feat: improve algebra functionality

src/AlgAss/AlgQuat.jl

@@ -502,7 +502,7 @@ function _is_principal_maximal_quaternion_generic_proper(a, M, side = :right)
 
     #@show B
 
-    v = _short_vectors_gram_integral(G, FlintZZ(B), hard = true)
+    v = _short_vectors_gram_integral(Vector, G, FlintZZ(B), hard = true)
 
     #if min == degree(base_ring(A))
     for w in v

src/AlgAss/Map.jl

@@ -202,14 +202,22 @@
   end
 end
 
-function hom(A::R, B::S, imgs::Vector) where {R <: AbstractAssociativeAlgebra, S <: AbstractAssociativeAlgebra}
+function hom(A::R, B::S, imgs::Vector; check = true) where {R <: AbstractAssociativeAlgebra, S <: AbstractAssociativeAlgebra}
   @assert length(imgs) == dim(A)
   bmat = basis_matrix(imgs)
-  return hom(A, B, bmat)
+  return hom(A, B, bmat; check = check)
 end
 
-function hom(A::R, B::S, M::T) where {R <: AbstractAssociativeAlgebra, S <: AbstractAssociativeAlgebra, T <: MatElem}
-  return AbsAlgAssMor{R, S, T}(A, B, M)
+function hom(A::R, B::S, M::T; check = true) where {R <: AbstractAssociativeAlgebra, S <: AbstractAssociativeAlgebra, T <: MatElem}
+  h = AbsAlgAssMor{R, S, T}(A, B, M)
+  if check
+    for a in basis(A), b in basis(A)
+      if h(a * b) != h(a) * h(b)
+        error("Data does not define an algebra homomorphism")
+      end
+    end
+  end
+  return h
 end
 
 function hom(A::R, B::S, M::T, N::T) where {R <: AbstractAssociativeAlgebra, S <: AbstractAssociativeAlgebra, T <: MatElem}

src/AlgAssRelOrd/Ideal.jl

@@ -611,28 +611,6 @@ end
 #
 ################################################################################
 
-@doc raw"""
-    *(a::AlgAssRelOrdIdl, x::AbsNumFieldOrderElem) -> AlgAssRelOrdIdl
-    *(x::AbsNumFieldOrderElem, a::AlgAssRelOrdIdl) -> AlgAssRelOrdIdl
-    *(a::AlgAssRelOrdIdl, x::RelNumFieldOrderElem) -> AlgAssRelOrdIdl
-    *(x::RelNumFieldOrderElem, a::AlgAssRelOrdIdl) -> AlgAssRelOrdIdl
-    *(a::AlgAssRelOrdIdl, x::Int) -> AlgAssRelOrdIdl
-    *(x::Int, a::AlgAssRelOrdIdl) -> AlgAssRelOrdIdl
-    *(a::AlgAssRelOrdIdl, x::ZZRingElem) -> AlgAssRelOrdIdl
-    *(x::ZZRingElem, a::AlgAssRelOrdIdl) -> AlgAssRelOrdIdl
-    *(a::AlgAssRelOrdIdl{S, T, U}, x::S) where { S, T, U } -> AlgAssRelOrdIdl{S, T, U}
-    *(x::S, a::AlgAssRelOrdIdl{S, T, U}) where { S, T, U } -> AlgAssRelOrdIdl{S, T, U}
-    *(a::AlgAssRelOrdIdl{S, T, U}, x::AbstractAssociativeAlgebraElem{S}) where { S, T, U }
-      -> AlgAssRelOrdIdl{S, T, U}
-    *(x::AbstractAssociativeAlgebraElem{S}, a::AlgAssRelOrdIdl{S, T, U}) where { S, T, U }
-      -> AlgAssRelOrdIdl{S, T, U}
-    *(a::AlgAssRelOrdIdl{S, T, U}, x::AlgAssRelOrdElem{S, T, U}) where { S, T, U }
-      -> AlgAssRelOrdIdl{S, T, U}
-    *(x::AlgAssRelOrdElem{S, T, U}, a::AlgAssRelOrdIdl{S, T}) where { S, T, U }
-      -> AlgAssRelOrdIdl{S, T, U}
-
-Returns the ideal $a*x$ respectively $x*a$.
-"""
 function *(a::AlgAssRelOrdIdl{S, T, U}, x::Union{ Int, ZZRingElem, AbsNumFieldOrderElem, RelNumFieldOrderElem, S }) where { S <: NumFieldElem, T, U }
   if iszero(x)
     return _zero_ideal(algebra(a))
@@ -750,7 +728,10 @@ Returns `true` if $a$ and $b$ are equal and `false` otherwise.
 """
 function ==(a::AlgAssRelOrdIdl{S, T, U}, b::AlgAssRelOrdIdl{S, T, U}) where { S, T, U }
   algebra(a) !== algebra(b) && return false
-  return basis_pmatrix(a, copy = false) == basis_pmatrix(b, copy = false)
+  apmat = basis_pmatrix(a, copy = false)
+  bpmat = basis_pmatrix(b, copy = false)
+  return _spans_subset_of_pseudohnf(apmat, bpmat; shape = :lowerleft) &&
+         _spans_subset_of_pseudohnf(bpmat, apmat; shape = :lowerleft)
 end
 
 ################################################################################
@@ -1793,3 +1774,18 @@ function _as_ideal_of_smaller_algebra(m::AbsAlgAssMor, I::AlgAssRelOrdIdl)
   return J
 end
 
+################################################################################
+#
+#  Extend
+#
+################################################################################
+
+function Base.:(*)(I::AlgAssRelOrdIdl, O::AlgAssRelOrd)
+  @assert algebra(I) === algebra(O)
+  return I * (one(algebra(O)) * O)
+end
+
+function Base.:(*)(O::AlgAssRelOrd, I::AlgAssRelOrdIdl)
+  @assert algebra(I) === algebra(O)
+  return (one(algebra(O)) * O) * I
+end

src/AlgAssRelOrd/Order.jl

@@ -369,8 +369,10 @@ end
 Returns `true` if $R$ and $S$ are equal and `false` otherwise.
 """
 function ==(R::AlgAssRelOrd, S::AlgAssRelOrd)
-  algebra(R) != algebra(S) && return false
-  return basis_pmatrix(R, copy = false) == basis_pmatrix(S, copy = false)
+  algebra(R) !== algebra(S) && return false
+  Rpmat = basis_pmatrix(R, copy = false)
+  Spmat = basis_pmatrix(S, copy = false)
+  return _spans_subset_of_pseudohnf(Rpmat, Spmat; shape = :lowerleft) && _spans_subset_of_pseudohnf(Spmat, Rpmat; shape = :lowerleft)
 end
 
 ################################################################################

src/EllCrv/Heights.jl

@@ -706,7 +706,7 @@ function CPS_height_bounds(E::EllipticCurve{T}) where T<:Union{QQFieldElem, AbsS
     ev_arch += 2*log(ev)
   end
 
-  non_arch_contribution = sum([CPS_non_archimedean(E, v, prec) for v in P])//d
+  non_arch_contribution = sum([CPS_non_archimedean(E, v, prec) for v in P];init = zero(ArbField(prec, cached = false)))//d
   return 1//(3*d) * dv_arch, 1//(3*d) * ev_arch + non_arch_contribution
 end
 

src/Sparse/Row.jl

@@ -801,7 +801,7 @@ end
 Returns $A \cdot A^t$.
 """
 function norm2(A::SRow{T}) where {T}
-  return sum([x * x for x in A.values])
+  return sum(T[x * x for x in A.values]; init = zero(base_ring(A)))
 end
 
 ################################################################################

test/AlgAss/AlgQuat.jl

@@ -11,4 +11,45 @@
       @test f(b) * f(bb) == f(b * bb)
     end
   end
+
+  let
+    K, a = quadratic_field(5)
+    A = Hecke.quaternion_algebra2(K, K(-1), K(-1))
+    O = maximal_order(A)
+    I = basis(A)[2] * O
+    fl, b = Hecke._is_principal_maximal_quaternion_generic_proper(I, O)
+    @assert fl
+    @test b * O == I
+  end
+
+  let
+    # from Swan '62
+    Qx, x = QQ["x"]
+    K, tau = number_field(x^4 - 4*x^2 + 2)
+    M = Array{elem_type(K), 3}(undef, 4, 4, 4)
+    M[:, :, 1] = K.([1 0 0 0; 0 -1 0 0; 0 0 -1 -tau; 0 0 0 -1])
+    M[:, :, 2] = K.([0 1 0 0; 1 tau 0 0; 0 0 0 1; 0 0 -1 0])
+    M[:, :, 3] = K.([0 0 1 0; 0 0 0 -1; 1 tau 0 0; 0 1 0 0])
+    M[:, :, 4] = K.([0 0 0 1; 0 0 1 tau; 0 -1 0 0; 1 0 0 0])
+    sqrt2 = sqrt(K(2))
+    B = associative_algebra(K, M)
+    zeta = basis(B)[2]
+    j = basis(B)[3]
+    i = tau^2 - 1 + (-tau^3 + 2*tau)*zeta
+    alpha = inv(sqrt2 * tau) * (sqrt2 + 1 + i)
+    beta = tau/sqrt2 * (1 + j)
+    gamma = alpha * beta
+    Gamma = Hecke._get_order_from_gens(B, [one(B), alpha, beta, alpha*beta])
+    @test is_maximal(Gamma)
+    P = Gamma * B(tau) + Gamma * beta
+    @test left_order(P) == Gamma
+    @test Gamma * P == P
+    @test P * Gamma != P
+    # we can only test right ideals so far ...
+    BOp, BtoBOp = opposite_algebra(B)
+    GammaOp = Hecke._get_order_from_gens(BOp, [BtoBOp(elem_in_algebra(x)) for x in absolute_basis(Gamma)])
+    POp = +([BtoBOp(x) * GammaOp for x in absolute_basis(P)]...)
+    fl, _ = Hecke._is_principal_maximal_quaternion_generic_proper(POp, GammaOp)
+    @test !fl
+  end
 end