Extra methods for the prime power case + change of the implementation of the structure of the congruence subgroup

lib/CongruenceGroups/congGroups.gd

@@ -113,6 +113,12 @@ DeclareAttribute( "IndexInAmbientGroup", IsHAPCongruenceSubgroup );
 ## is the cubic tree.
 DeclareAttribute( "StabilizerSubgroup", IsHAPCongruenceSubgroup );
 
+############################################################################
+##
+## ProjectiveSpave( <G> )
+##
+DeclareAttribute( "ProjectiveSpace", IsHAPCongruenceSubgroup );
+
 ############################################################################
 ##
 ## AmbientTransversal( <G> )

lib/CongruenceGroups/congGroups.gi

@@ -101,6 +101,7 @@ InstallMethod( CongruenceSubgroupGamma0, "for integer matrix group and positive
     ##################################################
     ##
     ## Remaining components to be computed in other functions.
+    ProjectiveSpace(G);
     AmbientTransversal(G);
     CosetPosFunction(G); #A generic method will be used to construct the
     CosetRepFunction(G); #these two functions except for cases with a

lib/CongruenceGroups/sl2methods.gi

@@ -23,6 +23,21 @@
      return ind;
      end);
 
+##########################################################################
+##
+## ProjectiveSpace( <G> )
+InstallMethod(ProjectiveSpace,
+     "Projective space",
+     [ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
+     function(G)
+     local n;
+        if DimensionOfMatrixGroup(G)>2 then TryNextMethod(); fi;
+
+        n := LevelOfCongruenceSubgroup(G);
+
+        return FiniteProjectiveLine(n);
+     end);
+
 ##########################################################################
 ##
 ## AmbientTransversal( <G> )

lib/Orru/sl2methodsExtra.gi

@@ -1,29 +1,113 @@
+# methods for power of primes
+
 InstallMethod(CosetPosFunction,
     "Returns cosetPos(g) function for the congruence subgroup G",
     [ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
     function(G)
-        local cosetPos, canonicalRep, n, ProjLine;
+        local cosetPos, canonicalRep, n, countRep;
 
         if DimensionOfMatrixGroup(G) <> 2 then
             TryNextMethod();
         fi;
 
         n := LevelOfCongruenceSubgroup(G);
 
-        if IsPrime(n) then
+        if not IsPrimePowerInt(n) then
             TryNextMethod();
         fi;
 
-        ProjLine := FiniteProjectiveLine(n);
+        if IsPrime(n) then
+            TryNextMethod();
+        fi;
 
         canonicalRep := function(g)
             local v, vv, U, d, dd, x, y;
             v := [g[1][1], g[2][1]];
             vv := List(v, x -> x mod n);
-            U := Units(Integers mod n);
+            U := Filtered([0..n],i->Gcd(i,n)=1);
+            if vv[1] mod n = 0 then
+                return [0,1];
+            elif vv[1] mod n in U then
+                return [1,(Inverse(vv[1]) mod n)*vv[2] mod n];
+            else
+                d := Gcd(vv[1],n);
+                dd := n/d;
+                x := vv[1]/d;
+                y := vv[2]/x mod dd;
+                while not Gcd(d,y) = 1 do
+                    y := y + dd;
+                od;
+                return [d, y];
+            fi;
+        end;
+
+        countRep := function(m)
+            local p, e, i, countp;
+
+            p := Set(Factors(m))[1];
+            e := Length(Factors(m));
+
+            countp := [1,m];
+
+            for i in [2..e] do
+                Add(countp, p^(e-i)*(p-1));
+            od;
+
+            return countp;
+        end;
+
+        cosetPos := function(g)
+            local w, count, e, U;
+            w := canonicalRep(g);
+
+            count := countRep(n);
+
+            if w[1] = 0 then
+                return 1;
+            elif w[1] = 1 then
+                U := [0..n-1];
+                return 1 + Position(U,w[2]);
+            else
+                e := Length(Factors(w[1]));
+                U := Filtered([0..n/w[1]], i-> Gcd(i,n/w[1]) = 1);
+                return Sum(count{[1..1 + e]}) + Position(U, w[2]);
+            fi;
+        end;
+
+        return cosetPos;
+    end);
+
+InstallMethod(CosetPosFunction,
+    "Returns cosetPos(g) function for the congruence subgroup G",
+    [ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
+    function(G)
+        local cosetPos, canonicalRep, n, ProjLine, U;
+
+        if DimensionOfMatrixGroup(G) <> 2 then
+            TryNextMethod();
+        fi;
+
+        n := LevelOfCongruenceSubgroup(G);
+
+        if IsPrime(n) then
+            TryNextMethod();
+        fi;
+
+        if IsPrimePowerInt(n) then
+            TryNextMethod();
+        fi;
+
+        ProjLine := ProjectiveSpace(G);
+
+        U := Filtered([0..n],i -> Gcd(i,n) = 1);
+
+        canonicalRep := function(g)
+            local v, vv, d, dd, x, y;
+            v := [g[1][1], g[2][1]];
+            vv := List(v, x -> x mod n);
             if vv[1] mod n = 0 then
                 return [0,1];
-            elif ZmodnZObj(vv[1],n) in U then
+            elif (vv[1] mod n) in U then
                 return [1,(Inverse(vv[1]) mod n)*vv[2] mod n];
             else
                 d := Gcd(vv[1],n);
@@ -62,7 +146,7 @@ InstallMethod(CosetRepFunction,
             TryNextMethod();
         fi;
 
-        ProjLine := FiniteProjectiveLine(n);
+        ProjLine := ProjectiveSpace(G);
 
         cosetOfInt := function(i)
             local a, c, b, d, gg;
@@ -106,7 +190,7 @@ InstallMethod(CosetRepFunction,
             TryNextMethod();
         fi;
 
-        ProjLine := FiniteProjectiveLine(n);
+        ProjLine := ProjectiveSpace(G);
 
         GG:=AmbientGroupOfCongruenceSubgroup(G);
 

lib/Orru/sl3methods.gi

@@ -2,6 +2,22 @@
 ##
 ## Methods for 3x3 congruence subgroups of SL3
 
+##########################################################################
+##
+## ProjectiveSpace( <G> )
+##
+
+InstallMethod(ProjectiveSpace,
+     "Projective space",
+     [ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
+     function(G)
+     local n;
+        if DimensionOfMatrixGroup(G)<>3 then TryNextMethod(); fi;
+
+        n := LevelOfCongruenceSubgroup(G);
+
+        return FiniteProjectivePlane(n);
+     end);
 ##########################################################################
 ##
 ## CosetPosFunction( <G> )
@@ -19,7 +35,7 @@ InstallMethod(CosetPosFunction,
 
         n := LevelOfCongruenceSubgroup(G);
 
-        ProjPlane := FiniteProjectivePlane(n);
+        ProjPlane := ProjectiveSpace(G);
 
         cosetPos := function(g)
             local v, vv, U, u, w;
@@ -59,7 +75,7 @@ InstallMethod(CosetRepFunction,
             return Inverse(Herm!.rowtrans);
         end;
 
-        ProjPlane := FiniteProjectivePlane(n);
+        ProjPlane := ProjectiveSpace(G);
 
         cosetOfInt:=function(i)
             local x,y,z;
@@ -94,7 +110,7 @@ function(G)
     fi;
 
     n := LevelOfCongruenceSubgroup(G);
-    ProjPlane := FiniteProjectivePlane(n);
+    ProjPlane := ProjectiveSpace(G);
 
     GG:=AmbientGroupOfCongruenceSubgroup(G);
     cosetPos:=CosetPosFunction(G);