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)); }