Added TraceForm Functionality to ConstructGeneratorsAndRelations

ModFrmHilD/CanonicalRing/CanonicalRing.m

@@ -183,6 +183,7 @@ intrinsic ConstructGeneratorsAndRelations(
   IdealClassesSupport:=false,
   Symmetric:=false,
   ComputeNewGenerators:=true,
+  NumberOfTraceForms := 0,
   PrecomputedGens:=AssociativeArray()
   ) -> Any
   {
@@ -216,6 +217,18 @@ intrinsic ConstructGeneratorsAndRelations(
   have_dim_formula := IdealClassesSupport eq NarrowClassGroupReps(M) and not Symmetric;
   KnownRelations := false;
 
+  // Computing Trace forms 
+  F := BaseField(M);
+  bound := 500;
+  I := IdealsUpTo(bound, F);
+  while #I lt NumberOfTraceForms do
+    bound *:= 2;
+    I := IdealsUpTo(bound, F);
+  end while;
+  // Ideals
+  TraceFormIdeals := (NumberOfTraceForms eq 0) select [] else I[1..NumberOfTraceForms];
+  PrecomputeTraceForms(M, TraceFormIdeals);
+
   /////////////////////
   // Generators of lowest weight.
 
@@ -228,6 +241,15 @@ intrinsic ConstructGeneratorsAndRelations(
   if IsDefined(PrecomputedGens, minimalGenWeight) then
       basis := PrecomputedGens[minimalGenWeight];
 
+  elif NumberOfTraceForms ne 0 then
+      Mk := HMFSpace(M, N, [minimalGenWeight : i in [1..n]]);
+      B := [ TraceForm(Mk,aa) : aa in TraceFormIdeals ];
+      basis := EisensteinBasis(Mk) cat Basis(B);
+      if #basis lt CuspDimension(Mk) then 
+          msg := "Not enough TraceForms";
+          error msg;
+      end if;
+
   elif ComputeNewGenerators then
       basis := Basis(HMFSpace(M, N, [minimalGenWeight : i in [1..n]]) : IdealClassesSupport := IdealClassesSupport, Symmetric:=Symmetric);
       assert not IsNull(basis);
@@ -319,6 +341,7 @@ intrinsic ConstructGeneratorsAndRelations(
                                   Alg := Alg,
                                   KnownMkDimension := knownDim,
                                   IdealClassesSupport := IdealClassesSupport,
+                                  TraceFormIdeals := TraceFormIdeals,
                                   Symmetric := Symmetric);
 
       newGens := basisWeightk[#weightedSymBasis + 1 .. #basisWeightk];
@@ -349,6 +372,7 @@ intrinsic ExtendBasis(forms::SeqEnum[ModFrmHilDElt], Mk :
                       Alg                 := "Standard",
                       KnownMkDimension    := false,
                       IdealClassesSupport := false,
+                      TraceFormIdeals := [],
                       Symmetric     := false) -> SeqEnum
 {Given a sequence Q of r linearly independent elements of a space M and a subspace V of M
 containing the elements of Q, extend the elements of Q to a basis for U; the basis is
@@ -362,12 +386,16 @@ intrinsic ExtendBasis(forms::SeqEnum[ModFrmHilDElt], Mk :
         // First try our luck with just Eisenstein series. If that fails, use the fallback.
 
         eisensteinbasis := EisensteinBasis(Mk : IdealClassesSupport:=IdealClassesSupport, Symmetric:=Symmetric);
-        moreforms := Basis(forms cat eisensteinbasis);
+        traceforms := [ TraceForm(Mk,aa) : aa in TraceFormIdeals ];
+        moreforms := Basis(forms cat eisensteinbasis cat traceforms );
         coeffs_matrix := CoefficientsMatrix(moreforms : IdealClasses:=IdealClassesSupport);
 
         // TODO: This double complement call can surely be optimized away.
         if Rank(coeffs_matrix) eq KnownMkDimension then
             return forms cat ComplementBasis(forms, Basis(moreforms) : Alg := Alg);
+        elif (#TraceFormIdeals ne 0) then 
+            msg := "Not enough TraceForms";
+            error msg;
         end if;
     end if;