object is defined in Algorithm 2 of [BH23], the glue kernel we aim to use
 # for gluing lattices in a p-admissible triples are actually submodules of these
 # V's.
-function _get_V(fq::TorQuadModuleMor, mu, p::IntegerUnion)
+function _get_V(fq::TorQuadModuleMor, mu::PolyRingElem, p::IntegerUnion)
   q = domain(fq)
   V, _ = primary_part(q, p)
   _, Vinq = sub(q, elem_type(q)[q(lift(divexact(order(g), p)*g)) for g in gens(V) if !(order(g)==1)])
-  fpV = restrict_endomorphism(fq, Vinq)
+  fpV = restrict_endomorphism(fq, Vinq; check = false)
   fpV = evaluate(mu, fpV)
   V, _ = kernel(fpV)
   Vinq = hom(V, q, elem_type(q)[q(lift(a)) for a in gens(V)])
@@ -98,7 +98,7 @@ function _get_V(fq::TorQuadModuleMor, mu, p::IntegerUnion)
 end
 
 # This is the rho function as defined in Definition 4.8 of BH23.
-function _rho_functor(q::TorQuadModule, p, l::Union{Integer, fmpz}; quad::Bool = (p == 2))
+function _rho_functor(q::TorQuadModule, p::IntegerUnion, l::IntegerUnion; quad::Bool = (p == 2))
   pq, pqtoq = primary_part(q, p)
   pq = rescale(pq, QQ(p)^(l-1))
   Nv = cover(pq)
@@ -127,12 +127,12 @@ end
 
 # A finite bilinear module over the 2-adic integers is even if all square are
 # zeros.
-function _is_even(T, p, l)
+function _is_even(T::TorQuadModule, p::IntegerUnion, l::IntegerUnion)
   B = gram_matrix_bilinear(_rho_functor(T, p, l; quad = false))
   return is_empty(B) || (all(is_zero, diagonal(B)) && all(is_integral, 2*B))
 end
 
-function _is_free(T, p, l)
+function _is_free(T::TorQuadModule, p::IntegerUnion, l::IntegerUnion)
   return _is_even(T, p, l-1) && _is_even(T, p, l+1)
 end
 
@@ -169,24 +169,24 @@ function _overlattice(gamma::TorQuadModuleMor,
     z = zero_matrix(QQ, 0, degree(A))
     glue = reduce(vcat, QQMatrix[matrix(QQ, 1, degree(A), g) for g in _glue]; init=z)
     glue = vcat(basis_matrix(A+B), glue)
-    glue = FakeFmpqMat(glue)
-    _B = hnf(glue)
-    _B = QQ(1, denominator(glue))*change_base_ring(QQ, numerator(_B))
+    Fakeglue = FakeFmpqMat(glue)
+    _FakeB = hnf(Fakeglue)
+    _B = QQ(1, denominator(Fakeglue))*change_base_ring(QQ, numerator(_FakeB))
     C = lattice(ambient_space(A), _B[end-rank(A)-rank(B)+1:end, :])
-    fC = block_diagonal_matrix([fA, fB])
+    fC = block_diagonal_matrix(QQMatrix[fA, fB])
     _B = solve_left(reduce(vcat, basis_matrix.([A,B])), basis_matrix(C))
     fC = _B*fC*inv(_B)
   else
     _glue = Vector{QQFieldElem}[lift(HAinD(a)) + lift(HBinD(gamma(a))) for a in gens(domain(gamma))]
     z = zero_matrix(QQ, 0, degree(cover(D)))
     glue = reduce(vcat, QQMatrix[matrix(QQ, 1, degree(cover(D)), g) for g in _glue]; init=z)
-    glue = vcat(block_diagonal_matrix(basis_matrix.([A, B])), glue)
-    glue = FakeFmpqMat(glue)
-    _B = hnf(glue)
-    _B = QQ(1, denominator(glue))*change_base_ring(QQ, numerator(_B))
+    glue = vcat(block_diagonal_matrix(basis_matrix.(ZZLat[A, B])), glue)
+    Fakeglue = FakeFmpqMat(glue)
+    _FakeB = hnf(Fakeglue)
+    _B = QQ(1, denominator(Fakeglue))*change_base_ring(QQ, numerator(_FakeB))
     C = lattice(ambient_space(cover(D)), _B[end-rank(A)-rank(B)+1:end, :])
-    fC = block_diagonal_matrix([fA, fB])
-    _B = solve_left(block_diagonal_matrix(basis_matrix.([A, B])), basis_matrix(C))
+    fC = block_diagonal_matrix(QQMatrix[fA, fB])
+    _B = solve_left(block_diagonal_matrix(basis_matrix.(ZZLat[A, B])), basis_matrix(C))
     fC = _B*fC*inv(_B)
   end
   @hassert :ZZLatWithIsom 1 fC*gram_matrix(C)*transpose(fC) == gram_matrix(C)
@@ -220,7 +220,7 @@ end
 
 function _on_subgroup_automorphic(T::TorQuadModule, g::AutomorphismGroupElem)
   q = domain(parent(g))
-  gene = TorQuadModuleElem[g(q(lift(t))) for t in gens(T)]
+  gene = elem_type(q)[g(q(lift(t))) for t in gens(T)]
   return sub(q, gene)[1]
 end
 
@@ -231,7 +231,7 @@ function stabilizer(O::AutomorphismGroup{TorQuadModule}, i::TorQuadModuleMor)
   q = domain(O)
   N, _ = sub(q, i.(gens(domain(i))))
   stab, _ = stabilizer(O, N, _on_subgroup_automorphic)
-  return sub(O, O.([h.X for h in gens(stab)]))
+  return sub(O, elem_type(O)[O(h) for h in gens(stab)])
 end
 
 function _subgroups_orbit_representatives_and_stabilizers(Vinq::TorQuadModuleMor,
@@ -248,7 +248,7 @@ function _subgroups_orbit_representatives_and_stabilizers(Vinq::TorQuadModuleMor
     return res
   end
 
-  fV = f isa TorQuadModuleMor ? restrict_endomorphism(f, Vinq) : restrict_endomorphism(hom(f), Vinq)
+  fV = f isa TorQuadModuleMor ? restrict_endomorphism(f, Vinq; check = false) : restrict_endomorphism(hom(f), Vinq; check = false)
   if ord == -1
     subs = collect(stable_submodules(V, TorQuadModuleMor[fV]))
   else
@@ -294,7 +294,7 @@ function _cokernel_as_Fp_vector_space(HinV::TorQuadModuleMor, p::IntegerUnion)
 
   function _VptoV(v::ModuleElem{FpFieldElem})
     x = lift.(v.v)
-    return sum(TorQuadModuleElem[x[i]*V[i] for i in 1:ngens(V)])
+    return sum(x[i]*V[i] for i in 1:n)
   end
 
   VtoVp = Hecke.MapFromFunc(V, Vp, _VtoVp, _VptoV)
@@ -308,15 +308,15 @@ end
 # Almost duplicate of an existing function: we do not always want to compute
 # stabilizers but just some orbit representatives
 function _orbit_representatives(G::MatrixGroup{E}, k::Int, O::AutomorphismGroup{TorQuadModule}) where E <: FinFieldElem
-  F = G.ring
-  n = G.deg
+  F = base_ring(G)
+  n = degree(G)
   q = GAP.Obj(order(F))
   V = VectorSpace(F, n)
   orbs = GAP.Globals.Orbits(G.X, GAP.Globals.Subspaces(GAP.Globals.GF(q)^n, k))
-  orbs = [GAP.Globals.BasisVectors(GAP.Globals.Basis(orb[1])) for orb in orbs]
-  orbs = [[[F(x) for x in v] for v in bas] for bas in orbs]
-  orbs = [sub(V, [V(v) for v in bas])[1] for bas in orbs]
-  return [(orbs[i], O) for i in 1:length(orbs)]
+  orbs1 = [GAP.Globals.BasisVectors(GAP.Globals.Basis(orb[1])) for orb in orbs]
+  orbs2 = [[[F(x) for x in v] for v in bas] for bas in orbs1]::Vector{Vector{Vector{elem_type(F)}}}
+  orbs3 = [sub(V, [V(v) for v in bas])[1] for bas in orbs2]
+  return [(orbs3[i], O) for i in 1:length(orbs3)]
 end
 
 # Given an abelian group injection V \to q where the group structure on V is
@@ -366,7 +366,7 @@ function _subgroups_orbit_representatives_and_stabilizers_elementary(Vinq::TorQu
 
   # In theory, V should contain H0 := p^l*pq where pq is the p-primary part of q
   all(a -> has_preimage(Vinq, (p^l)*pqtoq(a))[1], gens(pq)) || return res
-  H0, H0inq = sub(q, TorQuadModuleElem[q(lift((p^l)*a)) for a in gens(pq)])
+  H0, H0inq = sub(q, elem_type(q)[q(lift((p^l)*a)) for a in gens(pq)])
   @hassert :ZZLatWithIsom 1 is_invariant(f, H0inq)
 
   # H0 should be contained in the group we want. So either H0 is the only one
@@ -377,7 +377,7 @@ function _subgroups_orbit_representatives_and_stabilizers_elementary(Vinq::TorQu
     return res
   end
 
-  H0inV = hom(H0, V, TorQuadModuleElem[V(lift(a)) for a in gens(H0)])
+  H0inV = hom(H0, V, elem_type(V)[V(lift(a)) for a in gens(H0)])
   @hassert :ZZLatWithIsom 1 is_injective(H0inV)
 
   # Since V and H0 are elementary p-groups, they can be seen as finite
@@ -391,7 +391,7 @@ function _subgroups_orbit_representatives_and_stabilizers_elementary(Vinq::TorQu
   dim(Qp) == 0 && return res
 
   # We descend G to V for computing stabilizers later on
-  GV, GtoGV = restrict_automorphism_group(G, Vinq)
+  GV, GtoGV = restrict_automorphism_group(G, Vinq; check = false)
   if compute_stab
     satV, j = kernel(GtoGV)
   end
@@ -421,20 +421,20 @@ function _subgroups_orbit_representatives_and_stabilizers_elementary(Vinq::TorQu
 
   for (orb, stab) in orb_and_stab
     i = orb.map
-    gene_orb = elem_type(Qp)[Qp(vec(collect(i(v).v))) for v in gens(domain(i))]
-    gene_orb = elem_type(Vp)[preimage(VptoQp, v) for v in gene_orb]
+    gene_orbQp = elem_type(Qp)[Qp(vec(collect(i(v).v))) for v in gens(domain(i))]
+    gene_orbVp = elem_type(Vp)[preimage(VptoQp, v) for v in gene_orbQp]
 
-    gene_orb = vcat(gene_orb, gene_H0p)
-    gene_orb = elem_type(V)[preimage(VtoVp, Vp(v)) for v in gene_orb]
-    gene_orb = elem_type(q)[image(Vinq, v) for v in gene_orb]
-    orbq, orbqinq = sub(q, gene_orb)
+    gene_orbVp = vcat(gene_orbVp, gene_H0p)
+    gene_orbV = elem_type(V)[preimage(VtoVp, Vp(v)) for v in gene_orbVp]
+    gene_orbq = elem_type(q)[image(Vinq, v) for v in gene_orbV]
+    orbq, orbqinq = sub(q, gene_orbq)
     @hassert :ZZLatWithIsom 1 order(orbq) == ord
     # We keep only f-stable subspaces
     is_invariant(f, orbqinq) || continue
 
     if compute_stab
-      stabq = AutomorphismGroupElem{TorQuadModule}[GtoMGp\(s) for s in gens(stab)]
-      stabq, _ = sub(G, union!(stabq, gens(satV)))
+      stabq_gen = elem_type(G)[GtoMGp\(s) for s in gens(stab)]
+      stabq, _ = sub(G, union!(stabq_gen, gens(satV)))
       # Stabilizers should preserve the actual subspaces, by definition. so if we
       # have lifted since properly, this should hold..
       @hassert :ZZLatWithIsom 1 is_invariant(stabq, orbqinq)
@@ -472,7 +472,7 @@ function _classes_automorphic_subgroups(q::TorQuadModule,
   # primary part of H.
   #
   # First, we cut q as an orthogonal direct sum of its primary parts
-  pds = sort(prime_divisors(order(q)))
+  pds = sort!(prime_divisors(order(q)))
   if compute_stab
     blocks = TorQuadModuleMor[primary_part(q, pds[1])[2]]
     ni = Int[ngens(domain(blocks[1]))]
@@ -489,7 +489,8 @@ function _classes_automorphic_subgroups(q::TorQuadModule,
     blocks = TorQuadModuleMor[primary_part(q, p)[2] for p in pds]
     D, inj, proj = biproduct(domain.(blocks))
   end
-  _, phi = is_isometric_with_isometry(D, q)
+  phi = hom(D, q, TorQuadModuleElem[sum([blocks[i](proj[i](a)) for i in 1:length(pds)]) for a in gens(D)])
+  @hassert :ZZLatWithIsom 1 is_isometry(phi)
 
   list_can = Vector{Tuple{TorQuadModuleMor, AutomorphismGroup{TorQuadModule}}}[]
   # We collect the possible subgroups for each primary part, with the stabilizer
@@ -498,8 +499,8 @@ function _classes_automorphic_subgroups(q::TorQuadModule,
     qpinq = blocks[i]
     qp = domain(qpinq)
     T, _ = primary_part(H, p)
-    Oqp = restrict_automorphism_group(O, qpinq)[1]
-    fqp = restrict_endomorphism(f, qpinq)
+    Oqp, _ = restrict_automorphism_group(O, qpinq; check = false)
+    fqp = restrict_endomorphism(f, qpinq; check = false)
     if is_elementary(T, p)
       _, j = _get_V(id_hom(qp), minimal_polynomial(identity_matrix(QQ, 1)), p)
       sors = _subgroups_orbit_representatives_and_stabilizers_elementary(j, Oqp, order(T), fqp; compute_stab)
@@ -520,7 +521,7 @@ function _classes_automorphic_subgroups(q::TorQuadModule,
     embs = TorQuadModuleMor[l[1] for l in lis]
     embs = TorQuadModuleMor[hom(domain(embs[i]), q, TorQuadModuleElem[blocks[i](domain(blocks[i])(lift(embs[i](a)))) for a in gens(domain(embs[i]))]) for i in 1:length(lis)]
     H2, _proj = direct_product(domain.(embs)...)
-    _, H2inq = sub(q, TorQuadModuleElem[sum([embs[i](_proj[i](g)) for i in 1:length(lis)]) for g in gens(H2)])
+    _, H2inq = sub(q, elem_type(q)[sum([embs[i](_proj[i](g)) for i in 1:length(lis)]) for g in gens(H2)])
     if compute_stab
       stabs = AutomorphismGroup{TorQuadModule}[l[2] for l in lis]
       genestab = ZZMatrix[]
@@ -600,11 +601,11 @@ function _isomorphism_classes_primitive_extensions(N::ZZLat,
     HM = domain(HMinqM)
     OHM = orthogonal_group(HM)
 
-    actN = hom(stabN, OHN, elem_type(OHN)[OHN(restrict_automorphism(x, HNinqN)) for x in gens(stabN)])
-    actM = hom(stabM, OHM, elem_type(OHM)[OHM(restrict_automorphism(x, HMinqM)) for x in gens(stabM)])
+    actN = hom(stabN, OHN, elem_type(OHN)[OHN(restrict_automorphism(x, HNinqN; check = false); check = false) for x in gens(stabN)])
+    actM = hom(stabM, OHM, elem_type(OHM)[OHM(restrict_automorphism(x, HMinqM; check = false); check = false) for x in gens(stabM)])
     imM, _ = image(actM)
 
-    stabNphi = AutomorphismGroupElem{TorQuadModule}[OHM(compose(inv(phi), compose(hom(actN(g)), phi))) for g in gens(stabN)]
+    stabNphi = elem_type(OHM)[OHM(compose(inv(phi), compose(hom(actN(g)), phi)); check = false) for g in gens(stabN)]
     stabNphi, _ = sub(OHM, stabNphi)
 
     if is_elementary_with_prime(HM)[1]
@@ -862,7 +863,7 @@ function primitive_embeddings(G::ZZGenus, M::ZZLat; classification::Symbol = :su
       # $M\perp L2 \subset T \subset Mv\perp L1$.
       #
       # The upshot is that both $(T\cap S)/(M\perp L2)$ and
-      # $(T\cap Sv)/(M\perpL2) lies in the embedding of $qM \to D$. Their
+      # $(T\cap Sv)/(M\perp L2) lies in the embedding of $qM \to D$. Their
       # quotient is isometric to the quotient of their preimage.
       # The quotient of their preimage is precisely the subgroup of `qM`
       # we look for.
@@ -961,8 +962,8 @@ function admissible_equivariant_primitive_extensions(A::ZZLatWithIsom,
   if g == 0
     # Needed to compute the image of the stabilizer of the isometry we construct
     # (in the orthogonal group of the discriminant group of the new lattice).
-    geneA = AutomorphismGroupElem{TorQuadModule}[OqAinOD(OqA(a.X)) for a in gens(GA)]
-    geneB = AutomorphismGroupElem{TorQuadModule}[OqBinOD(OqB(b.X)) for b in gens(GB)]
+    geneA = elem_type(OD)[OqAinOD(OqA(a.X)) for a in gens(GA)]
+    geneB = elem_type(OD)[OqBinOD(OqB(b.X)) for b in gens(GB)]
     union!(geneA, geneB)
 
     # We compute the overlattice in this context
@@ -974,7 +975,7 @@ function admissible_equivariant_primitive_extensions(A::ZZLatWithIsom,
 
     qC2 = discriminant_group(C2)
     OqC2 = orthogonal_group(qC2)
-    phi2 = hom(qC2, D, TorQuadModuleElem[D(lift(x)) for x in gens(qC2)])
+    phi2 = hom(qC2, D, elem_type(D)[D(lift(x)) for x in gens(qC2)])
     @hassert :ZZLatWithIsom 1 is_isometry(phi2)
 
     # This is the new image of the stabilizer, just a direct product of
@@ -1029,35 +1030,23 @@ function admissible_equivariant_primitive_extensions(A::ZZLatWithIsom,
   for H1 in subsA, H2 in subsB
     ok, phi = is_anti_isometric_with_anti_isometry(domain(H1[1]), domain(H2[1]))
     !ok && continue
+
     SAinqA, stabA = H1
     SA = domain(SAinqA)
     SAinD = compose(SAinqA, qAinD)
     OSA = orthogonal_group(SA)
+    fSA = OSA(restrict_automorphism(fqA, SAinqA; check = false); check = false)
 
     SBinqB, stabB = H2
     SB = domain(SBinqB)
     SBinD = compose(SBinqB, qBinD)
     OSB = orthogonal_group(SB)
-    
+    fSB = OSB(restrict_automorphism(fqB, SBinqB; check = false); check = false)
+
     # we need a first admissible gluing. We know that such gluing exists because
     # we have an admissible triple as input and the glue kernels have been
     # chosen in such a way that their exist an admissible gluing between them.
-    phi = _find_admissible_gluing(SAinqA, SBinqB, phi, l, spec)
-
-    # we compute the image of the stabilizers in the respective OS* and we keep track
-    # of the elements of the stabilizers acting trivially in the respective S*
-    # (there are in the ker*).
-    actA = hom(stabA, OSA, elem_type(OSA)[OSA(restrict_automorphism(x, SAinqA)) for x in gens(stabA)])
-    imA, _ = image(actA)
-    kerA = AutomorphismGroupElem{TorQuadModule}[OqAinOD(x) for x in gens(kernel(actA)[1])]
-    push!(kerA, OqAinOD(one(OqA)))
-    fSA = OSA(restrict_automorphism(fqA, SAinqA))
-
-    actB = hom(stabB, OSB, elem_type(OSB)[OSB(restrict_automorphism(x, SBinqB)) for x in gens(stabB)])
-    imB, _ = image(actB)
-    kerB = AutomorphismGroupElem{TorQuadModule}[OqBinOD(x) for x in gens(kernel(actB)[1])]
-    push!(kerB, OqBinOD(one(OqB)))
-    fSB = OSB(restrict_automorphism(fqB, SBinqB))
+    phi = _find_admissible_gluing(SAinqA, SBinqB, phi, l, p, spec)
 
     # We want all isometries of SB which preserves p^l*q_B and such that they
     # define isometries of rho_{l+1}(B). If `spec == true`, then rho_{l+1}(B) is
@@ -1070,7 +1059,7 @@ function admissible_equivariant_primitive_extensions(A::ZZLatWithIsom,
     # phi might not "send" the restriction of fA to this of fB, but at least phi*fA*phi^-1
     # should be conjugate to fB inside O(SB, rho_l(qB)) for the gluing.
     # If not, we try the next potential pair.
-    fSAinOSB = OSB(compose(inv(phi), compose(hom(fSA), phi)))
+    fSAinOSB = OSB(compose(inv(phi), compose(hom(fSA), phi)); check = false)
     @hassert :ZZLatWithIsom 1 fSAinOSB in OSBrB  # Same as before, since phi is admissible, then the image of fSA should preserve rho_{l+1}(B)
     bool, g0 = is_conjugate_with_data(OSBrB, OSBrB(fSAinOSB), fSB)
     bool || continue
@@ -1078,13 +1067,22 @@ function admissible_equivariant_primitive_extensions(A::ZZLatWithIsom,
     # The new phi is "sending" the restriction of fA to this of fB
     # and it is still admissible. So we can glue SA and SB as wanted.
     phi = compose(phi, hom(OSB(g0)))
-    @hassert :ZZLatWithIsom OSBrB(compose(inv(phi), compose(hom(fSA), phi))) == fSB
+    @hassert :ZZLatWithIsom 1 OSBrB(compose(inv(phi), compose(hom(fSA), phi)); check = false) == fSB
+
+    # we compute the image of the stabilizers in the respective OS* and we keep track
+    # of the elements of the stabilizers acting trivially in the respective S*
+    # (there are in the ker*).
+    actA = hom(stabA, OSA, elem_type(OSA)[OSA(restrict_automorphism(x, SAinqA; check = false); check = false) for x in gens(stabA)])
+    imA, _ = image(actA)
+
+    actB = hom(stabB, OSB, elem_type(OSB)[OSB(restrict_automorphism(x, SBinqB; check = false); check = false) for x in gens(stabB)])
+    imB, _ = image(actB)
 
     # Now it is time to compute generators for O(SB, rho_l(qB), fB), and the induced
     # images of stabA|stabB for taking the double cosets next
     center, _ = centralizer(OSBrB, fSB)
     center, _ = sub(OSB, elem_type(OSB)[OSB(c) for c in gens(center)])
-    stabSAphi, _ = sub(OSB, elem_type(OSB)[OSB(compose(inv(phi), compose(hom(g), phi))) for g in gens(imA)])
+    stabSAphi, _ = sub(OSB, elem_type(OSB)[OSB(compose(inv(phi), compose(hom(g), phi)); check = false) for g in gens(imA)])
     stabSAphi, _ = intersect(center, stabSAphi)
     stabSB, _ = intersect(center, imB)
 
@@ -1108,29 +1106,13 @@ function admissible_equivariant_primitive_extensions(A::ZZLatWithIsom,
       # This is the type requirement: somehow, we want `(C2, fC2)` to be a "q-th root" of `(C, fC)`.
       !is_of_type(C2fC2^q, type(C)) && continue
 
-      # By the theory of primitive extensions, the discriminant group qC2 of C2
-      # is equal to H^{perp}/H where H is the graph of phig in D = qA + qB. We need
-      # to treat both at the same time to compute the image of the centralizer
-      # O(C2, fC2) in O(qC2, fqC2) using GA and GB.
-      ext = domain(extinD)
-      perp, j = orthogonal_submodule(D, ext)
-      disc = torsion_quadratic_module(cover(perp), cover(ext); modulus = modulus_bilinear_form(perp),
-                                                               modulus_qf = modulus_quadratic_form(perp))
+      disc, stab = _glue_stabilizers(phig, actA, actB, OqAinOD, OqBinOD, extinD)
+
       qC2 = discriminant_group(C2)
       OqC2 = orthogonal_group(qC2)
-      phi2 = hom(qC2, disc, TorQuadModuleElem[disc(lift(x)) for x in gens(qC2)])
+      phi2 = hom(qC2, disc, elem_type(disc)[disc(lift(x)) for x in gens(qC2)])
       @hassert :ZZLatWithIsom 1 is_isometry(phi2)              # In fact they are the same module so phi2, mathematically, is the identity.
 
-      # So now this new integer lattice with isometry `(C2, fC2)` is a good
-      # output. Just remain to compute GC2 in a smart way.
-      geneOSA =  AutomorphismGroupElem{TorQuadModule}[OSA(compose(phig, compose(hom(g1), inv(phig)))) for g1 in unique(gens(imB))]
-      im2_phi, _ = sub(OSA, geneOSA)
-      im3, _, _ = intersect(imA, im2_phi)
-      stab = AutomorphismGroupElem{TorQuadModule}[OqAinOD(actA\x) * OqBinOD(actB\(imB(compose(inv(phig), compose(hom(x), phig))))) for x in gens(im3)]
-      union!(stab, kerA)
-      union!(stab, kerB)
-      stab = TorQuadModuleMor[restrict_automorphism(g, j) for g in stab]
-      stab = TorQuadModuleMor[hom(disc, disc, [disc(lift(g(perp(lift(l))))) for l in gens(disc)]) for g in stab]
       stab = sub(OqC2, elem_type(OqC2)[OqC2(compose(phi2, compose(g, inv(phi2))); check = false) for g in stab])
 
       # If we have done good things, the action of fC2 on qC2 should centralize
@@ -1189,10 +1171,10 @@ function _compute_double_stabilizer(SBinqB::TorQuadModuleMor, l::IntegerUnion, s
   OSB = orthogonal_group(SB)
   p = elementary_divisors(SB)[1]
   rB = _rho_functor(qB, p, l+1)
-  rBtoSB = hom(rB, SB, TorQuadModuleElem[SB(QQ(p^l)*lift(a)) for a in gens(rB)])
+  rBtoSB = hom(rB, SB, elem_type(SB)[SB(QQ(p^l)*lift(a)) for a in gens(rB)])
   HB, HBinSB = sub(SB, rBtoSB.(gens(rB)))
   OSBHB, _ = stabilizer(OSB, HBinSB)
-  OHB, OSBHBtoOHB = restrict_automorphism_group(OSBHB, HBinSB)
+  OHB, OSBHBtoOHB = restrict_automorphism_group(OSBHB, HBinSB; check = false)
   K, _ = kernel(OSBHBtoOHB)
   if spec
     OHBrB, _ = stabilizer(OHB, gram_matrix_quadratic(rB), _on_modular_matrix_quad)
@@ -1203,56 +1185,6 @@ function _compute_double_stabilizer(SBinqB::TorQuadModuleMor, l::IntegerUnion, s
   return OSBrB
 end
 
-
-###############################################################################
-#
-#  Isometries between finite bilinear modules
-#
-###############################################################################
-
-# Test whether an abelian group isomorphism respect the finite bilinear product
-# of its domain (of type `TorQuadModule`).
-function _is_isometry_bilinear(f::TorQuadModuleMor)
-  !is_bijective(f) && return false
-  for a in gens(domain(f))
-    for b in gens(domain(f))
-      if f(a)*f(b) != a*b
-        return false
-      end
-    end
-  end
-  return true
-end
-
-# Test whether an abelian group isomorphism defines an anti isometry of the
-# finite bilinear product defined on its domain (of type `TorQuadModule`)
-function _is_anti_isometry_bilinear(f::TorQuadModuleMor)
-  !is_bijective(f) && return false
-  for a in gens(domain(f))
-    for b in gens(domain(f))
-      if f(a)*f(b) != -a*b
-        return false
-      end
-    end
-  end
-  return true
-end
-
-# Compute an anti-isometry between the two finite bilinear modules r1 and r2.
-function _anti_isometry_bilinear(r1::TorQuadModule, r2::TorQuadModule)
-  @hassert :ZZLatWithIsom is_semi_regular(r1) === is_semi_regular(r2) === true
-  hz = hom(r1, r2, zero_matrix(ZZ, ngens(r1), ngens(r2)))
-  r2m = rescale(r2, -1)
-  r2tor2m = hom(r2, r2m, identity_matrix(ZZ, ngens(r2)))
-  r1N, r1tor1N = normal_form(r1)
-  r2mN, r2mtor2mN = normal_form(r2m)
-  gram_matrix_bilinear(r1N) == gram_matrix_bilinear(r2mN) || return false, hz
-  T = hom(r1N, r2mN, identity_matrix(ZZ, ngens(r1N)))
-  T = compose(r1tor1N, compose(T, compose(inv(r2mtor2mN), inv(r2tor2m))))
-  @hassert :ZZLatWithIsom _is_anti_isometry_bilinear(T)
-  return true, T
-end
-
 ###############################################################################
 #
 #  Admissible gluings
@@ -1267,49 +1199,105 @@ function _find_admissible_gluing(SAinqA::TorQuadModuleMor,
                                  SBinqB::TorQuadModuleMor,
                                  phi::TorQuadModuleMor,
                                  l::IntegerUnion,
+                                 p::IntegerUnion,
                                  spec::Bool)
   SA = domain(SAinqA)
   SB = domain(SBinqB)
-  p = elementary_divisors(SA)[1]
   qA = codomain(SAinqA)
   qB = codomain(SBinqB)
-  pqA, pqAtoqA = primary_part(qA, p)
-  pqB, pqBtoqB = primary_part(qB, p)
-  rA = _rho_functor(qA, p, l+1)
-  rB = _rho_functor(qB, p, l+1)
-  rAtoSA = hom(rA, SA, TorQuadModuleElem[SA(QQ(p^l)*lift(a)) for a in gens(rA)])
-  HA, _ = sub(SA, rAtoSA.(gens(rA)))
-  rBtoSB = hom(rB, SB, TorQuadModuleElem[SB(QQ(p^l)*(lift(b))) for b in gens(rB)])
+  rA = _rho_functor(qA, p, l+1; quad = spec)
+  rB = _rho_functor(qB, p, l+1; quad = spec)
+  @hassert :ZZLatWithIsom modulus_quadratic_form(rA) == modulus_quadratic_form(rB)
+
+  rAtoSA = hom(rA, SA, elem_type(SA)[SA(QQ(p^l)*lift(a)) for a in gens(rA)])
+  HA, HAinSA = sub(SA, rAtoSA.(gens(rA)))
+  rAtoHA = hom(rA, HA, elem_type(HA)[HAinSA\(rAtoSA(a)) for a in gens(rA)])
+
+  rBtoSB = hom(rB, SB, elem_type(SB)[SB(QQ(p^l)*(lift(b))) for b in gens(rB)])
   HB, HBinSB = sub(SB, rBtoSB.(gens(rB)))
-  if spec
-    ok, phi_0 = is_anti_isometric_with_anti_isometry(rA, rB)
-    @hassert :ZZLatWithIsom 1 ok
-    @hassert :ZZLatWithIsom 1 is_anti_isometry(phi_0)
-  else
-    ok, phi_0 = _anti_isometry_bilinear(rA, rB)
-    @hassert :ZZLatWithIsom 1 ok
-    @hassert :ZZLatWithIsom 1 _is_anti_isometry_bilinear(phi_0)
-  end
-  phiHA, _ = sub(SB, TorQuadModuleElem[phi(SA(lift(a))) for a in gens(HA)])
+  rBtoHB = hom(rB, HB, elem_type(HB)[HBinSB\(rBtoSB(b)) for b in gens(rB)])
+
+  # We construct an abstract isometry between the rho functors: since they have
+  # the same modulus quadratic, either we see them as finite bilinear modules
+  # or both as finite quadratic modules depending on the value of spec
+  #
+  # Our goal would be to massage phi such that it maps HA to HB, and the
+  # restriction to rA and rB agrees with phi_0
+  ok, phi_0 = is_anti_isometric_with_anti_isometry(rA, rB)
+  @hassert :ZZLatWithIsom 1 ok
+  @hassert :ZZLatWithIsom 1 is_anti_isometry(phi_0)
+
+  # We first massage phi such that it maps HA to HB
+  phiHA, _ = sub(SB, elem_type(SB)[phi(SA(lift(a))) for a in gens(HA)])
   OSB = orthogonal_group(SB)
-  g = one(OSB)
-  for f in OSB
-    g = f
-    if cover(_on_subgroup_automorphic(phiHA, g)) == cover(HB)
-      break
-    end
-  end
+  G = GSetByElements(OSB, _on_subgroup_automorphic, TorQuadModule[HB])
+  ok, g = representative_action(G, phiHA, HB)
+  @hassert :ZZLatWithIsom 1 ok
   phi_1 = compose(phi, hom(g))
+  @hassert :ZZLatWithIsom 1 sub(SB, elem_type(SB)[phi_1(SA(lift(a))) for a in gens(HA)])[1] == HB
+
+  # Now we look at the restriction to rA and rB, and we modify a bit phi_1 by an
+  # element of OSBHB, which in particular preserves rB, in order to make phig
+  # and phi_0 agree
+  phi_1_res = hom(HA, HB, elem_type(HB)[HBinSB\(phi_1(HAinSA(a))) for a in gens(HA)])
+  phi_0_res = compose(inv(rAtoHA), compose(phi_0, rBtoHB))
   OSBHB, _ = stabilizer(OSB, HBinSB)
-  g = one(OSBHB)
-  for f in OSBHB
-    g = f
-    phif = compose(phi_1, hom(f))
-    psi = hom(rA, rB, TorQuadModuleElem[rBtoSB\(phif(rAtoSA(a))) for a in gens(rA)])
-    if matrix(psi) == matrix(phi_0)
-      break
-    end
-  end
+  OHB, OSBHBtoOHB = restrict_automorphism_group(OSBHB, HBinSB; check = false)
+  g = OSBHBtoOHB\(OHB(compose(inv(phi_1_res), phi_0_res); check = false))
   phig = compose(phi_1, hom(g))
+  @hassert :ZZLatWithIsom 1 matrix(hom(rA, rB, elem_type(rB)[rBtoSB\(phig(rAtoSA(a))) for a in gens(rA)])) == matrix(phi_0)
+  @hassert :ZZLatWithIsom 1 sub(SB, elem_type(SB)[phig(SA(lift(a))) for a in gens(HA)])[1] == HB
   return phig
 end
+
+###############################################################################
+#
+#  Extend stabilizers along equivariant primitive extensions
+#
+###############################################################################
+
+# Given an equivariant gluing phi between stable anti-isometric subgroups SA and
+# SB of qA and qB respectively, we know that the image of the centralizer of the
+# given equivariant primitive extensions in `O(D)` is given as a certain
+# product.
+#
+# Here actA and actB are respectively the orthogonal representations of the
+# stabilizers of SA and SB on SA and SB respectively. Then an isometry of OD
+# centralizing the extensions and comes from a global isometry if and only if it
+# can be written `(a, b)` where a and b are respectively centralizing elements
+# of the isometries of the two sublattices which preserve the glue groups, i.e.
+# elements of the domain of actA and actB respectively. We need to take care of
+# the kernel of such maps though, and compose the generators of their images.
+function _glue_stabilizers(phi::TorQuadModuleMor,
+                           actA::GAPGroupHomomorphism,
+                           actB::GAPGroupHomomorphism,
+                           OqAinOD::GAPGroupHomomorphism,
+                           OqBinOD::GAPGroupHomomorphism,
+                           graph::TorQuadModuleMor)
+  OD = codomain(OqAinOD)
+  OqA = domain(OqAinOD)
+  imA, _ = image(actA)
+  kerA = elem_type(OD)[OqAinOD(x) for x in gens(kernel(actA)[1])]
+  push!(kerA, OqAinOD(one(OqA)))
+
+  OqB = domain(OqBinOD)
+  imB, _ = image(actB)
+  kerB = elem_type(OD)[OqBinOD(x) for x in gens(kernel(actB)[1])]
+  push!(kerB, OqBinOD(one(OqB)))
+
+  ext = domain(graph)
+  perp, j = orthogonal_submodule(codomain(graph), ext)
+  disc = torsion_quadratic_module(cover(perp), cover(ext); modulus = QQ(1), modulus_qf = QQ(2))
+
+  OSA = codomain(actA)
+  geneOSA =  elem_type(OSA)[OSA(compose(phi, compose(hom(g1), inv(phi))); check = false) for g1 in unique(gens(imB))]
+  im2_phi, _ = sub(OSA, geneOSA)
+  im3, _, _ = intersect(imA, im2_phi)
+  stab = elem_type(OD)[OqAinOD(actA\x) * OqBinOD(actB\(imB(compose(inv(phi), compose(hom(x), phi)); check = false))) for x in gens(im3)]
+  union!(stab, kerA)
+  union!(stab, kerB)
+  stab = TorQuadModuleMor[restrict_automorphism(g, j; check = false) for g in stab]
+  stab = TorQuadModuleMor[hom(disc, disc, elem_type(disc)[disc(lift(g(perp(lift(l))))) for l in gens(disc)]) for g in stab]
+
+  return disc, stab
+end
diff --git a/experimental/QuadFormAndIsom/src/enumeration.jl b/experimental/QuadFormAndIsom/src/enumeration.jl
index 3892cd6058f0..97f6f702b87f 100644
--- a/experimental/QuadFormAndIsom/src/enumeration.jl
+++ b/experimental/QuadFormAndIsom/src/enumeration.jl
@@ -1,4 +1,4 @@
-##################################################################################
+#################################################################################
 #
 # This is an import to Oscar of the methods written following the paper [BH23] on
 # "Finite subgroups of automorphisms of K3 surfaces".
@@ -101,7 +101,7 @@ julia> is_admissible_triple(genus(F), genus(C), genus(Lf), 5)
 true
 ```
 """
-function is_admissible_triple(A::ZZGenus, B::ZZGenus, C::ZZGenus, p::Integer)
+function is_admissible_triple(A::ZZGenus, B::ZZGenus, C::ZZGenus, p::IntegerUnion)
   zg = genus(integer_lattice(; gram = matrix(QQ, 0, 0, [])))
   AperpB = direct_sum(A, B)
   (signature_tuple(AperpB) == signature_tuple(C)) || (return false)
@@ -122,7 +122,7 @@ function is_admissible_triple(A::ZZGenus, B::ZZGenus, C::ZZGenus, p::Integer)
   end
 
   # A+B and C must agree locally at every primes except p
-  for q in filter(qq -> qq != p, union!([2], primes(AperpB), primes(C)))
+  for q in filter(qq -> qq != p, union!(ZZRingElem[2], primes(AperpB), primes(C)))
     if local_symbol(AperpB, q) != local_symbol(C, q)
       return false
     end
@@ -223,18 +223,14 @@ function is_admissible_triple(A::ZZGenus, B::ZZGenus, C::ZZGenus, p::Integer)
     return false
   end 
 
-  qA, qB, qC = discriminant_group.([A, B, C])
+  qC = discriminant_group(C)
   spec = (p == 2) && (_is_free(qA, p, l+1)) && (_is_free(qB, p, l+1)) && (_is_even(qC, p, l))
-  rA = _rho_functor(qA, p, l+1)
-  rB = _rho_functor(qB, p, l+1)
-  if spec
-    return is_anti_isometric_with_anti_isometry(rA, rB)[1]
-  else
-    return _anti_isometry_bilinear(rA, rB)[1]
-  end
+  rA = _rho_functor(qA, p, l+1; quad = spec)
+  rB = _rho_functor(qB, p, l+1; quad = spec)
+  return is_anti_isometric_with_anti_isometry(rA, rB)[1]
 end
 
-function is_admissible_triple(A::T, B::T, C::T, p::Integer) where T <: Union{ZZLat, ZZLatWithIsom}
+function is_admissible_triple(A::T, B::T, C::T, p::IntegerUnion) where T <: Union{ZZLat, ZZLatWithIsom}
   return is_admissible_triple(genus(A), genus(B), genus(C), p)
 end
 
@@ -273,7 +269,7 @@ julia> admissible_triples(g, 2)
  (Genus symbol: II_(0, 0), Genus symbol: II_(5, 0) 2^-1_3 3^1)
 ```
 """
-function admissible_triples(G::ZZGenus, p::Integer; pA::Int = -1, pB::Int = -1)
+function admissible_triples(G::ZZGenus, p::IntegerUnion; pA::Int = -1, pB::Int = -1)
   @req is_prime(p) "p must be a prime number"
   @req is_integral(G) "G must be a genus of integral lattices"
   rG = rank(G)
@@ -319,7 +315,7 @@ function admissible_triples(G::ZZGenus, p::Integer; pA::Int = -1, pB::Int = -1)
   return L
 end
 
-admissible_triples(L::T, p::Integer; pA::Int = -1, pB::Int = -1) where T <: Union{ZZLat, ZZLatWithIsom} = admissible_triples(genus(L), p; pA, pB)
+admissible_triples(L::T, p::IntegerUnion; pA::Int = -1, pB::Int = -1) where T <: Union{ZZLat, ZZLatWithIsom} = admissible_triples(genus(L), p; pA, pB)
 
 ##################################################################################
 #
@@ -330,12 +326,13 @@ admissible_triples(L::T, p::Integer; pA::Int = -1, pB::Int = -1) where T <: Unio
 # we compute ideals of E/K whose absolute norm is equal to d
 
 function _ideals_of_norm(E::Field, d::QQFieldElem)
+  OE = maximal_order(E)
   if denominator(d) == 1
     return _ideals_of_norm(E, numerator(d))
   elseif numerator(d) == 1
-    return [inv(I) for I in _ideals_of_norm(E, denominator(d))]
+    return Hecke.fractional_ideal_type(OE)[inv(I) for I in _ideals_of_norm(E, denominator(d))]
   else
-    return [I*inv(J) for (I, J) in Hecke.cartesian_product_iterator([_ideals_of_norm(E, numerator(d)), _ideals_of_norm(E, denominator(d))]; inplace=false)]
+    return Hecke.fractional_ideal_type(OE)[I*inv(J) for (I, J) in Hecke.cartesian_product_iterator(Vector{Hecke.fractional_ideal_type(OE)}[_ideals_of_norm(E, numerator(d)), _ideals_of_norm(E, denominator(d))]; inplace=false)]
   end
 end
 
@@ -498,19 +495,19 @@ function representatives_of_hermitian_type(Lf::ZZLatWithIsom, m::Int = 1)
     @vprintln :ZZLatWithIsom 1 "$H"
     M, fM = trace_lattice_with_isometry(H)
     det(M) == d || continue
-    M = integer_lattice_with_isometry(M, fM)
-    @hassert :ZZLatWithIsom 1 is_of_hermitian_type(M)
-    @hassert :ZZLatWithIsom 1 order_of_isometry(M) == n*m
+    MfM = integer_lattice_with_isometry(M, fM; check = false)
+    @hassert :ZZLatWithIsom 1 is_of_hermitian_type(MfM)
+    @hassert :ZZLatWithIsom 1 order_of_isometry(MfM) == n*m
     if is_even(M) != is_even(Lf)
       continue
     end
-    if !is_of_same_type(M^m, Lf)
+    if !is_of_same_type(MfM^m, Lf)
       continue
     end
     gr = genus_representatives(H)
     for HH in gr
       M, fM = trace_lattice_with_isometry(HH)
-      push!(reps, integer_lattice_with_isometry(M, fM))
+      push!(reps, integer_lattice_with_isometry(M, fM; check = false))
     end
   end
   return reps
@@ -552,7 +549,7 @@ julia> is_of_same_type(Lf, reps[2]^2)
 true
 ```
 """
-function splitting_of_hermitian_prime_power(Lf::ZZLatWithIsom, p::Int; pA::Int = -1, pB::Int = -1)
+function splitting_of_hermitian_prime_power(Lf::ZZLatWithIsom, p::IntegerUnion; pA::Int = -1, pB::Int = -1)
   rank(Lf) == 0 && return ZZLatWithIsom[Lf]
 
   @req is_prime(p) "p must be a prime number"
@@ -615,7 +612,7 @@ julia> splitting_of_prime_power(Lf, 3, 1)
  Integer lattice with isometry of finite order 3
 ```
 """
-function splitting_of_prime_power(Lf::ZZLatWithIsom, p::Int, b::Int = 0)
+function splitting_of_prime_power(Lf::ZZLatWithIsom, p::IntegerUnion, b::Int = 0)
   if rank(Lf) == 0
     (b == 0) && return ZZLatWithIsom[Lf]
     return ZZLatWithIsom[]
@@ -668,29 +665,32 @@ of $(L, f)$.
 
 Note that `e` can be 0, while `d` has to be positive.
 """
-function splitting_of_pure_mixed_prime_power(Lf::ZZLatWithIsom, p::Int)
+function splitting_of_pure_mixed_prime_power(Lf::ZZLatWithIsom, _p::IntegerUnion)
   rank(Lf) == 0 && return ZZLatWithIsom[Lf]
 
+  n = order_of_isometry(Lf)
+
+  @req is_finite(n) "Isometry must be of finite order"
+
+  p = typeof(n)(_p)
   @req is_prime(p) "p must be a prime number"
-  @req is_finite(order_of_isometry(Lf)) "Isometry must be of finite order"
 
-  n = order_of_isometry(Lf)
   pd = prime_divisors(n)
 
   @req 1 <= length(pd) <= 2 && p in pd "Order must be divisible by p and have at most 2 prime divisors"
 
   if length(pd) == 2
     q = pd[1] == p ? pd[2] : pd[1]
-    d = valuation(n, p)
-    e = valuation(n, q)
+    d = valuation(n, p)::Int  # Weird ? Valuation is type stable and returns Int but it bugs here
+    e = valuation(n, q)::Int
   else
-    q = 1
-    d = valuation(n, p)
-    e = 0
+    q = one(n)
+    d = valuation(n, p)::Int
+    e = zero(n)
   end
 
   phi = minimal_polynomial(Lf)
-  chi = prod([cyclotomic_polynomial(p^d*q^i, parent(phi)) for i=0:e])
+  chi = prod(cyclotomic_polynomial(p^d*q^i, parent(phi)) for i=0:e; init = zero(phi))
 
   @req is_divisible_by(chi, phi) "Minimal polynomial is not of the correct form"
 
@@ -763,7 +763,7 @@ julia> all(LL -> is_of_same_type(Lf, LL^2), reps)
 true
 ```
 """
-function splitting_of_mixed_prime_power(Lf::ZZLatWithIsom, p::Int, b::Int = 1)
+function splitting_of_mixed_prime_power(Lf::ZZLatWithIsom, p::IntegerUnion, b::Int = 1)
   if rank(Lf) == 0
     b == 0 && return ZZLatWithIsom[Lf]
     return ZZLatWithIsom[]
@@ -788,7 +788,7 @@ function splitting_of_mixed_prime_power(Lf::ZZLatWithIsom, p::Int, b::Int = 1)
 
   x = gen(parent(minimal_polynomial(Lf)))
   B0 = kernel_lattice(Lf, x^(divexact(n, p)) - 1)
-  A0 = kernel_lattice(Lf, prod([cyclotomic_polynomial(p^d*q^i) for i in 0:e]))
+  A0 = kernel_lattice(Lf, prod(cyclotomic_polynomial(p^d*q^i) for i in 0:e))
   A = splitting_of_pure_mixed_prime_power(A0, p)
   isempty(A) && return reps
   B = splitting_of_mixed_prime_power(B0, p, 0)
@@ -817,13 +817,15 @@ Note that currently we support only orders which admit at most 2 prime divisors.
 function enumerate_classes_of_lattices_with_isometry(L::ZZLat, order::IntegerUnion)
   @req is_finite(order) && order >= 1 "order must be positive and finite"
   if order == 1
-    return representatives_of_hermitian_type(integer_lattice_with_isometry(L))
+    reps = representatives_of_hermitian_type(integer_lattice_with_isometry(L))
+    return reps
   end
   pd = prime_divisors(order)
   @req length(pd) in [1,2] "order must have at most two prime divisors"
   if length(pd) == 1
     v = valuation(order, pd[1])
-    return _enumerate_prime_power(L, pd[1], v)
+    reps = _enumerate_prime_power(L, pd[1], v)
+    return reps
   end
   p, q = sort!(pd)
   vp = valuation(order, p)
diff --git a/experimental/QuadFormAndIsom/src/hermitian_miranda_morrison.jl b/experimental/QuadFormAndIsom/src/hermitian_miranda_morrison.jl
index 6050908ddf19..950122405285 100644
--- a/experimental/QuadFormAndIsom/src/hermitian_miranda_morrison.jl
+++ b/experimental/QuadFormAndIsom/src/hermitian_miranda_morrison.jl
@@ -214,7 +214,7 @@ function _get_product_quotient(E::Hecke.NfRel, Fac::Vector{Tuple{NfOrdIdl, Int}}
 
   if length(Fac) == 0
     A = abelian_group()
-    function dlog_0(x::Vector); return id(A); end;
+    function dlog_0(x::Vector{<:Hecke.NfRelElem}); return id(A); end;
     function exp_0(x::GrpAbFinGenElem); return one(E); end;
     return A, dlog_0, exp_0
   end
@@ -230,25 +230,25 @@ function _get_product_quotient(E::Hecke.NfRel, Fac::Vector{Tuple{NfOrdIdl, Int}}
 
   G, proj, inj = biproduct(groups...)
 
-  function dlog(x::Vector)
+  function dlog(x::Vector{<:Hecke.NfRelElem})
     if length(x) == 1
-      return sum([inj[i](dlogs[i](x[1])) for i in 1:length(Fac)]) 
+      return sum(inj[i](dlogs[i](x[1])) for i in 1:length(Fac)) 
     else
       @hassert :ZZLatWithIsom 1 length(x) == length(Fac)
-      return sum([inj[i](dlogs[i](x[i])) for i in 1:length(Fac)])
+      return sum(inj[i](dlogs[i](x[i])) for i in 1:length(Fac))
     end
   end
 
   function exp(x::GrpAbFinGenElem)
-    v = Hecke.NfRelElem[exps[i](proj[i](x)) for i in 1:length(Fac)]
+    v = elem_type(E)[exps[i](proj[i](x)) for i in 1:length(Fac)]
     @hassert :ZZLatWithIsom 1 dlog(v) == x
     return v
   end
 
-  for i in 1:10
-    a = rand(G)
-    @hassert :ZZLatWithIsom 1 dlog(exp(a)) == a
-  end
+  @v_do :ZZLatWithIsom 1 for i in 1:10
+      a = rand(G)
+      @hassert :ZZLatWithIsom 1 dlog(exp(a)) == a
+    end
 
   return G, dlog, exp
 end
@@ -314,14 +314,15 @@ function _local_determinants_morphism(Lf::ZZLatWithIsom)
   qL, fqL = discriminant_group(Lf)
   OqL = orthogonal_group(qL)
   if rank(Lf) != degree(Lf)
-    Lf2 = integer_lattice_with_isometry(integer_lattice(gram = gram_matrix(Lf)), isometry(Lf); ambient_representation = false)
+    Lf2 = integer_lattice_with_isometry(integer_lattice(gram = gram_matrix(Lf)), isometry(Lf); ambient_representation = false, check = false)
     qL2, fqL2 = discriminant_group(Lf2)
     OqL2 = orthogonal_group(qL2)
     ok, phi12 = is_isometric_with_isometry(qL, qL2)
     @hassert :ZZLatWithIsom 1 ok
-    ok, g0 = is_conjugate_with_data(OqL, fqL, OqL(compose(phi12, compose(hom(fqL2), inv(phi12)))))
+    ok, g0 = is_conjugate_with_data(OqL, OqL(compose(phi12, compose(hom(fqL2), inv(phi12))); check = false), fqL)
     @hassert :ZZLatWithIsom 1 ok
     phi12 = compose(hom(OqL(g0)), phi12)
+    #@hassert :ZZLatWithIsom 1 matrix(compose(hom(fqL), phi12)) == matrix(compose(phi12, hom(fqL2)))
     @hassert :ZZLatWithIsom 1 is_isometry(phi12)
   else
     Lf2 = Lf
@@ -333,8 +334,11 @@ function _local_determinants_morphism(Lf::ZZLatWithIsom)
   # fqL, G is the group where we want to compute the image of O(L, f). This
   # group G corresponds to U(D_L) in the notation of BH23.
   G2, _ = centralizer(OqL2, fqL2)
-  G, _ = sub(OqL, elem_type(OqL)[OqL(compose(phi12, compose(hom(g), inv(phi12)))) for g in gens(G2)])
-  GtoG2 = hom(G, G2, gens(G), gens(G2))
+  gensG2 = gens(G2)
+  gensG = elem_type(OqL)[OqL(compose(phi12, compose(hom(g), inv(phi12))); check = false) for g in gensG2]
+  G, GinOqL = sub(OqL, gensG)
+  @hassert :ZZLatWithIsom 1 fqL in G
+  GtoG2 = hom(G, G2, gensG, gensG2; check = false)
 
   # This is the associated hermitian O_E-lattice to (L, f): we want to make qL
   # (aka D_L) correspond to the quotient D^{-1}H^#/H by the trace construction,
@@ -407,7 +411,7 @@ function _local_determinants_morphism(Lf::ZZLatWithIsom)
   RmodF, Flog, _ = _get_product_quotient(E, Fdata)
 
   A = elem_type(RmodF)[Flog(Fsharpexp(g)) for g in gens(RmodFsharp)]
-  f = hom(RmodFsharp, RmodF, A)
+  f = hom(RmodFsharp, RmodF, A; check = false)
   FmodFsharp, j = kernel(f)
 
   # Now according to Theorem 6.15 of BH23, it remains to quotient out the image
@@ -418,10 +422,10 @@ function _local_determinants_morphism(Lf::ZZLatWithIsom)
   OK = base_ring(OE)
   UOK, mUOK = unit_group(OK)
 
-  fU = hom(UOEabs, UOK, elem_type(UOK)[mUOK\norm(OE(mUOK(m))) for m in gens(UOK)])
+  fU = hom(UOEabs, UOK, elem_type(UOK)[mUOK\norm(OE(mUOK(m))) for m in gens(UOK)]; check = false)
   KU, jU = kernel(fU)
 
-  gene_norm_one = Hecke.NfRelElem[EabstoE(Eabs(mUOEabs(jU(k)))) for k in gens(KU)]
+  gene_norm_one = elem_type(E)[EabstoE(Eabs(mUOEabs(jU(k)))) for k in gens(KU)]
 
   # Now according to Theorem 6.15 of BH23, it remains to quotient out
   FOEmodFsharp, m = sub(RmodFsharp, elem_type(RmodFsharp)[Fsharplog(typeof(x)[x for i in 1:length(S)]) for x in gene_norm_one])
@@ -445,7 +449,7 @@ function _local_determinants_morphism(Lf::ZZLatWithIsom)
   # consider up to now, and then map the corresponding determinant adeles inside
   # Q. Since our matrices were approximate lifts of the generators of G, we can
   # create the map we wanted from those data.
-  for g in gens(G2)
+  for g in gensG2
     ds = elem_type(E)[]
     for p in S
       if !_is_special(H, p)
@@ -464,10 +468,11 @@ function _local_determinants_morphism(Lf::ZZLatWithIsom)
   end
 
   GSQ, SQtoGSQ, _ = Oscar._isomorphic_gap_group(SQ)
-  f2 = hom(G2, GSQ, gens(G2), SQtoGSQ.(imgs); check = false)
+  f2 = hom(G2, GSQ, gensG2, SQtoGSQ.(imgs); check = false)
   f = compose(GtoG2, f2)
 
-  return f
+  @hassert :ZZLatWithIsom 1 isone(f(fqL))
+  return f, GinOqL # Needs the second map to map the kernel of f into OqL
 end
 
 # We check whether for the prime ideal p E_O(L_p) != F(L_p).
@@ -506,7 +511,7 @@ function _is_special(L::HermLat, p::NfOrdIdl)
   u = elem_in_nf(uniformizer(P))
   s = involution(L)
   su = s(u)
-  H = block_diagonal_matrix([matrix(E, 2, 2, [0 u^(S[i]); su^(S[i]) 0]) for i in 1:length(S)])
+  H = block_diagonal_matrix(dense_matrix_type(E)[matrix(E, 2, 2, [0 u^(S[i]); su^(S[i]) 0]) for i in 1:length(S)])
   return is_locally_isometric(L, hermitian_lattice(E; gram = H), p)
 end
 
@@ -677,7 +682,7 @@ function _approximate_isometry(H::HermLat, H2::HermLat, g::AutomorphismGroupElem
   Bps = _local_basis_modular_submodules(H2, minimum(P), a, res)
   Bp = reduce(vcat, Bps)
   Gp = Bp*gram_matrix(ambient_space(H))*map_entries(involution(E), transpose(Bp))
-  Fp = block_diagonal_matrix([_transfer_discriminant_isometry(res, g, Bps[i], P, BHp_inv) for i in 1:length(Bps)])
+  Fp = block_diagonal_matrix(typeof(Gp)[_transfer_discriminant_isometry(res, g, Bps[i], P, BHp_inv) for i in 1:length(Bps)])
   # This is the local defect. By default, it should have scale P-valuations -a
   # and norm P-valuation e-1-a
   Rp = Gp - Fp*Gp*map_entries(involution(E), transpose(Fp))
@@ -717,7 +722,7 @@ function _local_basis_modular_submodules(H::HermLat, p::NfOrdIdl, a::Int, res::A
       L2 = restrict_scalars(H2, res)
       L2 = intersect(L, L2)
       B2 = basis_matrix(L2)
-      gene = [res(vec(collect(B2[i, :]))) for i in 1:nrows(B2)]
+      gene = Vector{elem_type(base_field(H))}[res(vec(collect(B2[i, :]))) for i in 1:nrows(B2)]
       H2 = lattice(ambient_space(H), gene)
       b = local_basis_matrix(H2, p; type = :submodule)
     end
@@ -736,7 +741,7 @@ end
 #    exactly what we look for (a unit at P with trace 1);
 #  - p is ramified, and then we cook up a good element. The actual code from
 #    that part is taken from the Sage implementation of Simon Brandhorst
-function _find_rho(P::Hecke.NfRelOrdIdl, e)
+function _find_rho(P::Hecke.NfRelOrdIdl, e::Int)
   OE = order(P)
   E = nf(OE)
   lp = prime_decomposition(OE, minimum(P))
diff --git a/experimental/QuadFormAndIsom/src/lattices_with_isometry.jl b/experimental/QuadFormAndIsom/src/lattices_with_isometry.jl
index bfce40849487..ea1a0e2ee3a9 100644
--- a/experimental/QuadFormAndIsom/src/lattices_with_isometry.jl
+++ b/experimental/QuadFormAndIsom/src/lattices_with_isometry.jl
@@ -614,7 +614,7 @@ function integer_lattice_with_isometry(L::ZZLat, f::QQMatrix; check::Bool = true
     @hassert :ZZLatWithIsom 1 basis_matrix(L)*f_ambient == f*basis_matrix(L)
   end
 
-  return ZZLatWithIsom(Vf, L, f, n)::ZZLatWithIsom
+  return ZZLatWithIsom(Vf, L, f, n)
 end
 
 @doc raw"""
@@ -646,7 +646,7 @@ function integer_lattice_with_isometry(L::ZZLat; neg::Bool = false)
   if neg
     f = -f
   end
-  return integer_lattice_with_isometry(L, f; check = false, ambient_representation = true)::ZZLatWithIsom
+  return integer_lattice_with_isometry(L, f; check = false, ambient_representation = true)
 end
 
 @doc raw"""
@@ -684,7 +684,7 @@ Integer lattice of rank 2 and degree 2
   [0   -1]
 ```
 """
-lattice(Vf::QuadSpaceWithIsom) = ZZLatWithIsom(Vf, lattice(space(Vf)), isometry(Vf), order_of_isometry(Vf))::ZZLatWithIsom
+lattice(Vf::QuadSpaceWithIsom) = ZZLatWithIsom(Vf, lattice(space(Vf)), isometry(Vf), order_of_isometry(Vf))
 
 @doc raw"""
     lattice(Vf::QuadSpaceWithIsom, B::MatElem{<:RationalUnion};
@@ -1437,7 +1437,7 @@ function discriminant_group(Lf::ZZLatWithIsom)
   q = discriminant_group(L)
   f = hom(q, q, elem_type(q)[q(lift(t)*f) for t in gens(q)])
   f = gens(Oscar._orthogonal_group(q, ZZMatrix[matrix(f)]; check = false))[1]
-  return (q, f)::Tuple{TorQuadModule, AutomorphismGroupElem{TorQuadModule}}
+  return (q, f)
 end
 
 @doc raw"""
@@ -1471,29 +1471,43 @@ julia> order(G)
   @req is_integral(Lf) "Underlying lattice must be integral"
   n = order_of_isometry(Lf)
   L = lattice(Lf)
-  f = ambient_isometry(Lf)
   if (n in [1, -1]) || (isometry(Lf) == -identity_matrix(QQ, rank(L)))
+    # Trivial cases: the lattice has rank 0, or is endowed with +- identity
     return image_in_Oq(L)
-  elseif is_definite(L) 
+  elseif rank(L) == euler_phi(n)
+    # this should also cover the case of rank 1
+    # The image of -id and multiplication by primitive root of unity
+    qL, fqL = discriminant_group(Lf)
+    OqL = orthogonal_group(qL)
+    UL = ZZMatrix[-matrix(one(OqL)), matrix(fqL)]
+    UL = elem_type(OqL)[OqL(m; check = false) for m in UL]
+    return sub(OqL, unique!(UL))
+  elseif is_definite(L) || rank(L) == 2
+    # We can compute the orthogonal groups and centralizer with GAP
     OL = orthogonal_group(L)
-    f = OL(f)
-    UL = QQMatrix[matrix(OL(s)) for s in gens(centralizer(OL, f)[1])]
+    f = isometry(Lf)
+    V = ambient_space(L)
+    B = basis_matrix(L)
+    B2 = orthogonal_complement(V, B)
+    C = vcat(B, B2)
+    f = block_diagonal_matrix(QQMatrix[f, identity_matrix(QQ, nrows(B2))])
+    f = inv(C)*f*C
+    f = OL(f; check = false)
+    UL = QQMatrix[matrix(s) for s in gens(centralizer(OL, f)[1])]
     qL = discriminant_group(L)
     UL = ZZMatrix[matrix(hom(qL, qL, elem_type(qL)[qL(lift(t)*g) for t in gens(qL)])) for g in UL]
     unique!(UL)
     OqL = orthogonal_group(qL)
     UL = elem_type(OqL)[OqL(m; check = false) for m in UL]
     return sub(OqL, UL)
-  elseif rank(L) == euler_phi(n)
-    qL = discriminant_group(L)
-    UL = ZZMatrix[matrix(hom(qL, qL, elem_type(qL)[qL(-lift(t)) for t in gens(qL)]))]
-    OqL = orthogonal_group(qL)
-    UL = elem_type(OqL)[OqL(m; check = false) for m in UL]
-    return sub(OqL, UL)
   else
     @req is_of_hermitian_type(Lf) "Not yet implemented for indefinite lattices with isometry which are not of hermitian type"
-    dets = Oscar._local_determinants_morphism(Lf)
-    return kernel(dets)
+    # indefinite of rank bigger or equal to 3
+    # We use Hermitian Miranda-Morrison (see [BH23, Part 6])
+    dets, j = Oscar._local_determinants_morphism(Lf)
+    _, jj = kernel(dets)
+    jj = compose(jj, j)
+    return image(jj)
   end
 end
 
@@ -1812,11 +1826,18 @@ true
 function coinvariant_lattice(L::ZZLat, G::MatrixGroup; ambient_representation::Bool = true)
   F = invariant_lattice(L, G; ambient_representation)
   C = orthogonal_submodule(L, F)
+  if !ambient_representation
+    V = ambient_space(L)
+    B = basis_matrix(L)
+    B2 = orthogonal_complement(V, B)
+    B3 = vcat(B, B2)
+  end
   gene = QQMatrix[]
   for g in gens(G)
     if !ambient_representation
-      m = solve(basis_matrix(L), matrix(g)*basis_matrix(L))
-      m = solve_left(basis_matrix(C), basis_matrix(C)*m)
+      g_ambient = block_diagonal_matrix(QQMatrix[matrix(g), identity_matrix(QQ, nrows(B2))])
+      g_ambient = inv(B3)*g_ambient*B3
+      m = solve_left(basis_matrix(C), basis_matrix(C)*g_ambient)
       push!(gene, m)
     else
       push!(gene, matrix(g))
@@ -1864,11 +1885,18 @@ with gram matrix
 function invariant_coinvariant_pair(L::ZZLat, G::MatrixGroup; ambient_representation::Bool = true)
   F = invariant_lattice(L, G; ambient_representation)
   C = orthogonal_submodule(L, F)
+  if !ambient_representation
+    V = ambient_space(L)
+    B = basis_matrix(L)
+    B2 = orthogonal_complement(V, B)
+    B3 = vcat(B, B2)
+  end
   gene = QQMatrix[]
   for g in gens(G)
     if !ambient_representation
-      m = solve(basis_matrix(L), matrix(g)*basis_matrix(L))
-      m = solve_left(basis_matrix(C), basis_matrix(C)*m)
+      g_ambient = block_diagonal_matrix(QQMatrix[matrix(g), identity_matrix(QQ, nrows(B2))])
+      g_ambient = inv(B3)*g_ambient*B3
+      m = solve_left(basis_matrix(C), basis_matrix(C)*g_ambient)
       push!(gene, m)
     else
       push!(gene, matrix(g))
diff --git a/src/GAP/gap_to_oscar.jl b/src/GAP/gap_to_oscar.jl
index 8837b3326eac..cc293a1502f5 100644
--- a/src/GAP/gap_to_oscar.jl
+++ b/src/GAP/gap_to_oscar.jl
@@ -45,13 +45,13 @@ function (F::FinField)(x::GAP.FFE)
         # value fits into an Int; telling Julia about this via a type assertion
         # results in slightly better code
         val = GAPWrap.INT_FFE_DEFAULT(x)
-        return F(val)
+        return F(val)::elem_type(F)
     end
 
     # Use `iso_oscar_gap` not `iso_gap_oscar` in order to make sure
     # that the result is in `F`, and in order to cache the isomorphism in `F`.
     iso = iso_oscar_gap(F)
-    return preimage(iso, x)
+    return preimage(iso, x)::elem_type(F)
 end
 
 # test code for producing a gap finite field element not stored as FFE:  `GAP.Globals.Z(65537)`
diff --git a/src/Groups/abelian_aut.jl b/src/Groups/abelian_aut.jl
index 7e1ac3fc9a4e..755763b5e566 100644
--- a/src/Groups/abelian_aut.jl
+++ b/src/Groups/abelian_aut.jl
@@ -42,7 +42,7 @@ Base.:^(x::AbTorElem,f::AutGrpAbTorElem) = apply_automorphism(f, x, true)
 # the _as_subgroup function needs a redefinition
 # to pass on the to_gap and to_oscar attributes to the subgroup
 function _as_subgroup(aut::AutomorphismGroup{S}, subgrp::GapObj) where S <: Union{TorQuadModule,GrpAbFinGen}
-  function img(x::S)
+  function img(x::AutomorphismGroupElem{S})
     return group_element(aut, x.X)
   end
   to_gap = get_attribute(aut, :to_gap)
diff --git a/src/Groups/gsets.jl b/src/Groups/gsets.jl
index 03e89f6251c4..a17c1b9f745d 100644
--- a/src/Groups/gsets.jl
+++ b/src/Groups/gsets.jl
@@ -947,14 +947,14 @@ ZZRingElem[12, 12, 16]
 ```
 """
 function orbit_representatives_and_stabilizers(G::MatrixGroup{E}, k::Int) where E <: FinFieldElem
-  F = G.ring
-  n = G.deg
+  F = base_ring(G)
+  n = degree(G)
   q = GAP.Obj(order(F))
   V = VectorSpace(F, n)
   orbs = GAP.Globals.Orbits(G.X, GAP.Globals.Subspaces(GAP.Globals.GF(q)^n, k))
   orbreps = [GAP.Globals.BasisVectors(GAP.Globals.Basis(orb[1])) for orb in orbs]
-  stabs = [Oscar._as_subgroup_bare(G, GAP.Globals.Stabilizer(G.X, v, GAP.Globals.OnSubspacesByCanonicalBasis)) for v in orbreps]
-  orbreps = [[[F(x) for x in v] for v in bas] for bas in orbreps]
-  orbreps = [sub(V, [V(v) for v in bas])[1] for bas in orbreps]
-  return [(orbreps[i], stabs[i]) for i in 1:length(stabs)]
+  stabs = [Oscar._as_subgroup_bare(G, GAP.Globals.Stabilizer(G.X, v, GAP.Globals.OnSubspacesByCanonicalBasis)) for v in orbreps]::Vector{typeof(G)}
+  orbreps1 = [[[F(x) for x in v] for v in bas] for bas in orbreps]::Vector{Vector{Vector{elem_type(F)}}}
+  orbreps2 = [sub(V, [V(v) for v in bas])[1] for bas in orbreps1]
+  return [(orbreps2[i], stabs[i]) for i in 1:length(stabs)]
 end
diff --git a/src/Groups/matrices/MatGrp.jl b/src/Groups/matrices/MatGrp.jl
index 03ea64d12404..67ddcc34fce3 100644
--- a/src/Groups/matrices/MatGrp.jl
+++ b/src/Groups/matrices/MatGrp.jl
@@ -467,7 +467,7 @@ transpose(x::MatrixGroupElem) = MatrixGroupElem(x.parent, transpose(x.elm))
 
 Return the base ring of the matrix group `G`.
 """
-base_ring(G::MatrixGroup) = G.ring
+base_ring(G::MatrixGroup{RE}) where RE <: RingElem = G.ring::parent_type(RE)
 
 """
     degree(G::MatrixGroup)