good reduction

src/AlgebraicGeometry/Schemes/CoveredSchemes/Morphisms/MorphismFromRationalFunctions/Methods.jl

@@ -593,13 +593,15 @@ end
 # But they can be set by the user so that certain checks of other methods 
 # are satisfied; i.e. the user has to take responsibility and confirm that 
 # they know what they're doing through these channels. 
-@attr Any function is_proper(phi::AbsCoveredSchemeMorphism)
+@attr Bool function is_proper(phi::AbsCoveredSchemeMorphism)
   error("no method implemented to check properness")
 end
 
-@attr Any function is_isomorphism(phi::AbsCoveredSchemeMorphism)
+@attr Bool function is_isomorphism(phi::AbsCoveredSchemeMorphism)
   error("no method implemented to check for being an isomorphism")
 end
+
+is_known(::typeof(is_isomorphism), phi::AbsCoveredSchemeMorphism) = has_attribute(phi, :is_isomorphism)
 
 ### Pullback of algebraic cycles along an isomorphism. 
 function pullback(phi::MorphismFromRationalFunctions, C::AbsAlgebraicCycle)
@@ -1073,6 +1075,7 @@ function _pushforward_prime_divisor(
   return PrimeIdealSheafFromChart(codomain(phi), cod_chart, JJ)
 end
 
+
 function compose(
     f::MorphismFromRationalFunctions,
     g::MorphismFromRationalFunctions
@@ -1088,7 +1091,13 @@ function compose(
   imgs_V = [b[V] for b in imgs]
   U = domain_chart(f)
   imgs_U = [evaluate(numerator(h), coordinate_images(f))//evaluate(denominator(h), coordinate_images(f)) for h in imgs_V]
-  return morphism_from_rational_functions(X, Z, U, codomain_chart(g), imgs_U; check=false)
+  fg =  morphism_from_rational_functions(X, Z, U, codomain_chart(g), imgs_U; check=false)
+  if is_known(is_isomorphism, f) && is_known(is_isomorphism, g)
+    if is_isomorphism(f) && is_isomorphism(g)
+      set_attribute!(fg, :is_isomorphism=>true)
+    end 
+  end
+  return fg 
 end
 
 #=

src/AlgebraicGeometry/Surfaces/EllipticSurface/Morphisms.jl

@@ -422,10 +422,33 @@ Return the pushforward ``f_*: V_1 \to V_2`` where ``V_i`` is the ambient quadrat
 
 This assumes that the image ``f_*(V_1)`` is contained in ``V_2``. If this is not the case, you will get  
 ``f_*`` composed with the orthogonal projection to ``V_2``. 
+
+# Algorithm
+If the attribute `good_reduction_map` has been set via the internal method `Oscar.set_good_reduction_map!`
+then the surfaces and the automorphism can be specialized and the computation carried out after specialization.
+This is much faster, especially when working over number fields and for complicated maps `f`.
+
+# Input
+The keyword argument `algorithm` can be 
+- `:default` -- use specialization if possible
+- `:specialization` -- use specialization and error if this is not possible
+- none of the above -- no specialization
 """
-function pushforward_on_algebraic_lattices(f::MorphismFromRationalFunctions{<:EllipticSurface, <:EllipticSurface})
-  imgs_divs = _pushforward_lattice_along_isomorphism(f)
-  M = matrix([basis_representation(codomain(f),i) for i in imgs_divs])
+function pushforward_on_algebraic_lattices(f::MorphismFromRationalFunctions{<:EllipticSurface, <:EllipticSurface}; algorithm=:default)
+  X1 = domain(f)
+  X2 = codomain(f)
+  can_specialize =  has_attribute(X1, :good_reduction_map) && has_attribute(X2, :good_reduction_map)
+  if algorithm == :specialization || (algorithm==:default && can_specialize)
+    match1 = good_reduction_algebraic_lattice(X1)
+    match2 = good_reduction_algebraic_lattice(X2)
+    f_red = good_reduction(f)
+    imgs_divs_red = _pushforward_lattice_along_isomorphism(f_red)
+    M_red = matrix([basis_representation(codomain(f_red), i) for i in imgs_divs_red])
+    M = match1 * M_red * inv(match2)
+  else
+    imgs_divs = _pushforward_lattice_along_isomorphism(f)
+    M = matrix([basis_representation(codomain(f),i) for i in imgs_divs])
+  end
   V1 = ambient_space(algebraic_lattice(domain(f))[3])
   V2 = ambient_space(algebraic_lattice(codomain(f))[3])
   # keep the check on since it is simple compared to all the other computations done here

src/AlgebraicGeometry/Surfaces/EllipticSurface/NeighborStep.jl

@@ -518,7 +518,7 @@ function transform_to_weierstrass(g::MPolyRingElem, x::MPolyRingElem, y::MPolyRi
   # In case of variables in the wrong order, switch and transform the result.
   if x == R[2] && y == R[1]
     switch = hom(R, R, reverse(gens(R)))
-    g_trans, trans = transform_to_weierstrass(switch(g), y, x, reverse(P))
+    g_trans, trans, inv_trans = transform_to_weierstrass(switch(g), y, x, reverse(P))
     new_trans = MapFromFunc(F, F, f->begin
                                 switch_num = switch(numerator(f))
                                 switch_den = switch(denominator(f))
@@ -528,7 +528,16 @@ function transform_to_weierstrass(g::MPolyRingElem, x::MPolyRingElem, y::MPolyRi
                                 switch(num)//switch(den)
                             end
                            )
-    return switch(g_trans), new_trans
+    new_inv_trans = MapFromFunc(F, F, f->begin
+                                switch_num = switch(numerator(f))
+                                switch_den = switch(denominator(f))
+                                interm_res = inv_trans(F(switch_num))//inv_trans(F(switch_den))
+                                num = numerator(interm_res)
+                                den = denominator(interm_res)
+                                switch(num)//switch(den)
+                            end
+                           )
+    return switch(g_trans), new_trans, new_inv_trans
   end
 
   g = inv(coeff(g,[0,2]))*g # normalise g
@@ -566,15 +575,20 @@ function transform_to_weierstrass(g::MPolyRingElem, x::MPolyRingElem, y::MPolyRi
   E = coeff(gx, 0)
   #E, D, C, B, A = coeff_gx
   if length(P)==3
+    @show "case 1"
     @req all(h->degree(h)<=3, coefficients(G)) "infinity (0:1:0) is not a point of this hypersurface"
     # y^2 = B*x^3+C*x^2+C*x+D
     x1 = F(inv(B)*x)
     y1 = F(inv(B)*y)
     trans = MapFromFunc(F, F, f->evaluate(numerator(f), [x1, y1])//evaluate(denominator(f), [x1, y1]))
+    x2 = F(B*x)
+    y2 = F(B*y)
+    inv_trans = MapFromFunc(F, F, f->evaluate(numerator(f), [x2, y2])//evaluate(denominator(f), [x2, y2]))
     f_trans = B^2*trans(F(g))
     result = numerator(B^2*f_trans)
-    return result, trans
+    return result, trans, inv_trans
   elseif !iszero(E)
+    @show "case 2"
     b = py
     a4, a3, a2, a1, a0 = A,B,C,D,E
     A = b
@@ -587,14 +601,15 @@ function transform_to_weierstrass(g::MPolyRingElem, x::MPolyRingElem, y::MPolyRi
     x1 = x1+px
 
     # TODO: The following are needed for the inverse. To be added eventually.
-    # x2 = (y-(A+B*x+C*x^2))//(D*x^2)
-    # y2 = x2//x
-    # x2 = evaluate(x2, [x-px, y])
-    # y2 = evaluate(y2, [x-px, y])
+    x2 = (y-(A+B*x+C*x^2))//(D*x^2)
+    y2 = x2//x
+    x2 = evaluate(x2, [x-px, y])
+    y2 = evaluate(y2, [x-px, y])
 
-    # @assert x == evaluate(x1, [x2, y2])
-    # @assert y == evaluate(y1, [x2, y2])
+    @assert x == evaluate(x1, [x2, y2])
+    @assert y == evaluate(y1, [x2, y2])
   else
+    @show "case 3"
     # TODO compute the inverse transformation (x2,y2)
     x1 = 1//x
     y1 = y//x^2
@@ -603,12 +618,15 @@ function transform_to_weierstrass(g::MPolyRingElem, x::MPolyRingElem, y::MPolyRi
     x1 = evaluate(x1, [-x//c, y//c])
     y1 = evaluate(y1, [-x//c, y//c])
     x1 = x1+px
+    x2 = x1 # A hack to allow for a coherent return format
+    y2 = y1 
     #@assert x == evaluate(x1, [x2, y2])
     #@assert y == evaluate(y1, [x2, y2])
   end
   @assert F === parent(x1) "something is wrong with caching of fraction fields"
   # TODO: eventually add the inverse.
   trans = MapFromFunc(F, F, f->evaluate(numerator(f), [x1, y1])//evaluate(denominator(f), [x1, y1]))
+  inv_trans = MapFromFunc(F, F, f->evaluate(numerator(f), [x2, y2])//evaluate(denominator(f), [x2, y2]))
   f_trans = trans(F(g))
   fac = [a[1] for a in factor(numerator(f_trans)) if isone(a[2]) && _is_in_weierstrass_form(a[1])]
   isone(length(fac)) || error("transform to weierstrass form did not succeed")
@@ -617,7 +635,7 @@ function transform_to_weierstrass(g::MPolyRingElem, x::MPolyRingElem, y::MPolyRi
   result = first(fac)
   result = inv(first(coefficients(coeff(result, gens(parent(result)), [3, 0]))))*result
 
-  return result, trans
+  return result, trans, inv_trans
 end
 
 function _is_in_weierstrass_form(f::MPolyRingElem)

src/Rings/mpoly-ideals.jl

@@ -525,7 +525,7 @@ Return the radical of `I`.
   * If the `base_ring` of `I` is a number field, then we first expand the minimal polynomials to reduce to a computation over the rationals.
   * For polynomial rings over the integers, the algorithm proceeds as suggested by Pfister, Sadiq, and Steidel. See [KL91](@cite), [Kem02](@cite), and [PSS11](@cite).
 
-When `preprocessing` is set to `true`, several preprocessing heuristics are applied. See the source code for more detailed information and options.
+When `eliminate_variables` is set to `true`, a preprocessing heuristic is applied in order to find variables which can be eliminated using `I`.
 
 # Examples
 ```jldoctest

src/Rings/mpolyquo-localizations.jl

@@ -1381,7 +1381,7 @@ end
                    <:MPolyQuoLocRing{<:Any, <:Any, <:Any, <:Any,
                                      <:MPolyPowersOfElement}})
   A = domain(f)
-  R = base_ring(f)
+  R = base_ring(A)
   g = hom(R, codomain(f), f.(gens(A)); check=false)
   K = kernel(g)
   return ideal(A, elem_type(A)[h for h in A.(gens(K)) if !is_zero(h)])

src/exports.jl

@@ -1372,6 +1372,7 @@ export projectivization
 export prune_with_map
 export pseudo_del_pezzo_polytope
 export pullback
+export pushforward_on_algebraic_lattices
 export pyramid
 export quadratic_form
 export quantum_automorphism_group
@@ -1440,6 +1441,7 @@ export rem_vertex!
 export rem_vertices!
 export renest
 export repres
+export mordell_weil_torsion
 export representative
 export representative_field
 export representative_patch
@@ -1461,6 +1463,8 @@ export reverse_direction!
 export revlex_basis_encoding
 export reynolds_operator
 export right_acting_group
+export translation_morphism
+export isomorphism_from_generic_fibers
 export right_coset
 export right_coset_action
 export right_cosets