/* ********************************************************************** ** hecke.m ** ** ** ** Computing Bianchi modular forms and the Hecke action for d=-2 ** ** ** ** Mehmet Haluk Sengun ** ** version of 13/08/2008 ** ** ** ********************************************************************** */ HECKE:=function(a,b,Data) K:=QuadraticField(-2); O:=MaximalOrder(K); z:=O!w; k:=Data`weight[1]; l:=Data`weight[2]; TA:=Data`ta; TB:=Data`tb; TU:=Data`tu; CA:=Data`ca; CB:=Data`cb; CU:=Data`cu; DER:=Data`der; INN:=Data`inn; s:=a+b*z; /* this is the prime for the Hecke Operator*/ cs:=a-b*z; /* this is the conjugate of s, also given by p/s */ p:=Norm(s,Integers()); /* norm over the integers */ c:=GF(p)! -(a/b) ; /* this is image of w under reduction modulo p */ sl:=Data`level; pl:=Norm(sl,Integers()); Ml:=map Integers() | x :-> x>; /* this is to coerce ul(i)'s and vl(i)'s into Integers */ L,ff:=ResidueClassField(O,sl*O); ms:=GF(pl)!ff(s); mcs:=GF(pl)!ff(cs); u:=mapGF(p) | x :-> -1/x>; /* this is the permutation of B on the transversal */ v:=mapGF(p) | x:-> x-c >; /* this is the permutation of U on the transversal */ ZZ:=mapGF(pl) | x:-> ms*x>; M:=map Integers() | x :-> x>; /* this is to coerce u(i)'s and v(i)'s into Integers */ c:=M(c); R:=PolynomialRing(K,2); Symm:=function(k,T) /* returns the action of the matrix T on Symm(k) as a matrix */ /* T:=Determinant(T)*T^-1; */ ST:=Matrix(K,k+1,k+1,[: i,j in [1..k+1]]); for i in [0..k] do Q:=(T[1,1]*x+T[1,2]*y)^(k-i)*(T[2,1]*x+T[2,2]*y)^(i); for j in [0..k] do ST[i+1,j+1]:=MonomialCoefficient(Q,x^(k-j)*y^(j)); end for; end for; return ST; end function; D:=Matrix(K,2,2,[1,0,0,s]); D0:=Matrix(K,2,2,[1,0,0,cs]); /* this is conjugate of \alpha^{\iota} */ SS:=MatrixAlgebra(K,(k+1)*(l+1)); IDD:=SS!1; NN:=SS!0; SDk:=Symm(k,D); SD0l:=Symm(l,D0); CD :=TensorProduct(SDk,SD0l); /* action of delta inverse on the module */ level:=pl; SSS:=MatrixAlgebra(K,(level+1)*(k+1)*(l+1)); N:=SSS!0; ID:=SSS!1; RRR:=[]; for i in [0..level-1] do Append(~RRR,Integers()!Eltseq(s*i-Ml(ZZ(i)))[1]); end for; III:=[]; for i in [0..level-1] do Append(~III,Integers()!Eltseq(s*i-Ml(ZZ(i)))[2]); end for; TD:=N; for i in [0..level-1] do TD:=InsertBlock(TD,CA^RRR[i+1]*CU^III[i+1]*CD,Ml(ZZ(i))*(k+1)*(l+1)+1,(i)*(k+1)*(l+1)+1); end for; TD:=InsertBlock(TD,CB*CD*CB,(level)*(k+1)*(l+1)+1,(level)*(k+1)*(l+1)+1); TAA:=function(m) W:=N; if m lt 0 then for j in [0..(-m-1)] do W:=W+TA^j; end for; W:= -W*TA^(m); end if; if m gt 0 then for j in [0..(m-1)] do W:=W+TA^j; end for; end if; if m eq 0 then W:=N; end if; return W; end function; /*================================================================= FUNCTION 2 uses TU's this will return 1+U+..+U^(k-1) or -U^k*(1+U+...+U^(-k-1)) if k is negative and 0 if k=0 =====================================*/ TUU:=function(m) W:=N; if m lt 0 then for j in [0..(-m-1)] do W:=W+TU^j; end for; W:= -W*TU^m; end if; if m gt 0 then for j in [0..(m-1)] do W:=W+TU^j; end for; end if; if m eq 0 then W:=N; end if; return W; end function; /*============================================ FUNCTION 3 word decomposition function, will work only for Euclidean fields this will be run in a for cycle with index i ======================================================*/ G:=function(i) S:=[]; /* this will return the decomposition */ B:=Matrix(O,2,2,[0,1,-1,0]); /* we need to redefine B inside Matrix(O) */ v:=O!M(u(i)); X:=Matrix(O,2,2,[v,(1+i*v)/s,-s,-i]); /* this is the i'th matrix of the image of B under the transfer map*/ if Abs(Norm(X[1,2])) gt Abs(Norm(X[2,2])) then X:=B*X; Append(~S,O.1); /* this will notify us that there's a B in the beginning of the decomposition, note that only in this case O.1=1, will be the first element of the sequence*/ end if; while Norm(X[1,2])ne 0 do /* this is the Euclidean Division */ r:=X[2,2] mod X[1,2]; q:=(X[2,2]-r)/X[1,2]; Append(~S,q); Q:=Matrix(O,2,2,[1,0,-q,1]); X:=B*Q*X; end while; if X[2,2] eq 1 then Append(~S,X[2,1]); end if; if X[2,2] eq -1 then Append(~S,-X[2,1]); end if; return S; end function; /* ==================================================== */ RR:=PolynomialRing(SSS,3); /* X,Y,Z represent f(A), f(B), f(U), coefficients are matrices */ POLY:=RR!0; MOLY:=TAA(-2*b)*TU^a*TD*X+TUU(a)*TD*Z; /* we need MOLY for image of U */ /* -----------------------BEGINNING of THE FOR CYCLE FOR IMAGE OF B ---------------------------*/ for i in [1..(p-1)] do /* we will get the summation that defines the Hecke operator's image on B */ t:=#G(i); /* ================================================================= FUNCTION 4 (uses function1 G) returns the real part of q_l, the l'th element of the word decomposition set S of the i'th matrix above ============================================================= */ R:=function(l) h:=Eltseq(G(i)[l])[1]; h:=Integers()!h; return h; end function; /* ================================================================= FUNCTION 5 (uses function1 G ) returns the imaginary part of q_l ============================================================== */ I:=function(l) h:=Eltseq(G(i)[l])[2]; h:=Integers()!h; return h; end function; /*=============================================================== this will return f(M_k) that appears in the image of B under the Hecke operator t will be #Euc but we have to be careful about the first element of G(i) ==========================================================*/ if G(i)[1] ne 1 then /* this is the case where the decomposition does NOT start with a B*/ g:=TAA(R(1))*TU^I(1)*X+TUU(I(1))*Z; /*this is to define s_0 */ for m in [2..t] do g:= (MonomialCoefficient(g,X)*TB*TA^R(m)*TU^I(m)+TAA(R(m))*TU^I(m))*X+ (MonomialCoefficient(g,Y)*TB*TA^R(m)*TU^I(m)+TA^R(m)*TU^I(m))*Y+ (MonomialCoefficient(g,Z)*TB*TA^R(m)*TU^I(m)+TUU(I(m)))*Z; end for; else g:=TAA(R(2))*TU^I(2)*X+TA^R(2)*TU^I(2)*Y+TUU(I(2))*Z; /*this is to define s_0 */ for m in [3..t] do g:= (MonomialCoefficient(g,X)*TB*TA^R(m)*TU^I(m)+TAA(R(m))*TU^I(m))*X+ (MonomialCoefficient(g,Y)*TB*TA^R(m)*TU^I(m)+TA^R(m)*TU^I(m))*Y+ (MonomialCoefficient(g,Z)*TB*TA^R(m)*TU^I(m)+TUU(I(m)))*Z; end for; end if; POLY:= POLY+MonomialCoefficient(g,X)*TD*TB*TA^i*X+ MonomialCoefficient(g,Y)*TD*TB*TA^i*Y+ MonomialCoefficient(g,Z)*TD*TB*TA^i*Z; /*this is the image of f(B) */ end for; /* ---------------------------------BEGINNING OF THE FOR CYCLE FOR IMAGE OF U ----------------------------------*/ /* =============================================================== FUNCTION 6 returns the real part of the upper corner of the k'th summand in image of U ======================================================*/ RE:=function(m) h:=Eltseq((M(v(m))-m+z)/s)[1]; h:=Integers()!h; return h; end function; /* =============================== FUNCTION 7 returns the imaginary part ==================================== */ IM:=function(m) h:=Eltseq((M(v(m))-m+z)/s)[2]; h:=Integers()!h; return h; end function; for j in [0..p-1] do MOLY:=MOLY+(TAA(RE(j))*TU^IM(j)*TB*TD*TB*TA^j)*X+(TA^RE(j)*TU^IM(j)*TB+ID)*(TD*TB*TA^j)*Y+(TUU(IM(j))*TB*TD*TB*TA^j)*Z; end for; /*------------------------------END OF THE FOR CYCLE FOR IMAGE OF U -------------------------------------- */ H11:=TAA(a)*(TU^-b*TB*TD*TB+TU^b*TD); /* this will be the block at index (1,1) */ H21:=(TA^a*TU^-b*TB+ID)*(TD*TB); H31:=TUU(-b)*TB*TD*TB+TUU(b)*TD; C1:=VerticalJoin(H11,H21); C1:=VerticalJoin(C1,H31); H12:=MonomialCoefficient(POLY,X); H22:=MonomialCoefficient(POLY,Y); H32:=MonomialCoefficient(POLY,Z); C2:=VerticalJoin(H12,H22); C2:=VerticalJoin(C2,H32); /* this is the second column of the matrix of the Hecke operator on E^3 */ H13:=MonomialCoefficient(MOLY,X); H23:=MonomialCoefficient(MOLY,Y); H33:=MonomialCoefficient(MOLY,Z); C3:=VerticalJoin(H13,H23); C3:=VerticalJoin(C3,H33); /* this is column 3 of the matrix that gives the action of the Hecke opearator given by a+bw */ HO:=HorizontalJoin(C1,C2); HO:=HorizontalJoin(HO,C3); V:=VectorSpace(K,3*(level+1)*(k+1)*(l+1)); DDER:=sub; BB:=ExtendBasis(DDER,V); V:=VectorSpaceWithBasis(BB); H:=Coordinates(V,BB[1]*HO); HH:=V![H[t]:t in [1..3*(level+1)*(k+1)*(l+1)]]; for t in [2..3*(level+1)*(k+1)*(l+1)] do H:=Coordinates(V,BB[t]*HO); e:=V![H[r]:r in [1..3*(level+1)*(k+1)*(l+1)]]; HH:=VerticalJoin(HH,e); end for; HH:=Transpose(HH); for i in [1..3*(level+1)*(k+1)*(l+1)-Dimension(DER)] do HH:=RemoveRowColumn(HH,Dimension(DER)+1,Dimension(DER)+1); end for; IINN:=sub; CC:=ExtendBasis(IINN,DDER); DDER:=VectorSpaceWithBasis(CC); Y:=Coordinates(DDER,ReduceVector(IINN,CC[Dimension(INN)+1]*HO)); YY:=VectorSpace(K,Dimension(DER))![Y[t]:t in [1..Dimension(DER)]]; for y in [Dimension(INN)+2..Dimension(DER)] do Y:=Coordinates(DDER,ReduceVector(IINN,CC[y]*HO)); e:=VectorSpace(K,Dimension(DER))![Y[t]:t in [1..Dimension(DER)]]; YY:=VerticalJoin(YY,e); end for; YY:=Matrix(YY); if Dimension(DER)-Dimension(INN) eq 1 then for i in [1..Dimension(INN)] do YY:=RemoveColumn(YY,1); end for; else YY:=Transpose(YY); for i in [1..Dimension(INN)] do YY:=RemoveRow(YY,1); end for; end if; EV:=SetToSequence(Eigenvalues(YY)); ES:=[**]; /* creates a list */ for i in [1..#EV] do Append(~ES,Eigenspace(YY,EV[i][1])); Append(~ES,EV[i][1]); end for; return FactoredCharacteristicPolynomial(YY),Eigenvalues(YY),ES; end function;