alghasse

default(realprecision,38);
default(parisize,"100M");

\\Tests related to hasse invariants in algebras

do(name, test) = {
 setrand(1);
 print(name,": ", iferr(test(), E, E));
}
gusuite(name, tests) = print("Suite: ", name); tests();

searchin(hf,pr,h) =
{
  my(i,n);
  for(i=1,#hf[1],
    if(hf[1][i]==pr, return(hf[2][i]==h))
  );
  return(h==0);
};

samehasse(hf1,hi1,hf2,hi2) =
{
  my(i,n);
  if(hi1 != hi2, return(0));

  n = #hf1[1];
  for(i=1,n,
    if(!searchin(hf2,hf1[1][i],hf1[2][i]), return(0));
  );

  n = #hf2[1];
  for(i=1,n,
    if(!searchin(hf1,hf2[1][i],hf2[2][i]), return(0));
  );

  return(1);
};

hassetriv(hf,hi) = samehasse(hf,hi,[[],Vecsmall([])],Vecsmall(vector(#hi,i,0)));
altriv(al) = hassetriv(alghassef(al),alghassei(al));
alsame(al,hf,hi) = samehasse(alghassef(al),alghassei(al),hf,hi);

alcheckhasse(al) =
{
  my(n,h);
  n = algdegree(al);
  h = Mod(0,n);
  for(i=1, #alghassef(al)[1],
      h += alghassef(al)[2][i]);
  for(i=1, #alghassei(al),
      h += (n/2)*alghassei(al)[i]);
  return(h == Mod(0,n));
};

al_cyclotomic(p,b,mo=1) =
{
  my(Q,F,pc,r);
  Q = nfinit(y);
  pc = polcyclo(p,x);
  F = rnfinit(Q,pc);
  r = lift(znprimroot(p));
  return(alginit(F, [Mod(Mod(1,y)*x^r,pc), b], 'x, mo));
};

hasse0() = gusuite("hasse sum to 0", ()->{
  do("cyclo construction", ()->al_cyclotomic(3,-175624635));
  do("cyclo ramified at infinity", ()->alcheckhasse(al_cyclotomic(3,-175624635)));
  do("cyclo unramified at infinity", ()->alcheckhasse(al_cyclotomic(3,2763764)));
  do("cyclo 5", ()->alcheckhasse(al_cyclotomic(5,7861623)));
  do("cyclo 5 bis", ()->alcheckhasse(al_cyclotomic(5,6569846846546548798*25)));
  do("cyclo 7 bis no mo", ()->alcheckhasse(al_cyclotomic(7,168656465154165487*7^3,0)));
  do("cyclo 11 no mo", ()->alcheckhasse(al_cyclotomic(11,87165765,0)));
  do("quat -1,-1 over Q", ()->alcheckhasse(alginit(nfinit(y),[-1,-1])));
  do("quat -1,-1 over Q(sqrt(2))", ()->alcheckhasse(alginit(nfinit(y^2-2),[-1,-1],x)));
  do("quat -1,-1 over Q(sqrt(60))", ()->alcheckhasse(alginit(nfinit(y^2-60),[-1,-1],x)));
});

alfromhasse() = gusuite("algebra from Hasse invariants", ()->{
  my(nf, pr7, pr13, finvm, finv, finv1, finv2, iinvm, iinv, iinv1, iinv2, d1,\
    d2, d, dm, mai, al);
  nf = nfinit(y^3+y^2-2*y-1);
  pr7 = idealprimedec(nf,7);
  pr13 = idealprimedec(nf,13);
  J = varhigher("J");

  al = alginit(nf,3,J);
  do("matrix algebra invariants", ()->altriv(al));

  d1 = 3;
  finv1 = [pr13,Vecsmall([1,1,1])];
  iinv1 = Vecsmall([0,0,0]);
  al = alginit(nf,[d1,finv1,iinv1],J);
  do("algebra 1 invariants", ()->alsame(al,finv1,iinv1));

  d2 = 2;
  finv2 = [pr7,Vecsmall([1])];
  iinv2 = Vecsmall([1,0,0]);
  al = alginit(nf,[d2,finv2,iinv2],J);
  do("algebra 2 invariants", ()->alsame(al,finv2,iinv2));

  nf = nfinit(y);
  p13 = idealprimedec(nf,13)[1];
  finv1 = [[p13],[Mod(1,2)]];
  iinv1 = Vecsmall([1]);
  do("test", ()->alginit(nf,[2,finv1,iinv1],J));

  d = 6;
  p3 = idealprimedec(nf,3)[1];
  p5 = idealprimedec(nf,5)[1];
  p7 = idealprimedec(nf,7)[1];
  p11 = idealprimedec(nf,11)[1];
  finv = [[p3,p5,p7,p11], Vecsmall([3,2,2,2])];
  iinv = Vecsmall([3]);
  al = alginit(nf,[d,finv,iinv],J,0);
  do("degree 6 algebra over Q", ()->alsame(al,finv,iinv));

  nf = nfinit(y^2-5);
  finv1 = [[],[]];
  iinv1 = Vecsmall([1,1]);
  do("trivial finite conditions", ()->my(al=alginit(nf,[2,finv1,iinv1]));
    [alghassei(al),alghassef(al)]);

});

all() = gusuite("all", ()->{
  hasse0();
  alfromhasse();
});

all();

\\better accessors for hasse invariants
print("better accessors");
setrand(1);
x='x;
nf = nfinit(y^3-y+1);
rnf = rnfinit(nf, polcyclo(5,x));
al = alginit(rnf, [x^2,-2],, 0);
alghasse(al,1)
alghasse(al,2)
alghasse(al,idealprimedec(nf,2)[1])
alghasse(al,idealprimedec(nf,3)[1])
alghasse(al,idealprimedec(nf,5)[1])
alghasse(al,idealprimedec(nf,5)[2])
algindex(al,1)
algindex(al,2)
algindex(al,idealprimedec(nf,2)[1])
algindex(al,idealprimedec(nf,3)[1])
algindex(al,idealprimedec(nf,5)[1])
algindex(al,idealprimedec(nf,5)[2])
algindex(al)
algisdivision(al,1)
algisdivision(al,2)
algisdivision(al,idealprimedec(nf,2)[1])
algisdivision(al,idealprimedec(nf,3)[1])
algisdivision(al,idealprimedec(nf,5)[1])
algisdivision(al,idealprimedec(nf,5)[2])
algisdivision(al)
algissplit(al,1)
algissplit(al,2)
algissplit(al,idealprimedec(nf,2)[1])
algissplit(al,idealprimedec(nf,3)[1])
algissplit(al,idealprimedec(nf,5)[1])
algissplit(al,idealprimedec(nf,5)[2])
algissplit(al)
algisramified(al,1)
algisramified(al,2)
algisramified(al,idealprimedec(nf,2)[1])
algisramified(al,idealprimedec(nf,3)[1])
algisramified(al,idealprimedec(nf,5)[1])
algisramified(al,idealprimedec(nf,5)[2])
algisramified(al)
algramifiedplaces(al)

print(" ");
al = 0;
al = alginit(rnf, [x^2,-1],, 0);
alghasse(al,1)
alghasse(al,2)
alghasse(al,idealprimedec(nf,2)[1])
alghasse(al,idealprimedec(nf,5)[1])
alghasse(al,idealprimedec(nf,5)[2])
algindex(al,1)
algindex(al,2)
algindex(al,idealprimedec(nf,2)[1])
algindex(al,idealprimedec(nf,5)[1])
algindex(al,idealprimedec(nf,5)[2])
algindex(al)
algisdivision(al,1)
algisdivision(al,2)
algisdivision(al,idealprimedec(nf,2)[1])
algisdivision(al,idealprimedec(nf,5)[1])
algisdivision(al,idealprimedec(nf,5)[2])
algisdivision(al)
algissplit(al,1)
algissplit(al,2)
algissplit(al,idealprimedec(nf,2)[1])
algissplit(al,idealprimedec(nf,5)[1])
algissplit(al,idealprimedec(nf,5)[2])
algissplit(al)
algisramified(al,1)
algisramified(al,2)
algisramified(al,idealprimedec(nf,2)[1])
algisramified(al,idealprimedec(nf,5)[1])
algisramified(al,idealprimedec(nf,5)[2])
algisramified(al)
algramifiedplaces(al)

print(" ");
al = 0;
al = alginit(rnf, [x^2, 1],, 0);
alghasse(al,1)
alghasse(al,2)
alghasse(al,idealprimedec(nf,2)[1])
alghasse(al,idealprimedec(nf,5)[2])
algindex(al,1)
algindex(al,2)
algindex(al,idealprimedec(nf,2)[1])
algindex(al,idealprimedec(nf,5)[2])
algindex(al)
algisdivision(al,1)
algisdivision(al,2)
algisdivision(al,idealprimedec(nf,2)[1])
algisdivision(al,idealprimedec(nf,5)[2])
algisdivision(al)
algissplit(al,1)
algissplit(al,2)
algissplit(al,idealprimedec(nf,2)[1])
algissplit(al,idealprimedec(nf,5)[2])
algissplit(al)
algisramified(al,1)
algisramified(al,2)
algisramified(al,idealprimedec(nf,2)[1])
algisramified(al,idealprimedec(nf,5)[2])
algisramified(al)
algramifiedplaces(al)

print("Hasse inv 0 bug");
setrand(1); a='a;
K=nfinit(a);PR=idealprimedec(K,2);A=alginit(K,[3,[PR,[0]],[0]],,0);
algdegree(A)
algdim(A)
algindex(A)
algisdivision(A)
algadd(A,[1,0,0,0,0,0,0,0,0]~,[1,2,3]~)
algsub(A,[1,0,0,0,0,0,0,0,0]~,[1,2,3]~)
algmul(A,[0,0,0,0,0,0,0,0,0]~,[1,2,3]~)

print("\ntests with splitting field that does not descend");
setrand(1);
nf = nfinit(y^2-5);
al = alginit(nf,[y,-1]);
algramifiedplaces(al)
al = alginit(nf,[-3+y,-1]);
algramifiedplaces(al)
al = alginit(nf,[-3+y,y]);
algramifiedplaces(al)
nf = nfinit(y^4 - 10*y^2 + 1);
p3 = idealprimedec(nf,3)[1];
p5 = idealprimedec(nf,5)[1];
al = alginit(nf, [2, [[p3,p5],[1/2,1/2]], [1/2,0,0,1/2]]);
algramifiedplaces(al)
al = alginit(nf, [2, [[p3],[1/2]], [0,1/2,0,0]]);
algramifiedplaces(al)
nf = nfinit(y^8 - 40*y^6 + 352*y^4 - 960*y^2 + 576);
p3 = idealprimedec(nf,3)[1];
al = alginit(nf, [2,[[p3],[1/2]], [1/2,0,0,1/2,1/2,0,0,0]]);
algramifiedplaces(al)

print("degree bug");
nf = nfinit(y); p2 = idealprimedec(nf,2)[1]; p3 = idealprimedec(nf,3)[1];
al = alginit(nf, [4,[[p2,p3],[1/2,1/2]],[0]]); algdegree(al)
{for(i=1,10,
  al = alginit(nf, [2,[[p2,p3],[0,0]],[0]]);
  print1(algdegree(al)," ");
)};
al = alginit(nf, [4,[[p2,p3],[0,0]],[0]]); algdegree(al)
nf = nfinit(z^2+1);
p31 = idealprimedec(nf,31)[1];
p2 = idealprimedec(nf,2)[1];
{for(i=1,10,
   al = alginit(nf, [2, [[p31,p2], [0, 0]], []]); print1(algdegree(al)," ");
)};