Attempt to improve some codes in QuadForm (#1254)

docs/src/quad_forms/Zgenera.md

@@ -101,8 +101,8 @@ ZZLocalGenus
 ### Creation
 
 ```@docs
-genus(L::ZZLat, p)
-genus(A::ZZMatrix, p)
+genus(::ZZLat, ::IntegerUnion)
+genus(::QQMatrix, ::IntegerUnion)
 ```
 
 ### Attributes

src/QuadForm/Herm/Genus.jl

@@ -129,7 +129,7 @@ $\mathfrak p$ of $\mathcal O_K$, return the rank of any hermitian lattice whose
 $\mathfrak p$-adic completion has local genus symbol `g`.
 """
 function rank(G::HermLocalGenus)
-  return reduce(+, (rank(G, i) for i in 1:length(G)), init = Int(0))
+  return reduce(+, rank(G, i) for i in 1:length(G); init = Int(0))
 end
 
 @doc raw"""
@@ -161,7 +161,7 @@ The returned value is $1$ or $-1$ depending on whether the determinant is a loca
 norm in `K`.
 """
 function det(G::HermLocalGenus)
-  return reduce(*, (det(G, i) for i in 1:length(G)), init = Int(1))
+  return reduce(*, det(G, i) for i in 1:length(G); init = Int(1))
 end
 
 @doc raw"""
@@ -631,7 +631,7 @@ function _genus(::Type{HermLat}, E::S, p::T, data::Vector{Tuple{Int, Int, Int}},
   z.is_dyadic = is_dyadic
   z.is_ramified = is_ramified
   z.is_split = is_split
-  keep = [i for (i,s) in enumerate(data) if s[2] != 0]  # We keep only blocks of non-zero rank
+  keep = Int[i for (i, s) in enumerate(data) if s[2] != 0]  # We keep only blocks of non-zero rank
   z.data = data[keep]
   return z
 end
@@ -791,7 +791,7 @@ function _genus(::Type{HermLat}, E::S, p::T, data::Vector{Tuple{Int, Int, Int}},
   z.is_split = false
   # We test the cheap thing
   @req z.is_dyadic && z.is_ramified "Prime must be dyadic and ramified"
-  keep = [i for (i,s) in enumerate(data) if s[2] != 0]    # We only keep the blocks of non-zero rank
+  keep = Int[i for (i, s) in enumerate(data) if s[2] != 0]    # We only keep the blocks of non-zero rank
   z.norm_val = norms[keep]
   z.data = data[keep]
   z.ni = _get_ni_from_genus(z)
@@ -980,7 +980,7 @@ function ==(G1::HermLocalGenus, G2::HermLocalGenus)
     # in particular normal.
 
     # Thus we only have to check Theorem 3.3.6 4.
-    return all(i -> det(G1, i) == det(G2, i), [i for i in 1:t if iseven(scale(G1,i))])
+    return all(det(G1, i) == det(G2, i) for i in 1:t if iseven(scale(G1, i)))
   end
 
   # Dyadic ramified case
@@ -1354,7 +1354,7 @@ function direct_sum(G1::HermGenus, G2::HermGenus)
   prim = eltype(primes(G1))[]
   P1 = Set(primes(G1))
   P2 = Set(primes(G2))
-  for p in union(P1, P2)
+  for p in union!(P1, P2)
     g1 = G1[p]
     g2 = G2[p]
     g3 = direct_sum(g1, g2)
@@ -1367,7 +1367,7 @@ function direct_sum(G1::HermGenus, G2::HermGenus)
   # For genera of hermitian lattices, are bad all primes dividing 2 and those
   # dividing the discriminant of the field extension (discriminant of the
   # maximal order of the top field in E/K).
-  bd = union(support(2*maximal_order(base_field(E))), support(discriminant(maximal_order(E))))
+  bd = union!(support(2*maximal_order(base_field(E))), support(discriminant(maximal_order(E))))
   # We keep only the local symbols which are defined over bad primes or which
   # are not unimodular.
   # Unimodular <=> There is only one Jordan block, and it is a scale valuation 0
@@ -1423,7 +1423,7 @@ function genus(L::Vector{<:HermLocalGenus}, signatures::Dict{<:InfPlc, Int})
   @req all(g -> rank(g) == r, L) "Local genus symbols must have the same rank"
   E = base_field(first(L))
   @req all(g -> base_field(g) == E, L) "Local genus symbols must be defined over the same extension E/K"
-  bd = union(support(2*maximal_order(base_field(E))), support(discriminant(maximal_order(E))))
+  bd = union!(support(2*maximal_order(base_field(E))), support(discriminant(maximal_order(E))))
   filter!(g -> (prime(g) in bd) || (scales(g) != Int[0]), L)
   return HermGenus(E, r, L, signatures)
 end
@@ -1448,7 +1448,7 @@ Return the global genus symbol `G` of the hermitian lattice `L`. `G` satisfies:
 end
 
 function _genus(L::HermLat)
-  bad = bad_primes(L, discriminant = true, dyadic = true)
+  bad = bad_primes(L; discriminant = true, dyadic = true)
 
   K = base_field(L)
   k = base_field(K)
@@ -1470,7 +1470,7 @@ end
 function _check_global_genus(LGS, signatures)
   _non_norm = _non_norm_primes(LGS, ignore_split = true)
   P = length(_non_norm)
-  I = length([(s, N) for (s, N) in signatures if isodd(mod(N, 2))])
+  I = count(v -> isodd(mod(v[2], 2)), signatures)
   if mod(P + I, 2) == 1
     return false
   end
@@ -1547,7 +1547,7 @@ function _hermitian_form_with_invariants(E, dim, P, N)
       push!(D, el)
     end
   end
-  push!(D, a * prod(D, init = one(E)))
+  push!(D, a * prod(D; init = one(E)))
   Dmat = diagonal_matrix(D)
   dim0, P0, N0 = _hermitian_form_invariants(Dmat)
   @assert dim == dim0
@@ -1601,9 +1601,9 @@ function representative(G::HermGenus)
   @vprintln :Lattice 1 "Finding maximal integral lattice"
   M = maximal_integral_lattice(V)
   lp = primes(G)
-  bd = union(support(2*fixed_ring(M)), support(discriminant(maximal_order(E))))
-  lp = union(lp, bd)
-  for p in lp
+  bd = union!(support(2*fixed_ring(M)), support(discriminant(maximal_order(E))))
+  union!(bd, lp)
+  for p in bd
     @vprintln :Lattice 1 "Finding representative for $g at $(prime(g))..."
     g = G[p]
     L = representative(g)
@@ -1694,14 +1694,14 @@ function hermitian_local_genera(E, p, rank::Int, det_val::Int, min_scale::Int, m
       dets = Vector{Int}[]
       for b in g
         if mod(b[1], 2) == 0
-          push!(dets, [1, -1])
+          push!(dets, Int[1, -1])
         end
         if mod(b[1], 2) == 1
-          push!(dets, [hyperbolic_det^(div(b[2], 2))])
+          push!(dets, Int[hyperbolic_det^(div(b[2], 2))])
         end
       end
 
-      for d in cartesian_product_iterator(dets, inplace = true)# Iterators.product(dets...)
+      for d in cartesian_product_iterator(dets)# Iterators.product(dets...)
         g2 = Vector{Tuple{Int, Int, Int}}(undef, length(g))
         for k in 1:n
           # Again the 0 for dummy purposes
@@ -1728,7 +1728,7 @@ function hermitian_local_genera(E, p, rank::Int, det_val::Int, min_scale::Int, m
     for b in g
       if mod(b[2], 2) == 1
         @assert iseven(b[1])
-        push!(det_norms, [[1, div(b[1], 2)], [-1, div(b[1], 2)]])
+        push!(det_norms, Vector{Int}[Int[1, div(b[1], 2)], Int[-1, div(b[1], 2)]])
       end
       if mod(b[2], 2) == 0
         dn = Vector{Int}[]
@@ -1748,7 +1748,7 @@ function hermitian_local_genera(E, p, rank::Int, det_val::Int, min_scale::Int, m
       end
     end
 
-    for dn in cartesian_product_iterator(det_norms, inplace = true)
+    for dn in cartesian_product_iterator(det_norms)
       g2 = Vector{Tuple{Int, Int, Int, Int}}(undef, length(g))
       for k in 1:n
         g2[k] = (g[k][1], g[k][2], dn[k][1], dn[k][2])
@@ -1783,25 +1783,20 @@ function hermitian_genera(E::Hecke.NfRel, rank::Int, signatures::Dict{<: InfPlc,
   @req rank >= 0 "Rank must be a non-negative integer"
   K = base_field(E)
   OE = maximal_order(E)
-  bd = union(support(2*maximal_order(K)), support(discriminant(OE)))
+  bd = union!(support(2*maximal_order(K)), support(discriminant(OE)))
   @req !iszero(max_scale) "max_scale must be a non-zero fractional ideal"
   @req !iszero(min_scale) "min_scale must be a non-zero fractional ideal"
-  @req all(v -> 0 <= v <= rank, collect(values(signatures))) "Incompatible signatures and rank"
-  primes = union(bd, support(norm(min_scale)))
-  union!(primes, support(norm(max_scale)))
-  for p in support(norm(determinant))
-    if !(p in primes)
-      push!(primes, p)
-    end
-  end
-  unique!(primes)
-  sort!(primes, by = (x -> minimum(x)))
+  @req all(v -> 0 <= v <= rank, values(signatures)) "Incompatible signatures and rank"
+  union!(bd, support(norm(min_scale)))
+  union!(bd, support(norm(max_scale)))
+  union!(bd, support(norm(determinant)))
+  sort!(bd; by = (x -> minimum(x)))
   local_symbols = Vector{local_genus_herm_type(E)}[]
 
   mins = norm(min_scale)
   maxs = norm(max_scale)
   ds = norm(determinant)
-  for p in primes
+  for p in bd
     det_val = valuation(ds, p)
     minscale_p = valuation(mins, p)
     maxscale_p = valuation(maxs, p)
@@ -1815,7 +1810,7 @@ function hermitian_genera(E::Hecke.NfRel, rank::Int, signatures::Dict{<: InfPlc,
   end
 
   res = genus_herm_type(E)[]
-  it = cartesian_product_iterator(local_symbols, inplace = true)
+  it = cartesian_product_iterator(local_symbols)
   for gs in it
     c = copy(gs)
     b = _check_global_genus(c, signatures)
@@ -1862,9 +1857,9 @@ function rescale(G::HermGenus, a::Union{FieldElem, RationalUnion})
   E = base_field(G)
   K = base_field(E)
   I = K(a)*maximal_order(K)
-  pd = union(primes(G), support(I))
-  bd = union(support(2*maximal_order(K)), support(discriminant(maximal_order(E))))
-  LGS = [rescale(G[p], a) for p in pd]
+  pd = union!(support(I), primes(G))
+  bd = union!(support(2*maximal_order(K)), support(discriminant(maximal_order(E))))
+  LGS = local_genus_herm_type(E)[rescale(G[p], a) for p in pd]
   filter!(g -> (prime(g) in bd) || (scales(g) != Int[0]), LGS)
   r = rank(G)
   sig = copy(signatures(G))

src/QuadForm/Herm/Lattices.jl

@@ -65,10 +65,10 @@ function hermitian_lattice(E::NumField, B::PMat; gram = nothing, check::Bool = t
     @assert gram isa MatElem
     @req is_square(gram) "gram must be a square matrix"
     @req ncols(B) == nrows(gram) "Incompatible arguments: the number of columns of B must correspond to the size of gram"
-    gram = map_entries(E, gram)
-    V = hermitian_space(E, gram)
+    gramE = map_entries(E, gram)
+    V = hermitian_space(E, gramE)
   end
-  return lattice(V, B, check = check)
+  return lattice(V, B; check)
 end
 
 @doc raw"""
@@ -84,7 +84,7 @@ matrix over `E` of size the number of columns of `basis`.
 By default, `basis` is checked to be of full rank. This test can be disabled by setting
 `check` to false.
 """
-hermitian_lattice(E::NumField, basis::MatElem; gram = nothing, check::Bool = true) = hermitian_lattice(E, pseudo_matrix(basis), gram = gram, check = check)
+hermitian_lattice(E::NumField, basis::MatElem; gram = nothing, check::Bool = true) = hermitian_lattice(E, pseudo_matrix(basis); gram, check)
 
 @doc raw"""
     hermitian_lattice(E::NumField, gens::Vector ; gram = nothing) -> HermLat
@@ -103,7 +103,7 @@ function hermitian_lattice(E::NumField, gens::Vector; gram = nothing)
   if length(gens) == 0
     @assert gram !== nothing
     pm = pseudo_matrix(matrix(E, 0, nrows(gram), []))
-    L = hermitian_lattice(E, pm, gram = gram, check = false)
+    L = hermitian_lattice(E, pm; gram, check = false)
     return L
   end
   @assert length(gens[1]) > 0
@@ -115,8 +115,8 @@ function hermitian_lattice(E::NumField, gens::Vector; gram = nothing)
     @assert gram isa MatElem
     @req is_square(gram) "gram must be a square matrix"
     @req length(gens[1]) == nrows(gram) "Incompatible arguments: the length of the elements of gens must correspond to the size of gram"
-    gram = map_entries(E, gram)
-    V = hermitian_space(E, gram)
+    gramE = map_entries(E, gram)
+    V = hermitian_space(E, gramE)
   end
   return lattice(V, gens)
 end
@@ -129,9 +129,9 @@ lattice inside the hermitian space over `E` with Gram matrix `gram`.
 """
 function hermitian_lattice(E::NumField; gram::MatElem)
   @req is_square(gram) "gram must be a square matrix"
-  gram = map_entries(E, gram)
-  B = pseudo_matrix(identity_matrix(E, ncols(gram)))
-  return hermitian_lattice(E, B, gram = gram, check = false)
+  gramE = map_entries(E, gram)
+  B = pseudo_matrix(identity_matrix(E, ncols(gramE)))
+  return hermitian_lattice(E, B; gram = gramE, check = false)
 end
 
 ################################################################################
@@ -227,8 +227,8 @@ function norm(L::HermLat)
   K = base_ring(G)
   R = base_ring(L)
   C = coefficient_ideals(L)
-  to_sum = sum(G[i, i] * C[i] * v(C[i]) for i in 1:length(C))
-  to_sum = to_sum + R * reduce(+, [tr(C[i] * G[i, j] * v(C[j])) for j in 1:length(C) for i in 1:(j-1)], init = ZZ(0)*inv(1*base_ring(R)))
+  to_sum = sum(G[i, i] * C[i] * v(C[i]) for i in 1:length(C); init = fractional_ideal(R, zero(K)))
+  to_sum = reduce(+, tr(C[i] * G[i, j] * v(C[j]))*R for j in 1:length(C) for i in 1:(j-1); init = to_sum)
   n = minimum(numerator(to_sum))//denominator(to_sum)
   L.norm = n
   return n
@@ -251,7 +251,7 @@ function scale(L::HermLat)
   for i in 1:d
     push!(to_sum, involution(L)(to_sum[i]))
   end
-  s = sum(to_sum, init = zero(base_field(L))*base_ring(L))
+  s = sum(to_sum; init = fractional_ideal(base_ring(L), zero(base_field(L))))
   L.scale = s
   return s
 end
@@ -270,7 +270,7 @@ function rescale(L::HermLat, a::Union{FieldElem, RationalUnion})
   K = fixed_field(L)
   b = base_field(L)(K(a))
   gramamb = gram_matrix(ambient_space(L))
-  return hermitian_lattice(base_field(L), pseudo_matrix(L),
+  return hermitian_lattice(base_field(L), pseudo_matrix(L);
                            gram = b * gramamb)
 end
 
@@ -472,12 +472,12 @@ function _ismaximal_integral(L::HermLat, p)
 
   absolute_map = absolute_simple_field(ambient_space(L))[2]
 
-  M = local_basis_matrix(L, p, type = :submodule)
+  M = local_basis_matrix(L, p; type = :submodule)
   G = gram_matrix(ambient_space(L), M)
   F, h = residue_field(R, D[1][1])
   hext = extend(h, E)
   sGmodp = map_entries(hext, s * G)
-  Vnullity, V = kernel(sGmodp, side = :left)
+  Vnullity, V = kernel(sGmodp; side = :left)
   if Vnullity == 0
     return true, zero_matrix(E, 0, 0)
   end
@@ -589,7 +589,7 @@ function _maximal_integral_lattice(L::HermLat, p, minimal = true)
     end
     # new we look for zeros of ax^2 + by^2
     kk, h = residue_field(R, P)
-    while sum(Int[S[i] * nrows(B[i]) for i in 1:length(B)]) > 1
+    while sum(S[i] * nrows(B[i]) for i in 1:length(B); init = 0) > 1
       k = 0
       for i in 1:(length(S) + 1)
         if S[i] == 1