##### # # This is GAP code # # The file is ~steger/donald/SU21/C10p2/padic.gap # # See file ~steger/donald/SU21/C10p2/D.txt for the definition of the # division algebra $\D$ and for all the notation. # ##### ##### # # $k=\QQ[\sqrt{2}]$ # Sk:=Sqrt(2); # Generating element kk:=Field(Sk); # The field itself kkBasis:=Basis(kk,[1,Sqrt(2)]); #Basis SzFcn:=x->x^2-2*One(x); # Polynomial which S satisfies IsZero(SzFcn(Sk)); # Check that it does #Function to reduce elements of~$k$ modulo an integer kkMod:=function(x,q) local c; c:=Coefficients(kkBasis,x); Apply(c,t->t mod q); return LinearCombination(kkBasis,c); end; #Function to reduce matrices over~$k$ modulo an integer kkModM:=function(mtx,q) return List(mtx,row->List(row,x->kkMod(x,q))); end; #Calculate the ``discriminant'' of the basis kkTraceMatrix:=List(kkBasis,x->List(kkBasis,y->Trace(kk,x*y))); Factors(Determinant(kkTraceMatrix)); Set(Factors(Determinant(kkTraceMatrix))) = [2]; ##### # # $\ell=\QQ[\sqrt{2},\sqrt{-5+2\sqrt{2}}]$ # xk:=Indeterminate(kk,"x"); ll:=FieldExtension(kk,xk^2 - (1+Sk)*xk + 2); # The field itself Ul:=RootOfDefiningPolynomial(ll); # Generating element, U Vl:=2*Ul-(1+Sqrt(2))*One(ll); # \sqrt{-5+2\sqrt{2}} UPl:=2/Ul; # U', The conjugate of U Sl:=Sk*One(ll); # \sqrt{2} llBasis:=Basis(ll,[One(ll),Ul]); #Basis over $k$ llPBasis:=Basis(ll,[One(ll),UPl]); # Alternative Basis over $k$ #Function to calculate coefficients over $\QQ$ llCoeff:=x->List(Coefficients(llBasis,x),y->Coefficients(kkBasis,y)); VzFcn:=x->x^2-(-5+2*Sqrt(2))*One(x); #Function satisfied by V UzFcn:=x->x^2-(1+Sqrt(2))*x+2*One(x); #Function satisfied by U #Check that S, V, U, U' are roots of their polynomials IsZero(SzFcn(Sl)); IsZero(VzFcn(Vl)); IsZero(UzFcn(Ul)); IsZero(UzFcn(UPl)); #Function to conjugate an element llConj:=x->LinearCombination(llPBasis,Coefficients(llBasis,x)); #Function to calculate the adjoint of a matrix llAdj:=m->TransposedMat(List(m,r->List(r,x->llConj(x)))); #Check that conjugates are as expected llConj(Ul) = UPl; llConj(UPl) = Ul; llConj(Vl) = -Vl; llConj(Sl) = Sl; #Function to reduce elements of~$\ell$ modulo an integer llMod:=function(x,q) return LinearCombination(llBasis, List(Coefficients(llBasis,x),c->kkMod(c,q)) ); end; #Function to reduce matrices over~$\ell$ modulo an integer llModM:=function(mtx,q) return List(mtx,row->List(row,x->llMod(x,q))); end; #Calculate the ``discriminant'' of the basis llTraceMatrix:=ListX(kkBasis,llBasis, function(x1,x2) return ListX(kkBasis,llBasis, function(y1,y2) return Trace(kk,Rationals,x1*y1*Trace(x2*y2)); end); end); Factors(Determinant(llTraceMatrix)); Set(Factors(Determinant(llTraceMatrix))) = [2,17]; ##### # # $k[\zeta_9+\zeta_9^-1] = \QQ[\sqrt{2},\zeta_9+\zeta_9^-1$, # which is the totally real part of # $m=k[\sqrt{-5+2\sqrt{2}},\zeta_9+\zeta_9^{-1}$ # SRe:=Sk; #Element generating $k$ WRe:=E(9)+E(9)^-1; #Element generating mmRe mmRe:=Field(WRe,SRe); # The field mmRe WzFcn:=x->x^3-3*x+One(x); # Polynomial of W IsZero(WzFcn(WRe)); # Check that it works #Basis over~$\QQ$ mmReBasis:=Basis(mmRe,[1,SRe,WRe,SRe*WRe,WRe^2,SRe*WRe^2]); #Automorphisms taking~$W$ to its two conjugates phiRe:=ANFAutomorphism(mmRe,31); phi2Re:=phiRe^2; #Polynomials which give $\phi$ and $\phi^2$, when applied to~$W$ phiOnW:= W -> -W^2-W+2*One(W); phi2OnW:= W -> W^2 - 2*One(W); #Check that the conjugates of W are conjugates IsZero(WzFcn(WRe^phiRe)); IsZero(WzFcn(WRe^phi2Re)); #Check that they are the right conjugates WRe^phiRe = phiOnW(WRe); WRe^phi2Re = phi2OnW(WRe); #Function to reduce elements of _mmRe_ modulo an integer mmReMod:=function(x,q) local c; c:=Coefficients(mmReBasis,x); Apply(c,t->t mod q); return LinearCombination(mmReBasis,c); end; #Calculate the ``discriminant'' of the basis mmReTraceMatrix:= List(mmReBasis,x->List(mmReBasis,y->Trace(mmRe,x*y))); Factors(Determinant(mmReTraceMatrix)); Set(Factors(Determinant(mmReTraceMatrix))) = [2,3]; ##### # # $m=\QQ[\sqrt{2},\sqrt{-5+2\sqrt{2}},\zeta_9+\zeta_9^{-1}$ # xRe:=Indeterminate(mmRe,"x"); mm:=FieldExtension(mmRe,xRe^2 - (1+SRe)*xRe + 2); #The field $m$ Um:=RootOfDefiningPolynomial(mm); #U, the generator of $m$ UPm:=2/Um; #U', the conjugate of U Vm:=2*Um-(1+Sqrt(2))*One(mm); #\sqrt{-5+2\sqrt{2}} Sm:=SRe*One(mm); # \sqrt{2} Wm:=WRe*One(mm); # \zeta_9+\zeta_9^{-1} mmBasis:=Basis(mm,[One(mm),Um]); #Basis over _mmRe_ mmPBasis:=Basis(mm,[One(mm),UPm]); #Alternative Basis over _mmRe_ #Function to calculate coefficients over~$\QQ$ mmCoeff:=x->List(Coefficients(mmBasis,x),y->Coefficients(mmReBasis,y)); #Check that S, U, UP, V, and W are roots of the right polynomials IsZero(SzFcn(Sm)); IsZero(UzFcn(Um)); IsZero(UzFcn(UPm)); IsZero(VzFcn(Vm)); IsZero(WzFcn(Wm)); #Automorphisms taking $W$ to its two conjugates phimm:=x->LinearCombination(mmBasis, List(Coefficients(mmBasis,x),y->y^phiRe)); phi2mm:=x->LinearCombination(mmBasis, List(Coefficients(mmBasis,x),y->y^phi2Re)); #Check that these fix S and V phimm(Sm) = Sm; phimm(Vm) = Vm; phi2mm(Sm)=Sm; phi2mm(Vm)=Vm; #Check that the conjugates are actually conjugates IsZero(WzFcn(phimm(Wm))); IsZero(WzFcn(phi2mm(Wm))); #Check that they can be calculated from W using the right polynomials phiOnW(Wm) = phimm(Wm); phi2OnW(Wm) = phi2mm(Wm); #Is an element of~$m$ actually in~$\ell$ Inl:=x->(phimm(x)=x); #Convert an element of~$m$ to~$\ell$ mTol:=function(x) if(not Inl(x)) then return fail; else return LinearCombination(llBasis,Coefficients(mmBasis,x)); fi; end; #Check that this works mTol(Sm)=Sl; mTol(Um)=Ul; mTol(Vm)=Vl; mTol(Wm)=fail; #Convert a matrix over~$m$ to a matrix over~$\ell$ mTolM:=mtx->List(mtx,row->List(row,x->mTol(x))); #Complex conjugation mmConj:=x->LinearCombination(mmPBasis,Coefficients(mmBasis,x)); #Adjoint of a matrix mmAdj:=m->TransposedMat(List(m,r->List(r,x->mmConj(x)))); #Check that conjugation acts as it should mmConj(Um) = UPm; mmConj(UPm) = Um; mmConj(Vm) = -Vm; mmConj(Sm) = Sm; mmConj(Wm) = Wm; #Norm from $m$ to $\ell$ NmTol:=x->mTol(x*phimm(x)*phi2mm(x)); #Is an element of~$m$ actually in _mmRe_ InmmRe:=x->IsZero(Coefficients(mmBasis,x)[2]); #Convert an element of~$m$ to _mmRe_ mToRe:=function(x) local c; c:=Coefficients(mmBasis,x); if(IsZero(c[2]))then return c[1]; else return fail; fi; end; #Check that this works mToRe(Sm)=SRe; mToRe(Um)=fail; mToRe(Vm)=fail; mToRe(Wm)=WRe; #Norm from $m$ to _mmRe_ NmTomRe:=x->mToRe(x*mmConj(x)); #Function to reduce elements of~$m$ modulo an integer mmMod:=function(x,q) return LinearCombination(mmBasis, List(Coefficients(mmBasis,x),c->mmReMod(c,q)) ); end; #Function to reduce matrices over~$m$ modulo an integer mmModM:=function(mtx,q) return List(mtx,row->List(row,x->mmMod(x,q))); end; #Calculate the ``discriminant'' of the basis mmTraceMatrix:=ListX(mmReBasis,mmBasis, function(x1,x2) return ListX(mmReBasis,mmBasis, function(y1,y2) return Trace(mmRe,Rationals,x1*y1*Trace(x2*y2)); end); end); Factors(Determinant(mmTraceMatrix)); Set(Factors(Determinant(mmTraceMatrix))) = [2,3,17]; ##### # # Utility functions for matrices # fOnM:=function(f,mtx) return List(mtx,row->List(row,x->f(x))); end; ###### # # The representation of~$\D$ as matrices over~$m$ # ###### WDM:=[[Wm,0,0],[0,phimm(Wm),0],[0,0,phi2mm(Wm)]]*One(mm); SDM:=Sm*One(WDM); UDM:=Um*One(WDM); UPDM:=2/UDM; VDM:=Vm*One(WDM); sigmaDM:=[[0,1,0],[0,0,1],[Um/Sm,0,0]]*One(mm); sigmaDMInv:=sigmaDM^-1; #Polynomial of which~$\sigma$ is a root sigmazmFcn:=sigma->sigma^3-(Um/Sm)*One(sigma); #Check that W, S, U, U', V, and $\sigma$ are roots of the right #polynomials. IsZero(WzFcn(WDM)); IsZero(SzFcn(SDM)); IsZero(UzFcn(UDM)); IsZero(UzFcn(UPDM)); IsZero(VzFcn(VDM)); IsZero(sigmazmFcn(sigmaDM)); #Check that W, S, V, and U all commute WDM*SDM=SDM*WDM; WDM*VDM=VDM*WDM; WDM*UDM=UDM*WDM; SDM*VDM=VDM*SDM; SDM*UDM=UDM*SDM; VDM*UDM=UDM*VDM; #Check that $\sigma$ commutes with S, V, and U sigmaDM*SDM=SDM*sigmaDM; sigmaDM*VDM=VDM*sigmaDM; sigmaDM*UDM=UDM*sigmaDM; #Check that~$\sigma$ conjugates~$W$ correctly sigmaDM*WDM=phiOnW(WDM)*sigmaDM; sigmaDM*phiOnW(WDM)=phi2OnW(WDM)*sigmaDM; sigmaDM*phi2OnW(WDM)=WDM*sigmaDM; ##### # # In the matrix representation of~$\D$ over~$m$, the matrix~$F$ so # that # # \iota(A) = F^{-1} A^* F # FDM:=-Sm*One(WDM)+(1-Sm)*WDM+WDM^2; # Check that FDM is self-adjoint mmAdj(FDM) = FDM; # Check that $\iota$ is acts correctly on S, U, V, W, $\sigma$, # and $sigma^-1$ mmAdj(SDM)*FDM = FDM*SDM; mmAdj(UDM)*FDM = FDM*UPDM; mmAdj(VDM)*FDM = -FDM*VDM; mmAdj(WDM)*FDM = FDM*WDM; mmAdj(sigmaDM)*FDM = FDM*(-Sm*One(WDM)+(-3+Sm)*WDM+(1-Sm)*WDM^2)*sigmaDMInv; mmAdj(sigmaDMInv)*FDM = FDM*((1-2*Sm)*One(WDM)+(1+2*Sm)*WDM+(1+Sm)*WDM^2)*sigmaDM; ##### # # We work out the correct integrality conditions at the 3-adic # place. # # See file ~steger/donald/SU21/C10p2/D.txt for the definition of the # division algebra $\D$ and for all the notation. # ##### ##### # # $\eta_3\in m$ so that # # N_{m/\ell} \eta_3 = U/\sqrt{2} \mod 3^12 # \eta_3 \bar\eta_3 = 1 \mod 3^12 # # This is an approximation $\mod 3^12$. # #First we find $\eta_3$ such that $N_{m/\ell}\eta_3=U/\sqrt{2} \mod 3$ etaList1:=[]; for c00 in [-1..1] do for c01 in [-1..1] do for c10 in [-1..1] do for c11 in [-1..1] do trial:=(c00+c01*Sm)+(c10+c11*Sm)*Um; test:=NmTol(trial)-Ul/Sl; if IsZero(llMod(test,3)) then Add(etaList1,[[c00,c01,c10,c11],trial]); fi; od; od; od; od; Length(etaList1); eta3Approx1:= 2*One(mm)+(2*One(mm)+2*Sm)*Um; #Next, we find $\eta_3 such that $N_{m/\ell}\eta_3=U/\sqrt{2} \mod 9$ and # $eta \bar\eta_3 = 1 \mod (W+1)^2$ etaList2:=[]; for c001 in [0..2] do for c011 in [0..2] do for c101 in [0..2] do for c111 in [0..2] do trial:=eta3Approx1+((c001+c011*Sm)+(c101+c111*Sm)*Um)*(Wm+1); lTest:=NmTol(trial)-Ul/Sl; ReTest:=NmTomRe(trial)-1; if IsIntegralCyclotomic(ReTest/(WRe+1)^2) and IsZero(llMod(lTest,9)) then Add(etaList2,[[c001,c011,c101,c111],trial]); fi; od; od; od; od; Length(etaList2); # Now we improve this value by successive approximations eta3ApproxY:=(3+Sm)+(1+Sm)*Wm+(4+4*Sm)*Um+(2+2*Sm)*Wm*Um; for j in [0..50] do pow3:=3^j; etaListX:=[]; for c00a in [0..2] do for c01a in [0..2] do for c00b in [0..2] do for c01b in [0..2] do trial:=eta3ApproxY +(c00a+c01a*Sm)*pow3*(Wm+1)^2+(c00b+c01b*Sm)*pow3*3; lTest:=NmTol(trial)-Ul/Sl; ReTest:=NmTomRe(trial)-1; if IsIntegralCyclotomic(ReTest/(3*(WRe+1)*pow3)) and IsZero(llMod(lTest,9*pow3)) then Add(etaListX,[[c00a,c01a,c00b,c01b],trial]); fi; od; od; od; od; #Print("pow3= 3^",j); #Print(" Length(etaListX)= ",Length(etaListX)); eta3ApproxX:=etaListX[1][2]; etaListY:=[]; for c00 in [0..2] do for c01 in [0..2] do for c10 in [0..2] do for c11 in [0..2] do trial:=eta3ApproxX +((c00+c01*Sm)+(c10+c11*Sm)*Um)*pow3*3*(Wm+1); lTest:=NmTol(trial)-Ul/Sl; ReTest:=NmTomRe(trial)-1; if IsIntegralCyclotomic(ReTest/(3*pow3*(WRe+1)^2)) and IsZero(llMod(lTest,27*pow3)) then Add(etaListY,[[c00,c01,c10,c11],trial]); fi; od; od; od; od; #Print(" Length(etaListY)= ",Length(etaListY),"\n"); eta3ApproxY:=etaListY[1][2]; od; eta3:= (205986+150195*Sm)+(110557+196619*Sm)*Wm+(399498+365303*Sm)*Wm^2+ (481942+177067*Sm)*Um+(481940+177065*Sm)*Wm*Um; ##### # # Matrices to conjugate the representation of~$\D$ as matrices # over~$m$ to the representation as matrices # over~$\ell_3=\ell\otimes\QQ_3=\QQ_3[\sqrt{2},\sqrt{-5+2\sqrt{2}}]$ # # These are approximations $\mod 3^12$ # C1:=[[eta3*phimm(eta3),0,0],[0,phimm(eta3),0],[0,0,1]]*One(mm); C1Inv:=mmModM(C1^-1,3^12); C2:=[ [1,1,1], [(Wm+1),phimm(Wm+1),phi2mm(Wm+1)], [(Wm+1)^2,phimm((Wm+1)^2),phi2mm((Wm+1)^2)] ]*One(mm); C2Inv:=C2^-1; #Check that the inverses work IsZero(mmModM(C1*C1Inv-One(C1),3^12)); IsOne(C2*C2Inv); ##### # # Conjugated copies of SDM, WDM, UDM, VDM, and sigmaDM lying in # $\ell_3$. # # These are approximations $\mod 3^10$. # SDM3:=mTolM(mmModM(C2*C1*SDM*C1Inv*C2Inv,3^10)); UDM3:=mTolM(mmModM(C2*C1*UDM*C1Inv*C2Inv,3^10)); UPDM3:=llModM(2/UDM3,3^10); VDM3:=mTolM(mmModM(C2*C1*VDM*C1Inv*C2Inv,3^10)); WDM3:=mTolM(mmModM(C2*C1*WDM*C1Inv*C2Inv,3^10)); sigmaDM3:=mTolM(mmModM(C2*C1*sigmaDM*C1Inv*C2Inv,3^10)); sigmaDM3Inv:=llModM(sigmaDM3^-1,3^10); #Check that sigmaDM3Inv is the inverse of sigmaDM3 IsZero(llModM(sigmaDM3Inv*sigmaDM3-One(sigmaDM3),3^10)); #Polynomial of which~$\sigma$ is a root sigmazlFcn:=sigma->sigma^3-(Ul/Sl)*One(sigma); #Check that W, S, U, U', V, and $\sigma$ are roots of the right #polynomials. IsZero(llModM(WzFcn(WDM3),3^10)); IsZero(llModM(SzFcn(SDM3),3^10)); IsZero(llModM(UzFcn(UDM3),3^10)); IsZero(llModM(UzFcn(UPDM3),3^10)); IsZero(llModM(VzFcn(VDM3),3^10)); IsZero(llModM(sigmazlFcn(sigmaDM3),3^10)); #Check that W, S, V, and U all commute IsZero(llModM(WDM3*SDM3-SDM3*WDM3,3^10)); IsZero(llModM(WDM3*VDM3-VDM3*WDM3,3^10)); IsZero(llModM(WDM3*UDM3-UDM3*WDM3,3^10)); IsZero(llModM(SDM3*VDM3-VDM3*SDM3,3^10)); IsZero(llModM(SDM3*UDM3-UDM3*SDM3,3^10)); IsZero(llModM(VDM3*UDM3-UDM3*VDM3,3^10)); #Check that $\sigma$ commutes with S, V, and U IsZero(llModM(sigmaDM3*SDM3-SDM3*sigmaDM3,3^10)); IsZero(llModM(sigmaDM3*VDM3-VDM3*sigmaDM3,3^10)); IsZero(llModM(sigmaDM3*UDM3-UDM3*sigmaDM3,3^10)); #Check that~$\sigma$ conjugates~$W$ correctly IsZero(llModM(sigmaDM3*WDM3-phiOnW(WDM3)*sigmaDM3,3^10)); IsZero(llModM(sigmaDM3*phiOnW(WDM3)-phi2OnW(WDM3)*sigmaDM3,3^10)); IsZero(llModM(sigmaDM3*phi2OnW(WDM3)-WDM3*sigmaDM3,3^10)); ##### # # In the 3-adic matrix representation, the element~$F$ such that # # \iota(A) = F^{-1} A^* F # # Thus, $\iota$-unitary elements will preserve the sesquilinear form # defined by $F$. # # This is an approximation $\mod 3^10$. # FDM3:=mTolM(mmModM(3*mmAdj(C2Inv)*mmAdj(C1Inv)*FDM*C1Inv*C2Inv,3^10)); # Check that FDM3 is self-adjoint llAdj(FDM3) = FDM3; # Check that $\iota$ is acts correctly on S, U, V, W, $\sigma$, # and $sigma^-1$ IsZero(llModM(llAdj(SDM3)*FDM3 - FDM3*SDM3,3^10)); IsZero(llModM(llAdj(UDM3)*FDM3 - FDM3*UPDM3,3^10)); IsZero(llModM(llAdj(VDM3)*FDM3 - -FDM3*VDM3,3^10)); IsZero(llModM(llAdj(WDM3)*FDM3 - FDM3*WDM3,3^10)); IsZero(llModM(llAdj(sigmaDM3)*FDM3 - FDM3*(-Sl*One(WDM3)+(-3+Sl)*WDM3+(1-Sl)*WDM3^2)*sigmaDM3Inv,3^10)); IsZero(llModM(llAdj(sigmaDM3Inv)*FDM3 - FDM3*((1-2*Sl)*One(WDM3)+(1+2*Sl)*WDM3+(1+Sl)*WDM3^2)*sigmaDM3,3^10)); #Check that FDM3 and its inverse are integral IsZero(llModM(3*FDM3,3)); IsZero(llModM(3*FDM3^-1,3)); IsZero(llMod(3*Determinant(FDM3),3)); not IsZero(llMod(Determinant(FDM3),3)); IsZero(llModM( FDM3 - One(FDM3)* [[3-3*Sqrt(2),-3*Sqrt(2),-1+2*Sqrt(2)], [-3*Sqrt(2),2-Sqrt(2),-1+Sqrt(2)], [-1+2*Sqrt(2),-1+Sqrt(2),1-Sqrt(2)]] ,3^10)); # The preceding checks show that the standard $\ell_3$ lattice # $L_0=\O_{\ell_3}^3$, is self-dual with respect to the form given # by~$F$. Note that # $\O_{\ell_3}=\ZZ_3[\sqrt{2},\sqrt{-5+2\sqrt{2}}}]$. # Consequently matrices which preserve that lattice, which is to say # matrices in $GL(3,\ZZ_3[\sqrt{2},\sqrt{-5+2\sqrt{2}}}]$, fix a # hyperspecial vertex in the 3-adic tree corresponding to the 3-adic # version of $PU(\D,\iota)$. # 36-element basis of $\D$ over $\QQ$ basisDM3:=ListX([0..1],[0..1],[0..2],[-1..1], function(i,j,k,l) return Sl^i*Ul^j*WDM3^k*sigmaDM3^l; end );; # For a $3\times 3$ matrix with coefficients in # $\O_\ell=\ZZ[\sqrt{2},\sqrt{-5+2\sqrt{2}}]$, for each of the # 9~entries this function calculate the four coefficients, each # modulo~$q$, and puts them together as a 36~element vector. CondCoeff3:=function(mtx,q) return Concatenation(ListX([1..3],[1..3], function(j,k) local c; c:=Concatenation(llCoeff(mtx[j][k])); Apply(c,x->x mod q); return c; end) ); end; # The element # # \sum c_{ijkl} \sqrt{2}^i U^j W^k \sigma^l # # in~$\D$ will have (approximate) 3-adic matrix representation # # Sum([1..36],ix->c[ix]*basis[ix]) # # and this 3-adic matrix representation will be 3-adically integral if # and only if the vector: # # CondMtxDM3Type1 * c # # consists of 3-adic integers. CondMtxDM3Type1:= TransposedMat(List(basisDM3,mtx->CondCoeff3(mtx,3^3)));; #CondMtxDM3Type1:=[ #[ 23, 1, 8, 13, 26, 22, 5, 1, 14, 4, 0, 23, 2, 0, 10, 4, 0, 11, 26, 0, 3, 16, # 0, 0, 14, 0, 0, 8, 0, 19, 4, 0, 20, 8, 0, 22 ], #[ 13, 0, 15, 8, 0, 0, 7, 0, 0, 4, 0, 23, 2, 0, 10, 4, 0, 11, 23, 1, 8, 13, # 26, 22, 5, 1, 14, 4, 0, 23, 2, 0, 10, 4, 0, 11 ], #[ 25, 0, 2, 26, 0, 22, 25, 0, 8, 17, 1, 14, 10, 26, 7, 26, 1, 11, 23, 0, 4, # 25, 0, 17, 23, 0, 16, 18, 0, 11, 12, 0, 7, 6, 0, 5 ], #[ 25, 0, 2, 26, 0, 22, 25, 0, 8, 9, 0, 19, 6, 0, 17, 3, 0, 16, 25, 0, 2, 26, # 0, 22, 25, 0, 8, 17, 1, 14, 10, 26, 7, 26, 1, 11 ], #[ 22, 0, 22, 4, 1, 4, 18, 25, 0, 7, 0, 14, 18, 0, 15, 11, 0, 4, 16, 0, 20, # 16, 0, 10, 15, 0, 20, 23, 0, 19, 0, 0, 12, 4, 0, 26 ], #[ 8, 0, 10, 8, 0, 5, 21, 0, 10, 25, 0, 23, 0, 0, 6, 2, 0, 13, 22, 0, 22, 4, # 1, 4, 18, 25, 0, 7, 0, 14, 18, 0, 15, 11, 0, 4 ], #[ 10, 0, 20, 18, 0, 6, 8, 0, 25, 7, 0, 19, 22, 1, 4, 24, 25, 12, 2, 0, 4, 0, # 0, 21, 25, 0, 14, 11, 0, 10, 25, 0, 16, 2, 0, 3 ], #[ 1, 0, 2, 0, 0, 24, 26, 0, 7, 19, 0, 5, 26, 0, 8, 1, 0, 15, 10, 0, 20, 18, # 0, 6, 8, 0, 25, 7, 0, 19, 22, 1, 4, 24, 25, 12 ], #[ 6, 0, 13, 17, 0, 2, 19, 1, 3, 0, 0, 20, 9, 0, 23, 7, 0, 17, 13, 0, 24, 25, # 0, 17, 14, 0, 17, 0, 0, 4, 0, 0, 19, 23, 0, 16 ], #[ 20, 0, 12, 26, 0, 22, 7, 0, 22, 0, 0, 2, 0, 0, 23, 25, 0, 8, 6, 0, 13, 17, # 0, 2, 19, 1, 3, 0, 0, 20, 9, 0, 23, 7, 0, 17 ], #[ 0, 0, 17, 9, 0, 2, 10, 0, 5, 6, 0, 1, 26, 0, 8, 4, 1, 0, 0, 0, 25, 0, 0, 4, # 2, 0, 19, 13, 0, 2, 16, 0, 25, 9, 0, 19 ], #[ 0, 0, 26, 0, 0, 2, 1, 0, 23, 20, 0, 1, 8, 0, 26, 18, 0, 23, 0, 0, 17, 9, 0, # 2, 10, 0, 5, 6, 0, 1, 26, 0, 8, 4, 1, 0 ], #[ 9, 0, 3, 18, 0, 9, 21, 24, 18, 6, 0, 6, 6, 0, 21, 6, 0, 18, 15, 0, 3, 3, 0, # 0, 9, 0, 24, 12, 0, 12, 12, 0, 15, 12, 0, 9 ], #[ 21, 0, 15, 15, 0, 0, 18, 0, 12, 6, 0, 6, 6, 0, 21, 6, 0, 18, 9, 0, 3, 18, # 0, 9, 21, 24, 18, 6, 0, 6, 6, 0, 21, 6, 0, 18 ], #[ 24, 0, 24, 24, 0, 3, 24, 0, 18, 0, 0, 21, 9, 0, 18, 12, 24, 18, 21, 0, 21, # 21, 0, 6, 21, 0, 9, 3, 0, 18, 18, 0, 12, 24, 0, 15 ], #[ 24, 0, 24, 24, 0, 3, 24, 0, 18, 15, 0, 9, 9, 0, 6, 12, 0, 21, 24, 0, 24, # 24, 0, 3, 24, 0, 18, 0, 0, 21, 9, 0, 18, 12, 24, 18 ], #[ 26, 1, 26, 22, 26, 4, 8, 1, 17, 25, 0, 2, 2, 0, 19, 4, 0, 8, 5, 0, 3, 4, 0, # 3, 20, 0, 3, 23, 0, 4, 4, 0, 11, 8, 0, 16 ], #[ 16, 0, 15, 2, 0, 15, 10, 0, 15, 25, 0, 2, 2, 0, 19, 4, 0, 8, 26, 1, 26, 22, # 26, 4, 8, 1, 17, 25, 0, 2, 2, 0, 19, 4, 0, 8 ], #[ 1, 0, 26, 26, 0, 4, 25, 0, 23, 2, 1, 23, 19, 26, 16, 2, 1, 5, 2, 0, 25, 25, # 0, 8, 23, 0, 19, 9, 0, 26, 0, 0, 19, 12, 0, 14 ], #[ 1, 0, 26, 26, 0, 4, 25, 0, 23, 18, 0, 13, 0, 0, 23, 6, 0, 7, 1, 0, 26, 26, # 0, 4, 25, 0, 23, 2, 1, 23, 19, 26, 16, 2, 1, 5 ], #[ 23, 0, 15, 9, 1, 5, 10, 1, 23, 9, 0, 16, 16, 0, 13, 7, 0, 12, 11, 0, 14, # 12, 0, 7, 22, 0, 17, 0, 0, 23, 23, 0, 8, 23, 0, 15 ], #[ 19, 0, 7, 6, 0, 17, 11, 0, 22, 0, 0, 25, 25, 0, 4, 25, 0, 21, 23, 0, 15, 9, # 1, 5, 10, 1, 23, 9, 0, 16, 16, 0, 13, 7, 0, 12 ], #[ 9, 0, 19, 19, 0, 7, 10, 0, 21, 5, 0, 9, 3, 1, 8, 22, 1, 23, 0, 0, 2, 2, 0, # 23, 2, 0, 6, 2, 0, 0, 25, 0, 17, 17, 0, 11 ], #[ 0, 0, 1, 1, 0, 25, 1, 0, 3, 1, 0, 0, 26, 0, 22, 22, 0, 19, 9, 0, 19, 19, 0, # 7, 10, 0, 21, 5, 0, 9, 3, 1, 8, 22, 1, 23 ], #[ 0, 0, 12, 12, 24, 0, 12, 24, 15, 12, 0, 0, 12, 0, 12, 18, 0, 21, 18, 0, 3, # 12, 0, 24, 3, 0, 21, 24, 0, 0, 24, 0, 24, 9, 0, 15 ], #[ 9, 0, 15, 6, 0, 12, 15, 0, 24, 12, 0, 0, 12, 0, 12, 18, 0, 21, 0, 0, 12, # 12, 24, 0, 12, 24, 15, 12, 0, 0, 12, 0, 12, 18, 0, 21 ], #[ 21, 0, 0, 21, 0, 21, 18, 0, 3, 9, 0, 12, 21, 24, 9, 12, 24, 24, 15, 0, 0, # 15, 0, 15, 9, 0, 6, 21, 0, 3, 15, 0, 0, 21, 0, 6 ], #[ 21, 0, 0, 21, 0, 21, 18, 0, 3, 24, 0, 15, 21, 0, 0, 24, 0, 3, 21, 0, 0, 21, # 0, 21, 18, 0, 3, 9, 0, 12, 21, 24, 9, 12, 24, 24 ], #[ 21, 0, 3, 3, 0, 21, 21, 24, 3, 0, 0, 21, 6, 0, 0, 12, 0, 9, 9, 0, 6, 24, 0, # 6, 0, 0, 9, 0, 0, 15, 12, 0, 0, 24, 0, 18 ], #[ 18, 0, 3, 12, 0, 3, 0, 0, 18, 0, 0, 21, 6, 0, 0, 12, 0, 9, 21, 0, 3, 3, 0, # 21, 21, 24, 3, 0, 0, 21, 6, 0, 0, 12, 0, 9 ], #[ 0, 0, 3, 24, 0, 0, 21, 0, 9, 21, 0, 12, 21, 0, 21, 3, 24, 3, 0, 0, 6, 21, # 0, 0, 15, 0, 18, 9, 0, 18, 12, 0, 6, 3, 0, 18 ], #[ 0, 0, 3, 24, 0, 0, 21, 0, 9, 18, 0, 9, 6, 0, 3, 15, 0, 9, 0, 0, 3, 24, 0, # 0, 21, 0, 9, 21, 0, 12, 21, 0, 21, 3, 24, 3 ], #[ 5, 1, 20, 19, 2, 1, 14, 4, 23, 25, 0, 2, 23, 0, 25, 19, 0, 8, 23, 0, 21, 7, # 0, 24, 20, 0, 24, 23, 0, 4, 19, 0, 23, 11, 0, 16 ], #[ 25, 0, 24, 17, 0, 12, 10, 0, 12, 25, 0, 2, 23, 0, 25, 19, 0, 8, 5, 1, 20, # 19, 2, 1, 14, 4, 23, 25, 0, 2, 23, 0, 25, 19, 0, 8 ], #[ 1, 0, 26, 2, 0, 1, 4, 0, 23, 8, 1, 17, 25, 2, 4, 26, 4, 11, 2, 0, 25, 4, 0, # 2, 8, 0, 19, 0, 0, 17, 15, 0, 1, 9, 0, 8 ], #[ 1, 0, 26, 2, 0, 1, 4, 0, 23, 0, 0, 22, 21, 0, 14, 18, 0, 4, 1, 0, 26, 2, 0, # 1, 4, 0, 23, 8, 1, 17, 25, 2, 4, 26, 4, 11 ] #]; ##### # # We work out the correct integrality conditions at the first # $17$-adic place of~$\ell$, denoted $17+$, and characterized # by the choice: # # \sqrt{2} = 6 \mod 17+ # # See file ~steger/donald/SU21/C10p2/D.txt for the definition of the # division algebra $\D$ and for all the notation. # ##### ##### # # S17 so that # # S17^2 = 2 \mod 17^{10}, S17 = 6 \mod 17 # # This is an approximation $\mod 17^{10}$ S17Approx:=6; for j in [1..10] do S17Approx:=(S17Approx - (S17Approx^2-2)/(2*S17Approx)) mod 17^10; od; S17:=292094863096; S17^2 mod 17^10 = 2; ##### # # $k_{17+}=\QQ_{17+}$, version for place $17+$ # # Everything is approximated $\mod 17^{10}$ Sk17p:=S17; kk17p:=Field(Sk17p); # The field itself kk17pBasis:=Basis(kk17p,[1]); #Basis #Function to reduce elements of~$k$ modulo an integer kk17pMod:=function(x,q) return x mod q; end; #Check that Sk17p satisfies the correct polynomial IsZero(kk17pMod(SzFcn(Sk17p),17^10)); #Function to reduce matrices over~$k_{17+}$ modulo an integer kk17pModM:=function(mtx,q) return List(mtx,row->List(row,x->kk17pMod(x,q))); end; #Calculate the ``discriminant'' of the basis kk17pTraceMatrix:= List(kk17pBasis,x->List(kk17pBasis,y->Trace(kk17p,x*y))); Factors(Determinant(kk17pTraceMatrix)); Set(Factors(Determinant(kk17pTraceMatrix))) = [1]; ##### # # $\ell_{17+}=\QQ_{17}[\sqrt{-5+2\cdot 6}]=\QQ_{17}[\sqrt{7}] # # Everything is approximated $\mod 17^{10}$. ll17p:=Field(Sqrt(7)); Sl17p:=Sk17p*One(ll17p); # \sqrt{2} ll17pBasis0:=Basis(ll17p,[1,Sqrt(7)]); # Basis #Function to reduce elements of $l_{17+}$_ modulo an integer ll17pMod:=function(x,q) local c; c:=Coefficients(ll17pBasis0,x); Apply(c,t->t mod q); return LinearCombination(ll17pBasis0,c); end; # Find an approximation $\mod 17^{10}$ to $\sqrt{-5+2\sqrt{2}}$ Vl17pApprox:=Sqrt(7); for j in [1..10] do Vl17pApprox:=ll17pMod(Vl17pApprox - (Vl17pApprox^2-(-5+2*Sl17p))/(2*Vl17pApprox),17^10); od; Vl17p:=1692262524954*Sqrt(7); # \sqrt{-5+2\sqrt{2}} Ul17p:=ll17pMod((1+Sl17p+Vl17p)/2,17^10); # Generating element, U UPl17p:=ll17pMod(2/Ul17p,17^10); # U', The conjugate of U ll17pBasis:=Basis(ll17p,[One(ll17p),Ul17p]); #Basis over $k_{17+}$ # Alternative Basis over $k_{17+}$ llP17pBasis:=Basis(ll17p,[One(ll17p),UPl17p]); #Function to calculate coefficients ll17pCoeff:=x->Coefficients(ll17pBasis,x); Vz17pFcn:=x->x^2-(-5+2*Sl17p)*One(x); #Function satisfied by V Uz17pFcn:=x->x^2-(1+Sl17p)*x+2*One(x); #Function satisfied by U #Check that S, V, U, U' are roots of their polynomials IsZero(ll17pMod(SzFcn(Sl17p),17^10)); IsZero(ll17pMod(Vz17pFcn(Vl17p),17^10)); IsZero(ll17pMod(Uz17pFcn(Ul17p),17^10)); IsZero(ll17pMod(Uz17pFcn(UPl17p),17^10)); #Function to conjugate an element ll17pConj:=x->x^ANFAutomorphism(ll17p,17); #Function to calculate the adjoint of a matrix ll17pAdj:=m->TransposedMat(List(m,r->List(r,x->ll17pConj(x)))); #Check that conjugates are as expected IsZero(ll17pMod(ll17pConj(Ul17p) - UPl17p,17^10)); IsZero(ll17pMod(ll17pConj(UPl17p) - Ul17p,17^10)); ll17pConj(Vl17p) = -Vl17p; ll17pConj(Sl17p) = Sl17p; #Function to reduce matrices over~$\ell_{17p}$ modulo an integer ll17pModM:=function(mtx,q) return List(mtx,row->List(row,x->ll17pMod(x,q))); end; ##### # # The polynomial $W^3-3W+1$ splits in $\ell_{17+}$. Indeed it splits # in $k_{17+}=\QQ_{17}$. # # Consequently $\mm_{17+}=\ell_{17+}[W]/(W^3-3W+1)$ is the direct sum # of 3~copies of $\ell_{17+}$. In these three copies we have, # respectively, $W=7 \mod 17$, $W=14 \mod 17$, and $W=13 \mod 17$. # Find $W$, $\mod 17^{10}$ Wl17pApprox:=7; for j in [1..10] do Wl17pApprox:=ll17pMod(Wl17pApprox - (WzFcn(Wl17pApprox)/(3*Wl17pApprox^2-3*One(Wl17pApprox))) ,17^10); od; Wl17p:=1108420179313; phiWl17p:=phiOnW(Wl17p) mod 17^10; phi2Wl17p:=phi2OnW(Wl17p) mod 17^10; # Check that $W$ and its conjugates are roots of the right polynomial # ($\mod 17^{10}$) IsZero(ll17pMod(WzFcn(Wl17p),17^10)); IsZero(ll17pMod(WzFcn(phiWl17p),17^10)); IsZero(ll17pMod(WzFcn(phi2Wl17p),17^10)); #Check that they are the right conjugates IsZero(ll17pMod(phiOnW(phiWl17p)-phi2Wl17p,17^10)); IsZero(ll17pMod(phiOnW(phi2Wl17p)-Wl17p,17^10)); ###### # # The representation of~$\D$ as matrices over~$\ell_{17+}$. # # We find matrices for $S$, $U$, $U'$, $V$, $W, and~$\sigma$, all of # which are approximations $\mod 17^{10}$. WDM17p:=[[Wl17p,0,0],[0,phiWl17p,0],[0,0,phi2Wl17p]]*One(ll17p); SDM17p:=Sl17p*One(WDM17p); UDM17p:=Ul17p*One(WDM17p); UPDM17p:=ll17pModM(2/UDM17p,17^10); VDM17p:=Vl17p*One(WDM17p); sigmaDM17p:= ll17pModM([[0,1,0],[0,0,1],[Ul17p/Sl17p,0,0]]*One(ll17p),17^10); sigmaDM17pInv:=ll17pModM(sigmaDM17p^-1,17^10); #Polynomial of which~$\sigma$ is a root sigmazl17pFcn:=sigma->sigma^3-(Ul17p/Sl17p)*One(sigma); #Check that W, S, U, U', V, and $\sigma$ are roots of the right #polynomials. IsZero(ll17pModM(WzFcn(WDM17p),17^10)); IsZero(ll17pModM(SzFcn(SDM17p),17^10)); IsZero(ll17pModM(Uz17pFcn(UDM17p),17^10)); IsZero(ll17pModM(Uz17pFcn(UPDM17p),17^10)); IsZero(ll17pModM(Vz17pFcn(VDM17p),17^10)); IsZero(ll17pModM(sigmazl17pFcn(sigmaDM17p),17^10)); #Check that W, S, V, and U all commute IsZero(ll17pModM(WDM17p*SDM17p-SDM17p*WDM17p,17^10)); IsZero(ll17pModM(WDM17p*VDM17p-VDM17p*WDM17p,17^10)); IsZero(ll17pModM(WDM17p*UDM17p-UDM17p*WDM17p,17^10)); IsZero(ll17pModM(SDM17p*VDM17p-VDM17p*SDM17p,17^10)); IsZero(ll17pModM(SDM17p*UDM17p-UDM17p*SDM17p,17^10)); IsZero(ll17pModM(VDM17p*UDM17p-UDM17p*VDM17p,17^10)); #Check that $\sigma$ commutes with S, V, and U IsZero(ll17pModM(sigmaDM17p*SDM17p-SDM17p*sigmaDM17p,17^10)); IsZero(ll17pModM(sigmaDM17p*VDM17p-VDM17p*sigmaDM17p,17^10)); IsZero(ll17pModM(sigmaDM17p*UDM17p-UDM17p*sigmaDM17p,17^10)); #Check that~$\sigma$ conjugates~$W$ correctly IsZero(ll17pModM(sigmaDM17p*WDM17p - phiOnW(WDM17p)*sigmaDM17p,17^10)); IsZero(ll17pModM(sigmaDM17p*phiOnW(WDM17p) - phi2OnW(WDM17p)*sigmaDM17p,17^10)); IsZero(ll17pModM(sigmaDM17p*phi2OnW(WDM17p) - WDM17p*sigmaDM17p,17^10)); ##### # # In the matrix representation of~$\D$ over~$\ell_{17+}$, the # matrix~$F$ so that # # \iota(A) = F^{-1} A^* F # FDM17p:= ll17pModM(-Sl17p*One(WDM17p)+(1-Sl17p)*WDM17p+WDM17p^2,17^10); # Check that FDM17p is self-adjoint ll17pAdj(FDM17p) = FDM17p; # Check that $\iota$ is acts correctly on S, U, V, W, $\sigma$, # and $sigma^-1$ ll17pAdj(SDM17p)*FDM17p = FDM17p*SDM17p; IsZero(ll17pModM(ll17pAdj(UDM17p)*FDM17p - FDM17p*UPDM17p,17^10)); IsZero(ll17pModM(ll17pAdj(VDM17p)*FDM17p - -FDM17p*VDM17p,17^10)); IsZero(ll17pModM(ll17pAdj(WDM17p)*FDM17p - FDM17p*WDM17p,17^10)); IsZero(ll17pModM(ll17pAdj(sigmaDM17p)*FDM17p - FDM17p* (-Sl17p*One(WDM17p)+(-3+Sl17p)*WDM17p+(1-Sl17p)*WDM17p^2) *sigmaDM17pInv ,17^10)); IsZero(ll17pModM(ll17pAdj(sigmaDM17pInv)*FDM17p - FDM17p* ((1-2*Sl17p)*One(WDM17p)+(1+2*Sl17p)*WDM17p+(1+Sl17p)*WDM17p^2) *sigmaDM17p ,17^10)); #Check that FDM17p and its inverse are integral IsZero(ll17pModM(17*FDM17p,17)); IsZero(ll17pModM(17*FDM17p^-1,17)); IsZero(ll17pMod(17*Determinant(FDM17p),17)); not IsZero(ll17pMod(Determinant(FDM17p),17)); # The preceding checks show that the standard $\ell_{17+}$-lattice # $L_0=\O_{\ell_{17+}}^3$, is self-dual with respect to the form given # by~$F$. Note that $\O_{\ell_{17+}=\ZZ_{17}[U]/(U^2-(1+\sqrt{2})U+2)$ # Consequently matrices which preserve that lattice, which is to say # matrices in $GL(3,\ZZ_{17}[U])$, fix a hyperspecial vertex in the # $17+$-adic tree corresponding to the $17+$-adic version of # $PU(\D,\iota)$. # 36-element basis of $\D$ over $\QQ$ basisDM17p:=ListX([0..1],[0..1],[0..2],[-1..1], function(i,j,k,l) return Sl17p^i*Ul17p^j*WDM17p^k*sigmaDM17p^l; end );; # For a $3\times 3$ matrix with coefficients in # $\O_{\ell_{17+}}=\ZZ_{17}[\sqrt{7}]$, for each of the 9~entries this # function calculate the four coefficients, each modulo~$q$, and puts # them together as a 18~element vector. CondCoeff17p:=function(mtx,q) return Concatenation(ListX([1..3],[1..3], function(j,k) local c; c:=ll17pCoeff(mtx[j][k]); Apply(c,x->x mod q); return c; end) ); end; # The element # # \sum c_{ijkl} \sqrt{2}^i U^j W^k \sigma^l # # in~$\D$, when reduced $\mod 17+$, will have (approximate) 17-adic # matrix representation # # Sum([1..36],ix->c[ix]*basis[ix]) # # and this $17$-adic matrix representation will be $17$-adically # integral if and only if the vector: # # CondMtxDM17pType1 * c # # consists of 17-adic integers. CondMtxDM17pType1:= TransposedMat(List(basisDM17p,mtx->CondCoeff17p(mtx,17^2)));; #CondMtxDM17pType1:=[ #[ 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 93, # 0, 0, 39, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 244, 0, 0, 93, 0, 0, 39, 0 ], #[ 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, # 93, 0, 0, 39, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 0, 244, 0, 0, 93, 0, 0, 39 ], #[ 123, 0, 0, 266, 0, 0, 9, 0, 0, 244, 0, 0, 93, 0, 0, 39, 0, 0, 245, 0, 0, # 168, 0, 0, 173, 0, 0, 2, 0, 0, 150, 0, 0, 268, 0, 0 ], #[ 167, 0, 0, 98, 0, 0, 125, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 288, 0, 0, 214, # 0, 0, 155, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 67, 0, 0, # 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 244, 0, 0, 67, 0, 0, 3, 0, 0 ], #[ 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 67, 0, # 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 244, 0, 0, 67, 0, 0, 3, 0 ], #[ 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 67, # 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 244, 0, 0, 67, 0, 0, 3 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 160, 0, 0, 23, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 0, 287, 0, 0, 25, 0, 0, 121 ], #[ 0, 0, 122, 0, 0, 209, 0, 0, 133, 0, 0, 123, 0, 0, 52, 0, 0, 217, 0, 0, 1, 0, # 0, 132, 0, 0, 84, 0, 0, 245, 0, 0, 261, 0, 0, 61 ], #[ 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 129, 0, # 0, 266, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 244, 0, 0, 129, 0, 0, 266, 0, 0 ], #[ 0, 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 129, # 0, 0, 266, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 244, 0, 0, 129, 0, 0, 266, 0 ] #]; ##### # # We work out the correct integrality conditions at the second # $17$-adic place of~$\ell$, denoted $17-$, and characterized # by the choice: # # \sqrt{2} = -6 \mod 17+ # # See file ~steger/donald/SU21/C10p2/D.txt for the definition of the # division algebra $\D$ and for all the notation. # ##### ##### # # $k_{17-}=\QQ_{17-}$, version for place $17-$ # # Everything is approximated $\mod 17^{10}$ Sk17m:=-S17; kk17m:=Field(Sk17m); # The field itself kk17mBasis:=Basis(kk17m,[1]); #Basis #Function to reduce elements of~$k$ modulo an integer kk17mMod:=function(x,q) return x mod q; end; #Check that Sk17m satisfies the correct polynomial IsZero(kk17mMod(SzFcn(Sk17m),17^10)); #Function to reduce matrices over~$k_{17-}$ modulo an integer kk17mModM:=function(mtx,q) return List(mtx,row->List(row,x->kk17mMod(x,q))); end; #Calculate the ``discriminant'' of the basis kk17mTraceMatrix:= List(kk17mBasis,x->List(kk17mBasis,y->Trace(kk17m,x*y))); Factors(Determinant(kk17mTraceMatrix)); Set(Factors(Determinant(kk17mTraceMatrix))) = [1]; ##### # # $\ell_{17-}=\QQ_{17}[\sqrt{-5+2\cdot 6}]=\QQ_{17}[\sqrt{7}] # # Everything is approximated $\mod 17^{10}$. ll17m:=Field(Sqrt(85)); Sl17m:=Sk17m*One(ll17m); # \sqrt{2} ll17mBasis0:=Basis(ll17m,[1,Sqrt(85)]); # Basis #Function to reduce elements of $l_{17-}$_ modulo an integer ll17mMod:=function(x,q) local c; c:=Coefficients(ll17mBasis0,x); Apply(c,t->t mod q); return LinearCombination(ll17mBasis0,c); end; # Find an approximation $\mod 17^{10}$ to $\sqrt{-5+2\sqrt{2}}$ Vl17mApprox:=Sqrt(85); for j in [1..10] do Vl17mApprox:=ll17mMod(Vl17mApprox - (Vl17mApprox^2-(-5+2*Sl17m))/(2*Vl17mApprox),17^10); od; Vl17m:=1861007208251*Sqrt(85); # \sqrt{-5+2\sqrt{2}} Ul17m:=ll17mMod((1+Sl17m+Vl17m)/2,17^10); # Generating element, U UPl17m:=ll17mMod(2/Ul17m,17^10); # U', The conjugate of U ll17mBasis:=Basis(ll17m,[One(ll17m),Ul17m]); #Basis over $k_{17-}$ # Alternative Basis over $k_{17-}$ llP17mBasis:=Basis(ll17m,[One(ll17m),UPl17m]); #Function to calculate coefficients ll17mCoeff:=x->Coefficients(ll17mBasis,x); Vz17mFcn:=x->x^2-(-5+2*Sl17m)*One(x); #Function satisfied by V Uz17mFcn:=x->x^2-(1+Sl17m)*x+2*One(x); #Function satisfied by U #Check that S, V, U, U' are roots of their polynomials IsZero(ll17mMod(SzFcn(Sl17m),17^10)); IsZero(ll17mMod(Vz17mFcn(Vl17m),17^10)); IsZero(ll17mMod(Uz17mFcn(Ul17m),17^10)); IsZero(ll17mMod(Uz17mFcn(UPl17m),17^10)); #Function to conjugate an element ll17mConj:=x->x^ANFAutomorphism(ll17m,6); #Function to calculate the adjoint of a matrix ll17mAdj:=m->TransposedMat(List(m,r->List(r,x->ll17mConj(x)))); #Check that conjugates are as expected IsZero(ll17mMod(ll17mConj(Ul17m) - UPl17m,17^10)); IsZero(ll17mMod(ll17mConj(UPl17m) - Ul17m,17^10)); ll17mConj(Vl17m) = -Vl17m; ll17mConj(Sl17m) = Sl17m; #Function to reduce matrices over~$\ell_{17m}$ modulo an integer ll17mModM:=function(mtx,q) return List(mtx,row->List(row,x->ll17mMod(x,q))); end; ##### # # The polynomial $W^3-3W+1$ splits in $\ell_{17-}$. Indeed it splits # in $k_{17-}=\QQ_{17}$. # # Consequently $\mm_{17-}=\ell_{17-}[W]/(W^3-3W+1)$ is the direct sum # of 3~copies of $\ell_{17-}$. In these three copies we have, # respectively, $W=7 \mod 17$, $W=14 \mod 17$, and $W=13 \mod 17$. Wl17m:=Wl17p; phiWl17m:=phiWl17p; phi2Wl17m:=phi2Wl17p; # Check that $W$ and its conjugates are roots of the right polynomial # ($\mod 17^{10}$) IsZero(ll17mMod(WzFcn(Wl17m),17^10)); IsZero(ll17mMod(WzFcn(phiWl17m),17^10)); IsZero(ll17mMod(WzFcn(phi2Wl17m),17^10)); #Check that they are the right conjugates IsZero(ll17mMod(phiOnW(phiWl17m)-phi2Wl17m,17^10)); IsZero(ll17mMod(phiOnW(phi2Wl17m)-Wl17m,17^10)); ###### # # The representation of~$\D$ as matrices over~$\ell_{17-}$. # # We find matrices for $S$, $U$, $U'$, $V$, $W, and~$\sigma$, all of # which are approximations $\mod 17^{10}$. WDM17m:=[[Wl17m,0,0],[0,phiWl17m,0],[0,0,phi2Wl17m]]*One(ll17m); SDM17m:=Sl17m*One(WDM17m); UDM17m:=Ul17m*One(WDM17m); UPDM17m:=ll17mModM(2/UDM17m,17^10); VDM17m:=Vl17m*One(WDM17m); sigmaDM17m:= ll17mModM([[0,1,0],[0,0,1],[Ul17m/Sl17m,0,0]]*One(ll17m),17^10); sigmaDM17mInv:=ll17mModM(sigmaDM17m^-1,17^10); #Polynomial of which~$\sigma$ is a root sigmazl17mFcn:=sigma->sigma^3-(Ul17m/Sl17m)*One(sigma); #Check that W, S, U, U', V, and $\sigma$ are roots of the right #polynomials. IsZero(ll17mModM(WzFcn(WDM17m),17^10)); IsZero(ll17mModM(SzFcn(SDM17m),17^10)); IsZero(ll17mModM(Uz17mFcn(UDM17m),17^10)); IsZero(ll17mModM(Uz17mFcn(UPDM17m),17^10)); IsZero(ll17mModM(Vz17mFcn(VDM17m),17^10)); IsZero(ll17mModM(sigmazl17mFcn(sigmaDM17m),17^10)); #Check that W, S, V, and U all commute IsZero(ll17mModM(WDM17m*SDM17m-SDM17m*WDM17m,17^10)); IsZero(ll17mModM(WDM17m*VDM17m-VDM17m*WDM17m,17^10)); IsZero(ll17mModM(WDM17m*UDM17m-UDM17m*WDM17m,17^10)); IsZero(ll17mModM(SDM17m*VDM17m-VDM17m*SDM17m,17^10)); IsZero(ll17mModM(SDM17m*UDM17m-UDM17m*SDM17m,17^10)); IsZero(ll17mModM(VDM17m*UDM17m-UDM17m*VDM17m,17^10)); #Check that $\sigma$ commutes with S, V, and U IsZero(ll17mModM(sigmaDM17m*SDM17m-SDM17m*sigmaDM17m,17^10)); IsZero(ll17mModM(sigmaDM17m*VDM17m-VDM17m*sigmaDM17m,17^10)); IsZero(ll17mModM(sigmaDM17m*UDM17m-UDM17m*sigmaDM17m,17^10)); #Check that~$\sigma$ conjugates~$W$ correctly IsZero(ll17mModM(sigmaDM17m*WDM17m - phiOnW(WDM17m)*sigmaDM17m,17^10)); IsZero(ll17mModM(sigmaDM17m*phiOnW(WDM17m) - phi2OnW(WDM17m)*sigmaDM17m,17^10)); IsZero(ll17mModM(sigmaDM17m*phi2OnW(WDM17m) - WDM17m*sigmaDM17m,17^10)); ##### # # In the matrix representation of~$\D$ over~$\ell_{17-}$, the # matrix~$F$ so that # # \iota(A) = F^{-1} A^* F # # This is an approximation $\mod 17^{10}$. FDM17m:= ll17mModM(-Sl17m*One(WDM17m)+(1-Sl17m)*WDM17m+WDM17m^2,17^10); # Check that FDM17m is self-adjoint ll17mAdj(FDM17m) = FDM17m; # Check that $\iota$ acts correctly on S, U, V, W, $\sigma$, and # $sigma^-1$ ll17mAdj(SDM17m)*FDM17m = FDM17m*SDM17m; IsZero(ll17mModM(ll17mAdj(UDM17m)*FDM17m - FDM17m*UPDM17m,17^10)); IsZero(ll17mModM(ll17mAdj(VDM17m)*FDM17m - -FDM17m*VDM17m,17^10)); IsZero(ll17mModM(ll17mAdj(WDM17m)*FDM17m - FDM17m*WDM17m,17^10)); IsZero(ll17mModM(ll17mAdj(sigmaDM17m)*FDM17m - FDM17m* (-Sl17m*One(WDM17m)+(-3+Sl17m)*WDM17m+(1-Sl17m)*WDM17m^2) *sigmaDM17mInv ,17^10)); IsZero(ll17mModM(ll17mAdj(sigmaDM17mInv)*FDM17m - FDM17m* ((1-2*Sl17m)*One(WDM17m)+(1+2*Sl17m)*WDM17m+(1+Sl17m)*WDM17m^2) *sigmaDM17m ,17^10)); #Check that FDM17m and its inverse are integral IsZero(ll17mModM(17*FDM17m,17)); IsZero(ll17mModM(17*FDM17m^-1,17)); IsZero(ll17mMod(17*Determinant(FDM17m),17)); not IsZero(ll17mMod(Determinant(FDM17m),17)); # The preceding checks show that the standard $\ell_{17-}$-lattice # $L_0=\O_{\ell_{17-}}^3$, is self-dual with respect to the form given # by~$F$. Note that $\O_{\ell_{17-}=\ZZ_{17}[U]/(U^2-(1+\sqrt{2})U+2)$ # Consequently matrices which preserve that lattice, which is to say # matrices in $GL(3,\ZZ_{17}[U])$, fix a vertex of Type~1 in the # $17-$-adic tree corresponding to the $17-$-adic version of # $PU(\D,\iota)$. # 36-element basis of $\D$ over $\QQ$ basisDM17m:=ListX([0..1],[0..1],[0..2],[-1..1], function(i,j,k,l) return Sl17m^i*Ul17m^j*WDM17m^k*sigmaDM17m^l; end );; # For a $3\times 3$ matrix with coefficients in # $\O_{\ell_{17-}}=\ZZ_{17}[U]$, for each of the 9~entries # this function calculate the four coefficients, each modulo~$q$, and # puts them together as a 18~element vector. CondCoeff17m:=function(mtx,q) return Concatenation(ListX([1..3],[1..3], function(j,k) local c; c:=ll17mCoeff(mtx[j][k]); Apply(c,x->x mod q); return c; end) ); end; # The element # # \sum c_{ijkl} \sqrt{2}^i U^j W^k \sigma^l # # in~$\D$, when reduced $\mod 17-$, will have (approximate) 17-adic # matrix representation # # Sum([1..36],ix->c[ix]*basis[ix]) # # and this $17$-adic matrix representation will be $17$-adically # integral if and only if the vector: # # CondMtxDM17mType1 * c # # consists of 17-adic integers. CondMtxDM17mType1:= TransposedMat(List(basisDM17m,mtx->CondCoeff17m(mtx,17^2)));; # Here we check that putting together the Type~1 conditions at~$17\pm$ # we obtain the simplest possible condition --- each of the $c_{ijkl}$ # must be a $17$-adic integer. CondMtxDM17pmType1:= Concatenation(CondMtxDM17pType1,CondMtxDM17mType1); ForAll(CondMtxDM17pmType1,row->ForAll(row,x->IsInt(x))); Determinant(CondMtxDM17pmType1) mod 17 <> 0; #CondMtxDM17mType1:=[ #[ 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 196, # 0, 0, 250, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 45, 0, 0, 196, 0, 0, 250, 0 ], #[ 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, # 196, 0, 0, 250, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 0, 45, 0, 0, 196, 0, 0, 250 ], #[ 168, 0, 0, 173, 0, 0, 259, 0, 0, 45, 0, 0, 196, 0, 0, 250, 0, 0, 46, 0, 0, # 271, 0, 0, 95, 0, 0, 2, 0, 0, 150, 0, 0, 268, 0, 0 ], #[ 122, 0, 0, 191, 0, 0, 164, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 288, 0, 0, 214, # 0, 0, 155, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 222, 0, 0, # 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 45, 0, 0, 222, 0, 0, 286, 0, 0 ], #[ 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 222, 0, # 0, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 45, 0, 0, 222, 0, 0, 286, 0 ], #[ 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 222, # 0, 0, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 45, 0, 0, 222, 0, 0, 286 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 244, 0, 0, 129, 0, 0, 266, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 0, 0, 287, 0, 0, 25, 0, 0, 121 ], #[ 0, 0, 167, 0, 0, 80, 0, 0, 156, 0, 0, 168, 0, 0, 212, 0, 0, 240, 0, 0, 1, 0, # 0, 132, 0, 0, 84, 0, 0, 46, 0, 0, 3, 0, 0, 107 ], #[ 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 160, 0, # 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 45, 0, 0, 160, 0, 0, 23, 0, 0 ], #[ 0, 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 160, # 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 132, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 45, 0, 0, 160, 0, 0, 23, 0 ] #]; ###### # # Now we conjugate these matrices to change to a vertex of second # type. # ##### ConjMtx17m:=ll17mModM( 17*[[1,0,0,],[0,1/Vl17m,4/Vl17m],[0,1,1]]*One(ll17m) ,17^11)/17; ConjMtx17mInv:=ll17mModM(17*ConjMtx17m^-1,17^11)/17; #Check that the inverse of ConjMtx17m is correct: IsZero(ll17mModM(ConjMtx17m*ConjMtx17mInv-One(ConjMtx17m),17^9)); ###### # # The ALTERNATIVE representation of~$\D$ as matrices # over~$\ell_{17-}$. # # We find the ALTERNATIVE matrices for $S$, $U$, $U'$, $V$, $W, # and~$\sigma$, all of which are approximations $\mod 17^{10}$. WDM17mA:=ll17mModM(17*ConjMtx17m*WDM17m*ConjMtx17mInv,17^11)/17; SDM17mA:=ll17mModM(ConjMtx17m*SDM17m*ConjMtx17mInv,17^10); UDM17mA:=ll17mModM(ConjMtx17m*UDM17m*ConjMtx17mInv,17^10); UPDM17mA:=ll17mModM(2/UDM17mA,17^10); VDM17mA:=ll17mModM(ConjMtx17m*VDM17m*ConjMtx17mInv,17^10); sigmaDM17mA:=ll17mModM(17*ConjMtx17m*sigmaDM17m*ConjMtx17mInv,17^11)/17; sigmaDM17mAInv:=ll17mModM(17*sigmaDM17mA^2/(Ul17m/Sl17m),17^11)/17; #Check that sigmaDM17mA and sigmaDM17mAInv are inverses IsZero(ll17mModM(sigmaDM17mA*sigmaDM17mAInv-One(sigmaDM17mA),17^9)); #Check that the ALTERNATIVE W, S, U, U', V, and $\sigma$ are roots of #the right polynomials. IsZero(ll17mModM(WzFcn(WDM17mA),17^9)); IsZero(ll17mModM(SzFcn(SDM17mA),17^9)); IsZero(ll17mModM(Uz17mFcn(UDM17mA),17^10)); IsZero(ll17mModM(Uz17mFcn(UPDM17mA),17^10)); IsZero(ll17mModM(Vz17mFcn(VDM17mA),17^10)); IsZero(ll17mModM(sigmazl17mFcn(sigmaDM17mA),17^9)); #Check that the ALTERNATIVE W, S, V, and U all commute IsZero(ll17mModM(WDM17mA*SDM17mA-SDM17mA*WDM17mA,17^9)); IsZero(ll17mModM(WDM17mA*VDM17mA-VDM17mA*WDM17mA,17^10)); IsZero(ll17mModM(WDM17mA*UDM17mA-UDM17mA*WDM17mA,17^9)); IsZero(ll17mModM(SDM17mA*VDM17mA-VDM17mA*SDM17mA,17^10)); IsZero(ll17mModM(SDM17mA*UDM17mA-UDM17mA*SDM17mA,17^10)); IsZero(ll17mModM(VDM17mA*UDM17mA-UDM17mA*VDM17mA,17^10)); #Check that the ALTERNATIVE $\sigma$ commutes with the ALTERNATIVE S, #V, and U IsZero(ll17mModM(sigmaDM17mA*SDM17mA-SDM17mA*sigmaDM17mA,17^9)); IsZero(ll17mModM(sigmaDM17mA*VDM17mA-VDM17mA*sigmaDM17mA,17^10)); IsZero(ll17mModM(sigmaDM17mA*UDM17mA-UDM17mA*sigmaDM17mA,17^9)); #Check that the ALTERNATIVE~$\sigma$ conjugates the ALTERNATIVE~$W$ #correctly IsZero(ll17mModM(sigmaDM17mA*WDM17mA - phiOnW(WDM17mA)*sigmaDM17mA,17^9)); IsZero(ll17mModM(sigmaDM17mA*phiOnW(WDM17mA) - phi2OnW(WDM17mA)*sigmaDM17mA,17^9)); IsZero(ll17mModM(sigmaDM17mA*phi2OnW(WDM17mA) - WDM17mA*sigmaDM17mA,17^9)); ##### # # In the ALTERNATIVE matrix representation of~$\D$ over~$\ell_{17-}$, # the matrix~$F$ so that # # \iota(A) = F^{-1} A^* F # # This is an approximation $\mod 17^{10}$. FDM17mA:=ll17mModM( 17*ll17mAdj(ConjMtx17mInv)*FDM17m*ConjMtx17mInv ,17^11)/17; FDM17mAInv:=ll17mModM(17*FDM17mA^-1,17^11)/17; #Check that FDM17mA and FDM17mAInv are inverses IsZero(ll17mModM(FDM17mA*FDM17mAInv-One(FDM17mA),17^10)); # Check that FDM17mA is self-adjoint IsZero(ll17mModM(ll17mAdj(FDM17mA) - FDM17mA,17^10)); # Check that $\iota$ acts correctly on the ALTERNATIVE S, U, V, W, # $\sigma$, and $sigma^-1$ IsZero(ll17mModM(ll17mAdj(SDM17mA)*FDM17mA - FDM17mA*SDM17mA,17^10)); IsZero(ll17mModM(ll17mAdj(UDM17mA)*FDM17mA - FDM17mA*UPDM17mA,17^10)); IsZero(ll17mModM(ll17mAdj(VDM17mA)*FDM17mA - -FDM17mA*VDM17mA,17^10)); IsZero(ll17mModM(ll17mAdj(WDM17mA)*FDM17mA - FDM17mA*WDM17mA,17^9)); IsZero(ll17mModM(ll17mAdj(sigmaDM17mA)*FDM17mA - FDM17mA* (-Sl17m*One(WDM17mA)+(-3+Sl17m)*WDM17mA+(1-Sl17m)*WDM17mA^2) *sigmaDM17mAInv ,17^9)); IsZero(ll17mModM(ll17mAdj(sigmaDM17mAInv)*FDM17mA - FDM17mA* ((1-2*Sl17m)*One(WDM17mA)+(1+2*Sl17m)*WDM17mA+(1+Sl17m)*WDM17mA^2) *sigmaDM17mA ,17^9)); # Check the degree of integrality of FDM17mA, the matrix of the # new sesquilinear form, and of its inverse. Note that _Vl17m_ # $=\sqrt{-5+2\sqrt{2}}$ is a uniformizer for $\ell_{17-}$. #Check that FDM17mA is integral $\mod 17-$. IsZero(ll17mModM(17*FDM17mA,17)); #Check that Vl17m*FDM17mA^-1 is integral $\mod 17-$ IsZero(ll17mModM(17*Vl17m*FDM17mA^-1,17)); #Check that Determinant(FDM17mA) has the same valuation as~$17$ IsIntegralCyclotomic(Determinant(FDM17mA)/17); not IsIntegralCyclotomic(Determinant(FDM17mA)/(17*Vl17m)); # Since FDM17mA is integral, $L_0 \subseteq L_0^*$ # Since FDM17mA^-1*Vl17m is integral, $_Vl17m_ L_0^* \subseteq L_0$. # Since Determinant(FDM17mA) has the same $17-$-adic valuation # as~$17$, $[L_0^*:L_0] = 289$ # The preceding checks show that the standard $\ell_{17-}$ lattice # $L_0=\O_{\ell_{17-}}^3$ and its dual lattice relative to the # form~$F$ are neighbors in the $\tilde A_2$~building of # $PGL_3(\ell_{17-})$. Note that # $\O_{\ell_{17-}=\ZZ_{17}[U]/(U^2-(1+\sqrt{2})U+2)$. # Consequently matrices which preserve that lattice, which is to say # matrices in $GL(3,\ZZ_{17}[U]$, fix a vertex of Type~2 in the # $17-$-adic tree corresponding to the $17-$-adic reduction of # $PU(\D,\iota)$. # 36-element basis of $\D$ over $\QQ$ basisDM17mA:=ListX([0..1],[0..1],[0..2], [sigmaDM17mAInv,sigmaDM17mA^0,sigmaDM17mA^1], function(i,j,k,sigmaDM17mATol) return Sl17m^i*Ul17m^j*WDM17mA^k*sigmaDM17mATol; end );; # For a $3\times 3$ matrix with coefficients in # $\O_{\ell_{17-}}=\ZZ_{17}[U]$, for each of the 9~entries this # function calculate the four coefficients, each modulo~$q$, and puts # them together as a 18~element vector. CondCoeff17m:=function(mtx,q) return Concatenation(ListX([1..3],[1..3], function(j,k) local c; c:=ll17mCoeff(mtx[j][k]); Apply(c,x->x mod q); return c; end) ); end; # The element # # \sum c_{ijkl} \sqrt{2}^i U^j W^k \sigma^l # # in~$\D$, when reduced $\mod 17-$, will have (approximate) 17-adic # matrix representation # # Sum([1..36],ix->c[ix]*basis[ix]) # # and this $17$-adic matrix representation will be $17$-adically # integral if and only if the vector: # # CondMtxDM17mType2 * c # # consists of 17-adic integers. CondMtxDM17mType2:=TransposedMat( List(basisDM17mA,mtx->CondCoeff17m(17*mtx,17*17^2)/17));; #CondMtxDM17mType2:=[ #[ 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 0, 0, 196, # 0, 0, 250, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 75, 0, 0, 134, 0, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 45, 0, 0, 196, 0, 0, 250, 0 ], #[ 55, 0, 208, 79, 0, 283, 145, 0, 128, 177, 0, 194, 270, 0, 100, 20, 0, 275, # 163, 0, 112, 87, 0, 19, 167, 0, 269, 162, 0, 60, 12, 0, 165, 33, 0, 237 ], #[ 56, 0, 192, 154, 0, 239, 279, 0, 7, 30, 0, 81, 227, 0, 6, 263, 0, 161, 208, # 0, 259, 283, 0, 62, 128, 0, 26, 194, 0, 177, 100, 0, 270, 275, 0, 20 ], #[ 233, 0, 194, 135, 0, 100, 10, 0, 275, 274, 0, 0, 31, 0, 0, 13, 0, 0, 81, 0, # 60, 6, 0, 165, 161, 0, 237, 192, 0, 0, 239, 0, 0, 7, 0, 0 ], #[ 152, 0, 0, 129, 0, 0, 138, 0, 0, 0, 0, 194, 0, 0, 100, 0, 0, 275, 193, 0, 0, # 25, 0, 0, 141, 0, 0, 0, 0, 60, 0, 0, 165, 0, 0, 237 ], #[ 4370/17, 0, 387/17, 269/17, 0, 3399/17, 672/17, 0, 718/17, 91/17, 0, # 2832/17, 1104/17, 0, 4193/17, 1516/17, 0, 2931/17, 708/17, 0, 364/17, # 545/17, 0, 74/17, 1051/17, 0, 231/17, 2650/17, 0, 569/17, 4885/17, 0, # 3436/17, 1172/17, 0, 3290/17 ], #[ 2411/17, 0, 3497/17, 4361/17, 0, 360/17, 4155/17, 0, 991/17, 543/17, 0, # 1143/17, 4644/17, 0, 2041/17, 4241/17, 0, 64/17, 3588/17, 0, 2172/17, # 14/17, 0, 3195/17, 4327/17, 0, 3268/17, 4205/17, 0, 4617/17, 4368/17, 0, # 3989/17, 3862/17, 0, 568/17 ], #[ 95, 1, 193, 113, 245, 220, 177, 279, 122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 229, # 45, 15, 172, 43, 74, 162, 128, 288, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 95, 1, 193, 113, 245, 220, 177, 279, 122, 0, 0, # 0, 0, 0, 0, 0, 0, 0, 229, 45, 15, 172, 43, 74, 162, 128, 288 ], #[ 2017/17, 0, 181/17, 653/17, 4699/17, 1548/17, 3831/17, 3623/17, 4689/17, # 2123/17, 0, 3245/17, 3374/17, 3085/17, 4545/17, 308/17, 3582/17, 2770/17, # 3776/17, 0, 4677/17, 3953/17, 4242/17, 1456/17, 3908/17, 2062/17, 2925/17, # 1032/17, 0, 2392/17, 4151/17, 972/17, 1647/17, 277/17, 1084/17, 1247/17 ], #[ 1395/17, 0, 834/17, 3226/17, 914/17, 184/17, 4759/17, 3122/17, 3528/17, # 2896/17, 0, 4732/17, 4260/17, 214/17, 3365/17, 1082/17, 1290/17, 224/17, # 4397/17, 0, 3717/17, 381/17, 4427/17, 1633/17, 2318/17, 4371/17, 1833/17, # 1137/17, 0, 236/17, 960/17, 671/17, 3457/17, 1005/17, 2851/17, 1988/17 ], #[ 1, 0, 0, 82, 0, 0, 77, 0, 0, 0, 0, 244, 0, 0, 129, 0, 0, 266, 45, 0, 0, 222, # 0, 0, 286, 0, 0, 0, 0, 287, 0, 0, 25, 0, 0, 121 ], #[ 0, 0, 167, 0, 0, 80, 0, 0, 156, 1, 0, 168, 82, 0, 212, 77, 0, 240, 0, 0, 1, # 0, 0, 132, 0, 0, 84, 45, 0, 46, 222, 0, 3, 286, 0, 107 ], #[ 208, 0, 81, 1, 4, 284, 132, 278, 168, 194, 0, 95, 176, 126, 276, 112, 87, # 90, 112, 0, 177, 45, 180, 64, 160, 83, 46, 60, 0, 229, 117, 179, 282, 127, # 158, 4 ], #[ 192, 0, 97, 201, 226, 151, 233, 101, 244, 81, 0, 208, 288, 285, 5, 157, 11, # 121, 259, 0, 30, 86, 55, 148, 81, 210, 287, 177, 0, 112, 244, 109, 225, # 129, 206, 243 ], #[ 194, 1, 96, 176, 258, 69, 112, 171, 167, 0, 0, 0, 0, 0, 0, 0, 0, 0, 60, 45, # 274, 117, 50, 215, 127, 181, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 194, 1, 96, 176, 258, 69, 112, 171, 167, 0, 0, 0, # 0, 0, 0, 0, 0, 0, 60, 45, 274, 117, 50, 215, 127, 181, 1 ] #]; #17*CondMtxDM17mType2:=[ #[ 0, 17, 0, 0, 1275, 0, 0, 2278, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 765, 0, 0, # 3332, 0, 0, 4250, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 1275, 0, 0, 2278, 0, 0, 0, 0, 0, 0, # 0, 0, 0, 0, 0, 765, 0, 0, 3332, 0, 0, 4250, 0 ], #[ 935, 0, 3536, 1343, 0, 4811, 2465, 0, 2176, 3009, 0, 3298, 4590, 0, 1700, # 340, 0, 4675, 2771, 0, 1904, 1479, 0, 323, 2839, 0, 4573, 2754, 0, 1020, # 204, 0, 2805, 561, 0, 4029 ], #[ 952, 0, 3264, 2618, 0, 4063, 4743, 0, 119, 510, 0, 1377, 3859, 0, 102, 4471, # 0, 2737, 3536, 0, 4403, 4811, 0, 1054, 2176, 0, 442, 3298, 0, 3009, 1700, # 0, 4590, 4675, 0, 340 ], #[ 3961, 0, 3298, 2295, 0, 1700, 170, 0, 4675, 4658, 0, 0, 527, 0, 0, 221, 0, # 0, 1377, 0, 1020, 102, 0, 2805, 2737, 0, 4029, 3264, 0, 0, 4063, 0, 0, # 119, 0, 0 ], #[ 2584, 0, 0, 2193, 0, 0, 2346, 0, 0, 0, 0, 3298, 0, 0, 1700, 0, 0, 4675, # 3281, 0, 0, 425, 0, 0, 2397, 0, 0, 0, 0, 1020, 0, 0, 2805, 0, 0, 4029 ], #[ 4370, 0, 387, 269, 0, 3399, 672, 0, 718, 91, 0, 2832, 1104, 0, 4193, 1516, # 0, 2931, 708, 0, 364, 545, 0, 74, 1051, 0, 231, 2650, 0, 569, 4885, 0, # 3436, 1172, 0, 3290 ], #[ 2411, 0, 3497, 4361, 0, 360, 4155, 0, 991, 543, 0, 1143, 4644, 0, 2041, # 4241, 0, 64, 3588, 0, 2172, 14, 0, 3195, 4327, 0, 3268, 4205, 0, 4617, # 4368, 0, 3989, 3862, 0, 568 ], #[ 1615, 17, 3281, 1921, 4165, 3740, 3009, 4743, 2074, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 3893, 765, 255, 2924, 731, 1258, 2754, 2176, 4896, 0, 0, 0, 0, 0, 0, 0, # 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1615, 17, 3281, 1921, 4165, 3740, 3009, 4743, # 2074, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3893, 765, 255, 2924, 731, 1258, 2754, # 2176, 4896 ], #[ 2017, 0, 181, 653, 4699, 1548, 3831, 3623, 4689, 2123, 0, 3245, 3374, 3085, # 4545, 308, 3582, 2770, 3776, 0, 4677, 3953, 4242, 1456, 3908, 2062, 2925, # 1032, 0, 2392, 4151, 972, 1647, 277, 1084, 1247 ], #[ 1395, 0, 834, 3226, 914, 184, 4759, 3122, 3528, 2896, 0, 4732, 4260, 214, # 3365, 1082, 1290, 224, 4397, 0, 3717, 381, 4427, 1633, 2318, 4371, 1833, # 1137, 0, 236, 960, 671, 3457, 1005, 2851, 1988 ], #[ 17, 0, 0, 1394, 0, 0, 1309, 0, 0, 0, 0, 4148, 0, 0, 2193, 0, 0, 4522, 765, # 0, 0, 3774, 0, 0, 4862, 0, 0, 0, 0, 4879, 0, 0, 425, 0, 0, 2057 ], #[ 0, 0, 2839, 0, 0, 1360, 0, 0, 2652, 17, 0, 2856, 1394, 0, 3604, 1309, 0, # 4080, 0, 0, 17, 0, 0, 2244, 0, 0, 1428, 765, 0, 782, 3774, 0, 51, 4862, 0, # 1819 ], [ 3536, 0, 1377, 17, 68, 4828, 2244, 4726, 2856, 3298, 0, 1615, # 2992, 2142, 4692, 1904, 1479, 1530, 1904, 0, 3009, 765, 3060, 1088, 2720, # 1411, 782, 1020, 0, 3893, 1989, 3043, 4794, 2159, 2686, 68 ], #[ 3264, 0, 1649, 3417, 3842, 2567, 3961, 1717, 4148, 1377, 0, 3536, 4896, # 4845, 85, 2669, 187, 2057, 4403, 0, 510, 1462, 935, 2516, 1377, 3570, # 4879, 3009, 0, 1904, 4148, 1853, 3825, 2193, 3502, 4131 ], #[ 3298, 17, 1632, 2992, 4386, 1173, 1904, 2907, 2839, 0, 0, 0, 0, 0, 0, 0, 0, # 0, 1020, 765, 4658, 1989, 850, 3655, 2159, 3077, 17, 0, 0, 0, 0, 0, 0, 0, # 0, 0 ], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 3298, 17, 1632, 2992, 4386, 1173, 1904, 2907, # 2839, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1020, 765, 4658, 1989, 850, 3655, 2159, # 3077, 17 ] #];