cache automorphism groups of number fields (#2436)

* cache automorphism groups of number fields

and related:
 - aut grp local fields
 - IdelClassGmodule
 - allow for use of lll-order by changeing ideal parents
 - galois group via cohomolgy: make sure the rcf is normal

* add tests...

* remove debugging assertion

* and another border case

* more trivia

experimental/GModule/GModule.jl

@@ -915,10 +915,28 @@ function hom_base(C::_T, D::_T) where _T <: GModule{<:Any, <:Generic.FreeModule{
     for i=1:length(z1)
       push!(t, hom_base(z1[i], z2[i]))
     end
-    tt = [Hecke.modular_lift([t[i][j] for i=1:length(z1)], me) for j=1:length(t[1])]
-    if length(tt) == 0
+    #should actually compute an rref of the hom base to make sure
+    if any(x->length(x) != length(t[1]), t)
+      #bad prime...
+      continue
+    end
+    if length(t[1]) == 0
       return []
     end
+    pv = [findfirst(!iszero, x) for x = t[1]]
+    if any(!isequal(pv[1]), pv)
+      continue
+    end
+    for i=1:length(pv)
+      for j=1:length(t)
+        if !isone(t[j][i][pv[i]])
+          t[j][i] *= inv(t[j][i][pv[i]])
+        end
+      end
+    end
+    @assert all(i->all(x->x[pv[i]] == 1, t[i]), 1:length(pv))
+
+    tt = [Hecke.modular_lift([t[i][j] for i=1:length(z1)], me) for j=1:length(t[1])]
     @assert base_ring(tt[1]) == k
     if isone(pp)
       pp = ZZRingElem(p)
@@ -929,7 +947,7 @@ function hom_base(C::_T, D::_T) where _T <: GModule{<:Any, <:Generic.FreeModule{
       pp *= p
       S = []
       for t = T
-        fl, s = induce_rational_reconstruction(t, pp)
+        fl, s = induce_rational_reconstruction(t, pp)# , ErrorTolerant = true)
         fl || break
         push!(S, s)
       end
@@ -992,7 +1010,7 @@ function hom_base(C::_T, D::_T) where _T <: GModule{<:Any, <:Generic.FreeModule{
           reco *= 2
         end
         for t = T
-          fl, s = induce_rational_reconstruction(t, pp)
+          fl, s = induce_rational_reconstruction(t, pp, ErrorTolerant = true)
           fl || break
           push!(S, s)
         end
@@ -1228,22 +1246,22 @@ function Hecke.induce_crt(a::Generic.MatSpaceElem{nf_elem}, b::Generic.MatSpaceE
   return c
 end
 
-function Hecke.induce_rational_reconstruction(a::Generic.MatSpaceElem{nf_elem}, pg::ZZRingElem)
+function Hecke.induce_rational_reconstruction(a::Generic.MatSpaceElem{nf_elem}, pg::ZZRingElem; ErrorTolerant::Bool = false)
   c = parent(a)()
   for i=1:nrows(a)
     for j=1:ncols(a)
-      fl, c[i,j] = rational_reconstruction(a[i,j], pg)
+      fl, c[i,j] = rational_reconstruction(a[i,j], pg)#, ErrorTolerant = ErrorTolerant)
       fl || return fl, c
     end
   end
   return true, c
 end
 
-function Hecke.induce_rational_reconstruction(a::ZZMatrix, pg::ZZRingElem)
+function Hecke.induce_rational_reconstruction(a::ZZMatrix, pg::ZZRingElem; ErrorTolerant::Bool = false)
   c = zero_matrix(QQ, nrows(a), ncols(a))
   for i=1:nrows(a)
     for j=1:ncols(a)
-      fl, n, d = rational_reconstruction(a[i,j], pg, ErrorTolerant = true)
+      fl, n, d = rational_reconstruction(a[i,j], pg, ErrorTolerant = ErrorTolerant)
       fl || return fl, c
       c[i,j] = n//d
     end

experimental/GModule/GaloisCohomology.jl

@@ -13,19 +13,47 @@ export local_index
 Oscar.elem_type(::Type{Hecke.NfMorSet{T}}) where {T <: Hecke.LocalField} = Hecke.LocalFieldMor{T, T}
 parent(f::Hecke.LocalFieldMor) = Hecke.NfMorSet(domain(f))
 
+_can_cache_aut(::FlintPadicField) = nothing
+function _can_cache_aut(k) 
+  a = get_attribute(k, :AutGroup)
+  if a === nothing
+    a = Dict{Tuple, Map}()
+    set_attribute!(k, :AutGroup => a)
+  end
+  return a
+end
 #not type restricted: used for number fields as well as
 #LocalFields
 #could be extended to allow more group types
 function Oscar.automorphism_group(::Type{PermGroup}, k)
+  a = _can_cache_aut(k)
+  if a !== nothing && haskey(a, (PermGroup, ))
+    mG = a[(PermGroup, )]
+    return domain(mG), mG
+  end
   G, mG = automorphism_group(k)
   mH = isomorphism(PermGroup, G)
-  return codomain(mH), inv(mH)*mG
+  mmH = inv(mH)*mG
+  if a !== nothing 
+    a[(PermGroup, )] = mmH
+  end
+  return codomain(mH), mmH
 end
 
 function Oscar.automorphism_group(::Type{PermGroup}, K, k)
+  a = _can_cache_aut(k)
+  if a !== nothing && haskey(a, (PermGroup, k))
+    mG = a[(PermGroup, k)]
+    return domain(mG), mG
+  end
+
   G, mG = automorphism_group(K, k)
   mH = isomorphism(PermGroup, G)
-  return codomain(mH), inv(mH)*mG
+  mmH = inv(mH)*mG
+  if a !== nothing 
+    a[(PermGroup, k)] = mmH
+  end
+  return codomain(mH), mmH
 end
 
 
@@ -45,11 +73,17 @@ end
 The natural `ZZ[G]` module where `G`, the
   automorphism group, acts on the ideal group defining the class field.
 """
-function Oscar.gmodule(R::ClassField, mG = automorphism_group(PermGroup, base_field(R))[2])
+function Oscar.gmodule(R::ClassField, mG = automorphism_group(PermGroup, base_field(R))[2]; check::Bool = true)
   k = base_field(R)
   G = domain(mG)
   mR = R.rayclassgroupmap
   mq = R.quotientmap
+  if check
+    c, mc = conductor(R)
+    @req all(x -> c == Hecke.induce_image(mG(x), c), gens(G)) "field is not normal"
+    s1 = Set(mc)
+    @req all(x -> s1 == Set(Hecke.induce_image(mG(x), y) for y = s1), gens(G)) "field is not normal"
+  end
 
   ac = Hecke.induce_action(mR, [image(mG, g) for g = gens(G)], mq)
   return GModule(G, ac)
@@ -655,8 +689,13 @@ or Ali,
 Find a gmodule C s.th. C is cohomology-equivalent to the cohomology
 of the idel-class group. The primes in `s` will always be used.
 """
-function idel_class_gmodule(k::AnticNumberField, s::Vector{Int} = Int[])
-  @vprint :GaloisCohomology 1 "Ideal class group cohomology for $k\n"
+function idel_class_gmodule(k::AnticNumberField, s::Vector{Int} = Int[]; redo::Bool=false)
+  @vprint :GaloisCohomology 2 "Ideal class group cohomology for $k\n"
+  I = get_attribute(k, :IdelClassGmodule)
+  if !redo && I !== nothing
+    return I
+  end
+
   I = IdelParent()
   I.k = k
   G, mG = automorphism_group(PermGroup, k)
@@ -799,7 +838,7 @@ function idel_class_gmodule(k::AnticNumberField, s::Vector{Int} = Int[])
     W, pro, inj = direct_product(V, F, task = :both)
     @assert isdefined(W, :hnf)
 
-    ac = GrpAbFinGenMap(pro[1]*psi*sigma*pseudo_inv(psi)*inj[1])+ GrpAbFinGenMap(pro[2]*hom(F, W, [inj[1](preimage(psi, x[i])) - inj[2](F[i-1]) for i=2:length(x)]))
+    ac = GrpAbFinGenMap(pro[1]*psi*sigma*pseudo_inv(psi)*inj[1])+ GrpAbFinGenMap(pro[2]*hom(F, W, GrpAbFinGenElem[inj[1](preimage(psi, x[i])) - inj[2](F[i-1]) for i=2:length(x)]))
     Et = gmodule(G_inf, [ac])
     @assert is_consistent(Et)
     mq = pseudo_inv(psi)*inj[1]
@@ -857,6 +896,7 @@ function idel_class_gmodule(k::AnticNumberField, s::Vector{Int} = Int[])
   @hassert :GaloisCohomology 1 is_consistent(q)
   I.mq = mq
   I.data = (q, F[1])
+  set_attribute!(k, :IdelClassGmodule=>I)
   return I
 end
 
@@ -905,6 +945,12 @@ end
 function Oscar.ideal(I::IdelParent, a::GrpAbFinGenElem; coprime::Union{NfOrdIdl, Nothing})
   a = preimage(I.mq, a)
   zk = maximal_order(I.k)
+  o_zk = zk
+  if coprime !== nothing && order(coprime) !== zk
+    o_zk = order(coprime)
+    coprime = zk*coprime
+    @assert order(coprime) === zk
+  end
   id = 1*zk
   for p = I.S
     lp = prime_decomposition(zk, minimum(p))
@@ -922,7 +968,7 @@ function Oscar.ideal(I::IdelParent, a::GrpAbFinGenElem; coprime::Union{NfOrdIdl,
       end
     end
   end
-  return id
+  return o_zk*id
 end
 
 function Oscar.galois_group(A::ClassField)
@@ -964,6 +1010,14 @@ function Oscar.galois_group(A::ClassField, ::QQField; idel_parent::IdelParent =
   mR = A.rayclassgroupmap
   mQ = A.quotientmap
   zk = order(m0)
+  @req order(automorphism_group(nf(zk))[1]) == degree(zk) "base field must be normal"
+  if !Hecke.is_normal(A)
+    A = normal_closure(A)
+    mR = A.rayclassgroupmap
+    mQ = A.quotientmap
+    m0, m_inf = defining_modulus(A)
+    @assert length(m_inf) == 0
+  end
   qI = cohomology_group(idel_parent, 2)
   q, mq = snf(qI[1])
   a = qI[2](image(mq, q[1])) # should be a 2-cycle in q

test/Experimental/gmodule.jl

@@ -64,3 +64,40 @@ end
   @test order(preimage(q[2], c)) == 6
 end
 
+@testset "GlobalFundClass" begin
+  k, a = quadratic_field(-101)
+  H = hilbert_class_field(k)
+  G, _ = galois_group(H, QQ)
+  @test describe(G) == "D28"
+
+  r = ray_class_field(49*maximal_order(k), n_quo = 7)
+  G, _ = galois_group(r, QQ)
+  @test describe(G) == "C7 x D14"
+
+  lp = collect(keys(factor(7*maximal_order(k))))
+  s = ray_class_field(lp[1])
+  G, _ = galois_group(s, QQ)
+  @test describe(G) == "C6 x D42"
+
+  @test degree(normal_closure(s)) == 126
+end
+
+@testset "BrauerGroup" begin
+  G = transitive_group(24, 201)
+  T = character_table(G)
+  R = gmodule(T[9])
+  S = gmodule(CyclotomicField, R)
+
+  @test schur_index(T[9]) == 2
+
+  B, mB = relative_brauer_group(base_ring(S), character_field(S))
+  b = B(S)
+
+  C = grunwald_wang(Dict(2*ZZ => 2), Dict(complex_embeddings(QQ)[1] => 2))
+  @test degree(C) == 2
+  K, m1, m2 = compositum(base_ring(S), absolute_simple_field(number_field(C))[1])
+
+  SS = Oscar.GModuleFromGap.gmodule_over(m2, gmodule(K, S))
+
+  @test degree(base_ring(SS)) == 2
+end