GEN
nffactormod(GEN nf, GEN x, GEN pr)
{
long j, l, vx = varn(x), vn;
pari_sp av = avma;
GEN F, E, rep, xrd, modpr, T, p;
nf = checknf(nf);
vn = varn(nf[1]);
if (typ(x)!=t_POL) pari_err(typeer,"nffactormod");
if (varncmp(vx,vn) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nffactormod");
modpr = nf_to_ff_init(nf, &pr, &T, &p);
xrd = modprX(x, nf, modpr);
rep = FqX_factor(xrd,T,p);
settyp(rep, t_MAT);
F = gel(rep,1); l = lg(F);
E = gel(rep,2); settyp(E, t_COL);
for (j = 1; j < l; j++) {
gel(F,j) = modprX_lift(gel(F,j), modpr);
gel(E,j) = stoi(E[j]);
}
return gerepilecopy(av, rep);
}
/* K a bnf, compute Cl'(K) = ell-Sylow of Cl(K) / (places above ell).
* Return [D, u, R0, U0, ordS]
* - D: cyclic factors for Cl'(K)
* - u: generators of cyclic factors (all coprime to ell)
* - R0: subgroup isprincipal(<S>) (divides K.cyc)
* - U0: generators of R0 are of the form S . U0
* - ordS[i] = order of S[i] in CL(K) */
static GEN
CL_prime(GEN K, GEN ell, GEN Sell)
{
GEN g, ordS, R0, U0, U, D, u, cyc = bnf_get_cyc(K);
long i, l, lD, lS = lg(Sell);
g = leafcopy(bnf_get_gen(K));
l = lg(g);
for (i = 1; i < l; i++)
{
GEN A = gel(g,i), a = gcoeff(A,1,1);
long v = Z_pvalrem(a, ell, &a);
if (v) gel(g,i) = hnfmodid(A, a); /* make coprime to ell */
}
R0 = cgetg(lS, t_MAT);
ordS = cgetg(lS, t_VEC);
for (i = 1; i < lS; i++)
{
gel(R0,i) = isprincipal(K, gel(Sell,i));
gel(ordS,i) = charorder(cyc, gel(R0,i)); /* order of Sell[i] */
}
R0 = shallowconcat(R0, diagonal_shallow(cyc));
/* R0 = subgroup generated by S in Cl(K) [ divides diagonal(K.cyc) ]*/
R0 = ZM_hnfall(R0, &U0, 2); /* [S | cyc] * U0 = R0 in HNF */
D = ZM_snfall(R0, &U,NULL);
D = RgM_diagonal_shallow(D);
lD = lg(D);
u = ZM_inv(U, gen_1); settyp(u, t_VEC);
for (i = 1; i < lD; i++) gel(u,i) = idealfactorback(K,g,gel(u,i),1);
setlg(U0, l);
U0 = rowslice(U0,1,lS-1); /* restrict to 'S' part */
return mkvec5(D, u, R0, U0, ordS);
}
GEN
gtrans(GEN x)
{
long i, dx, lx;
GEN y;
switch(typ(x))
{
case t_VEC: y = gcopy(x); settyp(y,t_COL); break;
case t_COL: y = gcopy(x); settyp(y,t_VEC); break;
case t_MAT:
lx = lg(x); if (lx==1) return cgetg(1,t_MAT);
dx = lgcols(x); y = cgetg(dx,t_MAT);
for (i = 1; i < dx; i++) gel(y,i) = row_transposecopy(x,i);
break;
default: pari_err_TYPE("gtrans",x); return NULL;
}
return y;
}
static GEN
bnflog_i(GEN bnf, GEN ell)
{
long prec0, prec;
GEN nf, US, vdegS, S, T, M, CLp, CLt, Ftilde, vtG, ellk;
GEN D, Ap, cycAp, bnfS;
long i, j, lS, lvAp;
checkbnf(bnf);
nf = checknf(bnf);
S = idealprimedec(nf, ell);
bnfS = bnfsunit0(bnf, S, nf_GENMAT, LOWDEFAULTPREC); /* S-units */
US = leafcopy(gel(bnfS,1));
prec0 = maxss(30, vtilde_prec(nf, US, ell));
US = shallowconcat(bnf_get_fu(bnf), US);
settyp(US, t_COL);
T = padicfact(nf, S, prec0);
lS = lg(S); Ftilde = cgetg(lS, t_VECSMALL);
for (j = 1; j < lS; j++) Ftilde[j] = ftilde(nf, gel(S,j), gel(T,j));
CLp = CL_prime(bnf, ell, S);
cycAp = gel(CLp,1);
Ap = gel(CLp,2);
for(;;)
{
CLt = CL_tilde(nf, US, ell, T, Ftilde, &vtG, prec0);
if (CLt) break;
prec0 <<= 1;
T = padicfact(nf, S, prec0);
}
prec = ellexpo(cycAp, ell) + ellexpo(CLt,ell) + 1;
if (prec == 1) return mkvec3(cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC));
vdegS = get_vdegS(Ftilde, ell, prec0);
ellk = powiu(ell, prec);
lvAp = lg(Ap);
if (lvAp > 1)
{
GEN Kcyc = bnf_get_cyc(bnf);
GEN C = zeromatcopy(lvAp-1, lS-1);
GEN Rell = gel(CLp,3), Uell = gel(CLp,4), ordS = gel(CLp,5);
for (i = 1; i < lvAp; i++)
{
GEN a, b, bi, A = gel(Ap,i), d = gel(cycAp,i);
bi = isprincipal(bnf, A);
a = vecmodii(ZC_Z_mul(bi,d), Kcyc);
/* a in subgroup generated by S = Rell; hence b integral */
b = hnf_invimage(Rell, a);
b = vecmodii(ZM_ZC_mul(Uell, ZC_neg(b)), ordS);
A = mkvec2(A, cgetg(1,t_MAT));
A = idealpowred(nf, A, d);
/* find a principal representative of A_i^cycA_i up to elements of S */
a = isprincipalfact(bnf,gel(A,1),S,b,nf_GENMAT|nf_FORCE);
if (!gequal0(gel(a,1))) pari_err_BUG("bnflog");
a = famat_mul_shallow(gel(A,2), gel(a,2)); /* principal part */
if (lg(a) == 1) continue;
for (j = 1; j < lS; j++)
gcoeff(C,i,j) = vtilde(nf, a, gel(T,j), gel(vdegS,j), ell, prec0);
}
C = gmod(gneg(C),ellk);
C = shallowtrans(C);
M = mkmat2(mkcol2(diagonal_shallow(cycAp), C), mkcol2(gen_0, vtG));
M = shallowmatconcat(M); /* relation matrix */
}
else
M = vtG;
M = ZM_hnfmodid(M, ellk);
D = matsnf0(M, 4);
if (lg(D) == 1 || !dvdii(gel(D,1), ellk))
pari_err_BUG("bnflog [missing Z_l component]");
D = vecslice(D,2,lg(D)-1);
return mkvec3(D, CLt, ellsylow(cycAp, ell));
}
