Merge pull request #464 from abhijit-mudigonda/cubic-tests

basis tests for HMFs over cubic fields

ModFrmHilD/Restriction.m

@@ -1,4 +1,4 @@
-intrinsic RestrictionToDiagonal(f::ModFrmHilDElt,M::ModFrmHilDGRng,bb::RngQuadIdl) -> ModFrmElt
+intrinsic RestrictionToDiagonal(f::ModFrmHilDElt,M::ModFrmHilDGRng,bb::RngOrdIdl) -> ModFrmElt
   {Given an HMF f of parallel weight k, returns the classical modular curve of weight nk and level obtained from restricting
   the component bb of the HMF to the diagonal.}
   require #SequenceToSet(Weight(f)) eq 1: "Only defined for parallel weight.";
@@ -36,41 +36,29 @@ intrinsic RestrictionToDiagonal(f::ModFrmHilDElt,M::ModFrmHilDGRng,bb::RngQuadId
   return modForms!(denom*(restriction +O(q^(prec))));
 end intrinsic;
 
-/*
-// Totally Positive Elements in an Ideal
-   From a basis {a,b} for the ideal bb where
-     Tr(a) = n and Tr(b) = 0.
-   Elements in ideal will look like xa + yb where x,y in ZZ
-   and have embedding xa_1 + yb_1 and xa_2 + yb_2.
-   All totally positive elements of given trace t will satisfy
-   1)    t = Tr(xa+yb)    <=>   t = xn
-   2)    xa+yb totally positive       <=>   y > -x*a_1/b_1   and  y > -x*a_2/b_2
-   Eq 1) determines the value for x,
-     and Eq 2) allows us to loop over values of y
-*/
 intrinsic PositiveElementsOfTrace(aa::RngOrdFracIdl, t::RngIntElt) -> SeqEnum[RngOrdFracIdl]
-  {Given aa a fractional ideal and t a nonnegative integer,
-   returns the totally positive elements of aa with trace t.}
+  {
+    Given aa a fractional ideal and t a nonnegative integer,
+    returns the totally positive elements of aa with trace t.
+  }
   basis := TraceBasis(aa);
-  places := InfinitePlaces(NumberField(Parent(basis[1])));
-  smallestTrace := Integers()!Trace(basis[1]);
-  T := [];
-  if t mod smallestTrace eq 0 then
-    x := t div smallestTrace;
-    a_1 := Evaluate(basis[1],places[1]);
-    a_2 := Evaluate(basis[1],places[2]);
-    b_1 := Evaluate(basis[2],places[1]);
-    b_2 := Evaluate(basis[2],places[2]);
-    assert b_1 lt 0 and b_2 gt 0; // if this assumption changes, the inequalities get swapped
-    // at place 2, x*a2 + y*b2 >= 0 => y >= -x*a2/b2
-    lower_bnd := Ceiling(-x*a_2/b_2);
-    // at place 1, x*a1 + y*b1 >= 0 => y <= -x*a1/b1
-    upper_bnd := Floor(-x*a_1/b_1);
-    for y in [lower_bnd .. upper_bnd] do
-      Append(~T, x*basis[1]+y*basis[2]);
-    end for;
-  end if;
-  return T;
-end intrinsic;
+  smallest_trace := Integers()!Trace(basis[1]);
+  if (t mod smallest_trace) eq 0 then
+    F := NumberField(Parent(basis[1]));
+    n := Degree(F);
 
+    B := Matrix([[Evaluate(b, v) : v in InfinitePlaces(F)] : b in basis]);
+    B := t * B^-1;
 
+    // drop the first coordinate, it'll always be t / smallest_trace
+    vertices := [Rationalize(Vector([v[i] : i in [2 .. n]])) : v in Rows(B)];
+    assert #vertices[1] eq n-1 and #vertices eq n;
+    P := Polytope(vertices);
+    pts := InteriorPoints(P);
+    // put t / smallest_trace back in each vector
+    x := t div smallest_trace;
+    return [x * basis[1] + &+[Eltseq(pt)[i] * basis[i+1] : i in [1 .. n-1]] : pt in pts];
+  else
+    return [];
+  end if;
+end intrinsic;

Tests/cubic_cuspforms.m

@@ -0,0 +1,26 @@
+// Checks that the restrictions of parallel weight cusp forms over
+// cubic fields are classical cusp forms. 
+//
+// TODO abhijitm - this test will be made more robust shortly, once we can compute
+// parallel weight k forms for k > 2. 
+
+R<x> := PolynomialRing(Rationals());
+F := NumberField(x^3-x^2-2*x+1);
+ZF := Integers(F);
+
+M := GradedRingOfHMFs(F, 150);
+N := 3*ZF;
+M222 := HMFSpace(M, N, [2,2,2]);
+S222 := CuspFormBasis(M222);
+assert #S222 eq 1;
+f := S222[1];
+
+// The restriction of a parallel weight 2 form f to the diagonal g(z) := f(z,z,z)
+// should be a weight 6 classical modular form of level 3:
+g := RestrictionToDiagonal(f, M, 1*ZF);
+
+// the space of level 3 weight 6 forms has dimension 1
+// https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3/6/a/a/
+h := Basis(CuspForms(Gamma0(3), 6))[1];
+
+assert Degree(F) * h eq g;

Tests/cubic_mult.m

@@ -0,0 +1,23 @@
+// tests multiplication for HMFs over cubic fields 
+// 
+// checks that the square of an Eisenstein series of parallel weight 1
+// lies in the computed space of [2,2,2] forms. 
+//
+// TODO abhijitm - this test will be made more robust once we can compute
+// cusp spaces for weights beyond parallel weight 2. 
+
+R<x> := PolynomialRing(Rationals());
+F := NumberField(x^3-x^2-2*x+1);
+ZF := Integers(F);
+
+M := GradedRingOfHMFs(F, 150);
+N := 3*ZF;
+M222 := HMFSpace(M, N, [2,2,2]);
+B222 := Basis(M222);
+H := HeckeCharacterGroup(N, [1,2,3]);
+chi := H.1;
+assert Order(chi) eq 2;
+M111 := HMFSpace(M, N, [1,1,1], chi);
+E111 := EisensteinBasis(M111);
+E111_squared := [f^2 : f in E111];  
+assert #Intersection(E111_squared, B222) eq #E111_squared;

Tests/pos_elts_of_trace.m

@@ -0,0 +1,59 @@
+// we use the old PositiveElementsOfTrace function to test
+// the new one
+
+/*
+   Totally Positive Elements in an Ideal
+   From a basis {a,b} for the ideal bb where
+     Tr(a) = n and Tr(b) = 0.
+   Elements in ideal will look like xa + yb where x,y in ZZ
+   and have embedding xa_1 + yb_1 and xa_2 + yb_2.
+   All totally positive elements of given trace t will satisfy
+   1)    t = Tr(xa+yb)    <=>   t = xn
+   2)    xa+yb totally positive       <=>   y > -x*a_1/b_1   and  y > -x*a_2/b_2
+   Eq 1) determines the value for x,
+   and Eq 2) allows us to loop over values of y
+*/
+
+function pos_elts_of_trace_quadratic(aa, t)
+  // aa - RngOrdFracIdl
+  // t - RngIntElt
+  // returns - SeqEnum[RngOrdFracIdl]
+  basis := TraceBasis(aa);
+  F := NumberField(Parent(basis[1]));
+  assert Degree(F) eq 2; // only works for quadratic fields
+  places := InfinitePlaces(F);
+  smallestTrace := Integers()!Trace(basis[1]);
+  T := [];
+  if t mod smallestTrace eq 0 then
+    x := t div smallestTrace;
+    a_1 := Evaluate(basis[1],places[1]);
+    a_2 := Evaluate(basis[1],places[2]);
+    b_1 := Evaluate(basis[2],places[1]);
+    b_2 := Evaluate(basis[2],places[2]);
+    assert b_1 lt 0 and b_2 gt 0; // if this assumption changes, the inequalities get swapped
+    // at place 2, x*a2 + y*b2 >= 0 => y >= -x*a2/b2
+    lower_bnd := Ceiling(-x*a_2/b_2);
+    // at place 1, x*a1 + y*b1 >= 0 => y <= -x*a1/b1
+    upper_bnd := Floor(-x*a_1/b_1);
+    for y in [lower_bnd .. upper_bnd] do
+      Append(~T, x*basis[1]+y*basis[2]);
+    end for;
+  end if;
+  return T;
+end function;
+
+max_N_norm := 20;
+max_x := 5;
+F := QuadraticField(5);
+for N in [I : I in IdealsUpTo(max_N_norm, F) | not IsZero(I)] do
+  basis := TraceBasis(N);
+  smallest_trace := Integers()!Trace(basis[1]);
+  for x in [1 .. max_x] do 
+    assert SequenceToSet(PositiveElementsOfTrace(N, x * smallest_trace)) \
+      eq SequenceToSet(pos_elts_of_trace_quadratic(N, x * smallest_trace));
+    if smallest_trace ne 1 then
+      assert #PositiveElementsOfTrace(N, x * smallest_trace - 1) eq 0; 
+    end if;
+  end for;
+end for;
+