// PPM FLATTENERS (final formula)
FTYPE Ficalc(int dir, FTYPE *V, FTYPE *P, FTYPE **ypl)
{
FTYPE ftilde( int dir, int shift, FTYPE *P, FTYPE *V, FTYPE **ypl );
int signdP;
FTYPE Fi;
signdP = (P[1] - P[-1] > 0) * 2 - 1;
// FLASH Equation 48
Fi = max( ftilde(dir, 0, V,P,ypl), ftilde(dir, -signdP, V,P,ypl) );
return(Fi);
}
GEN
bnflogdegree(GEN nf, GEN A, GEN ell)
{
pari_sp av = avma;
GEN AZ, A0Z, NA0;
long vAZ;
if (typ(ell) != t_INT) pari_err_TYPE("bnflogdegree", ell);
nf = checknf(nf);
A = idealhnf(nf, A);
AZ = gcoeff(A,1,1);
vAZ = Z_pvalrem(AZ, ell, &A0Z);
if (is_pm1(A0Z))
NA0 = gen_1;
else
(void)Z_pvalrem(idealnorm(nf,A), ell, &NA0);
if (vAZ)
{
GEN Aell = ZM_hnfmodid(A, powiu(ell,vAZ));
GEN S = idealprimedec(nf, ell), T;
long l, i, s = 0;
T = padicfact(nf, S, 100);
l = lg(S);
for (i = 1; i < l; i++)
{
GEN P = gel(S,i);
long v = idealval(nf, Aell, P);
if (v) s += v * ftilde(nf, P, gel(T,i));
}
if (s) NA0 = gmul(NA0, gpowgs(ell1(ell), s));
}
return gerepileupto(av, NA0);
}
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));
}
