Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /*******************************************************************/
16 : /* */
17 : /* COMPUTATION OF STARK UNITS OF TOTALLY REAL FIELDS */
18 : /* */
19 : /*******************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_stark
24 :
25 : /* ComputeCoeff */
26 : typedef struct {
27 : GEN L0, L1, L11, L2; /* VECSMALL of p */
28 : GEN L1ray, L11ray; /* precomputed isprincipalray(pr), pr | p */
29 : GEN rayZ; /* precomputed isprincipalray(i), i < condZ */
30 : long condZ; /* generates cond(bnr) \cap Z, assumed small */
31 : } LISTray;
32 :
33 : /* Char evaluation */
34 : typedef struct {
35 : long ord;
36 : GEN *val, chi;
37 : } CHI_t;
38 :
39 : /* RecCoeff */
40 : typedef struct {
41 : GEN M, beta, B, U, nB;
42 : long v, G, N;
43 : } RC_data;
44 :
45 : /********************************************************************/
46 : /* Miscellaneous functions */
47 : /********************************************************************/
48 : static GEN
49 19572 : chi_get_c(GEN chi) { return gmael(chi,1,2); }
50 : static long
51 56539 : chi_get_deg(GEN chi) { return itou(gmael(chi,1,1)); }
52 :
53 : /* Compute the image of logelt by character chi, zeta_ord(chi)^n; return n */
54 : static ulong
55 15344 : CharEval_n(GEN chi, GEN logelt)
56 : {
57 15344 : GEN gn = ZV_dotproduct(chi_get_c(chi), logelt);
58 15344 : return umodiu(gn, chi_get_deg(chi));
59 : }
60 : /* Compute the image of logelt by character chi, as a complex number */
61 : static GEN
62 15288 : CharEval(GEN chi, GEN logelt)
63 : {
64 15288 : ulong n = CharEval_n(chi, logelt), d = chi_get_deg(chi);
65 15288 : long nn = Fl_center(n,d,d>>1);
66 15288 : GEN x = gel(chi,2);
67 15288 : x = gpowgs(x, labs(nn));
68 15288 : if (nn < 0) x = conj_i(x);
69 15288 : return x;
70 : }
71 :
72 : /* return n such that C(elt) = z^n */
73 : static ulong
74 699164 : CHI_eval_n(CHI_t *C, GEN logelt)
75 : {
76 699164 : GEN n = ZV_dotproduct(C->chi, logelt);
77 699164 : return umodiu(n, C->ord);
78 : }
79 : /* return C(elt) */
80 : static GEN
81 697351 : CHI_eval(CHI_t *C, GEN logelt)
82 : {
83 697351 : return C->val[CHI_eval_n(C, logelt)];
84 : }
85 :
86 : static void
87 4228 : init_CHI(CHI_t *c, GEN CHI, GEN z)
88 : {
89 4228 : long i, d = chi_get_deg(CHI);
90 4228 : GEN *v = (GEN*)new_chunk(d);
91 4228 : v[0] = gen_1;
92 4228 : if (d != 1)
93 : {
94 4228 : v[1] = z;
95 36617 : for (i=2; i<d; i++) v[i] = gmul(v[i-1], z);
96 : }
97 4228 : c->chi = chi_get_c(CHI);
98 4228 : c->ord = d;
99 4228 : c->val = v;
100 4228 : }
101 : /* as t_POLMOD */
102 : static void
103 2597 : init_CHI_alg(CHI_t *c, GEN CHI) {
104 2597 : long d = chi_get_deg(CHI);
105 : GEN z;
106 2597 : switch(d)
107 : {
108 0 : case 1: z = gen_1; break;
109 973 : case 2: z = gen_m1; break;
110 1624 : default: z = mkpolmod(pol_x(0), polcyclo(d,0));
111 : }
112 2597 : init_CHI(c,CHI, z);
113 2597 : }
114 : /* as t_COMPLEX */
115 : static void
116 1631 : init_CHI_C(CHI_t *c, GEN CHI) {
117 1631 : init_CHI(c,CHI, gel(CHI,2));
118 1631 : }
119 :
120 : typedef struct {
121 : long r; /* rank = lg(gen) */
122 : GEN j; /* current elt is gen[1]^j[1] ... gen[r]^j[r] */
123 : GEN cyc; /* t_VECSMALL of elementary divisors */
124 : } GROUP_t;
125 :
126 : static int
127 693091 : NextElt(GROUP_t *G)
128 : {
129 693091 : long i = 1;
130 693091 : if (G->r == 0) return 0; /* no more elt */
131 764652 : while (++G->j[i] == G->cyc[i]) /* from 0 to cyc[i]-1 */
132 : {
133 72373 : G->j[i] = 0;
134 72373 : if (++i > G->r) return 0; /* no more elt */
135 : }
136 692279 : return i; /* we have multiplied by gen[i] */
137 : }
138 :
139 : /* enumerate all group elements; trivial elt comes last */
140 : GEN
141 42364 : cyc2elts(GEN cyc)
142 : {
143 : long i, n;
144 : GEN z;
145 : GROUP_t G;
146 :
147 42364 : G.cyc = typ(cyc)==t_VECSMALL? cyc: gtovecsmall(cyc);
148 42364 : n = zv_prod(G.cyc);
149 42364 : G.r = lg(cyc)-1;
150 42364 : G.j = zero_zv(G.r);
151 :
152 42364 : z = cgetg(n+1, t_VEC);
153 42364 : gel(z,n) = leafcopy(G.j); /* trivial elt comes last */
154 672322 : for (i = 1; i < n; i++)
155 : {
156 629958 : (void)NextElt(&G);
157 629958 : gel(z,i) = leafcopy(G.j);
158 : }
159 42364 : return z;
160 : }
161 :
162 : /* nchi: a character given by a vector [d, (c_i)], e.g. from char_normalize
163 : * such that chi(x) = e((c . log(x)) / d) where log(x) on bnr.gen */
164 : static GEN
165 3899 : get_Char(GEN nchi, long prec)
166 3899 : { return mkvec2(nchi, rootsof1_cx(gel(nchi,1), prec)); }
167 :
168 : /* prime divisors of conductor */
169 : static GEN
170 448 : divcond(GEN bnr) {GEN bid = bnr_get_bid(bnr); return gel(bid_get_fact(bid),1);}
171 :
172 : /* vector of prime ideals dividing bnr but not bnrc */
173 : static GEN
174 161 : get_prdiff(GEN D, GEN Dc)
175 : {
176 161 : long n, i, l = lg(D);
177 161 : GEN diff = cgetg(l, t_COL);
178 448 : for (n = i = 1; i < l; i++)
179 287 : if (!tablesearch(Dc, gel(D,i), &cmp_prime_ideal)) gel(diff,n++) = gel(D,i);
180 161 : setlg(diff, n); return diff;
181 : }
182 :
183 : #define ch_prec(x) realprec(gel(x,1))
184 : #define ch_C(x) gel(x,1)
185 : #define ch_bnr(x) gel(x,2)
186 : #define ch_3(x) gel(x,3)
187 : #define ch_q(x) gel(x,3)[1]
188 : #define ch_CHI(x) gel(x,4)
189 : #define ch_diff(x) gel(x,5)
190 : #define ch_CHI0(x) gel(x,6)
191 : #define ch_small(x) gel(x,7)
192 : #define ch_comp(x) gel(x,7)[1]
193 : #define ch_phideg(x) gel(x,7)[2]
194 : static long
195 1393 : ch_deg(GEN dtcr) { return chi_get_deg(ch_CHI(dtcr)); }
196 :
197 : /********************************************************************/
198 : /* 1rst part: find the field K */
199 : /********************************************************************/
200 : static GEN AllStark(GEN data, long flag, long prec);
201 :
202 : /* Columns of C [HNF] give the generators of a subgroup of the finite abelian
203 : * group A [ in terms of implicit generators ], compute data to work in A/C:
204 : * 1) order
205 : * 2) structure
206 : * 3) the matrix A ->> A/C
207 : * 4) the subgroup C */
208 : static GEN
209 1302 : InitQuotient(GEN C)
210 : {
211 1302 : GEN U, D = ZM_snfall_i(C, &U, NULL, 1), h = ZV_prod(D);
212 1302 : return mkvec5(h, D, U, C, cyc_normalize(D));
213 : }
214 :
215 : /* lift chi character on A/C [Qt from InitQuotient] to character on A [cyc]*/
216 : static GEN
217 3465 : LiftChar(GEN Qt, GEN cyc, GEN chi)
218 : {
219 3465 : GEN ncyc = gel(Qt,5), U = gel(Qt,3), nchi = char_normalize(chi, ncyc);
220 3465 : GEN c = ZV_ZM_mul(gel(nchi,2), U), d = gel(nchi,1);
221 3465 : return char_denormalize(cyc, d, c);
222 : }
223 :
224 : /* Let C be a subgroup, system of representatives of the quotient */
225 : static GEN
226 322 : subgroup_classes(GEN C)
227 : {
228 322 : GEN U, D = ZM_snfall_i(C, &U, NULL, 1), e = cyc2elts(D);
229 322 : long i, l = lg(e);
230 :
231 322 : if (ZM_isidentity(U))
232 1946 : for (i = 1; i < l; i++) (void)vecsmall_to_vec_inplace(gel(e,i));
233 : else
234 : {
235 14 : GEN Ui = ZM_inv(U,NULL);
236 84 : for (i = 1; i < l; i++) gel(e,i) = ZM_zc_mul(Ui, gel(e,i));
237 : }
238 322 : return e;
239 : }
240 :
241 : /* Let s: A -> B given by [P,cycA,cycB] A and B, compute the kernel of s. */
242 : GEN
243 448 : abmap_kernel(GEN S)
244 : {
245 448 : GEN U, P = gel(S,1), cycA = gel(S,2), DB = diagonal_shallow(gel(S,3));
246 448 : long nA = lg(cycA)-1, rk;
247 :
248 448 : rk = nA + lg(DB) - lg(ZM_hnfall_i(shallowconcat(P, DB), &U, 1));
249 448 : return ZM_hnfmodid(matslice(U, 1,nA, 1,rk), cycA);
250 : }
251 : /* let H be a subgroup of A; return s(H) */
252 : GEN
253 1463 : abmap_subgroup_image(GEN S, GEN H)
254 1463 : { return ZM_hnfmodid(ZM_mul(gel(S,1), H), gel(S,3)); }
255 :
256 : /* Let m and n be two moduli such that n|m and let C be a congruence
257 : group modulo n, compute the corresponding congruence group modulo m
258 : ie the kernel of the map Clk(m) ->> Clk(n)/C */
259 : static GEN
260 448 : ComputeKernel(GEN bnrm, GEN bnrn, GEN dtQ)
261 : {
262 448 : pari_sp av = avma;
263 448 : GEN S = bnrsurjection(bnrm, bnrn);
264 448 : GEN P = ZM_mul(gel(dtQ,3), gel(S,1));
265 448 : return gerepileupto(av, abmap_kernel(mkvec3(P, gel(S,2), gel(dtQ,2))));
266 : }
267 :
268 : static long
269 1169 : cyc_is_cyclic(GEN cyc) { return lg(cyc) <= 2 || equali1(gel(cyc,2)); }
270 :
271 : /* Let H be a subgroup of cl(bnr)/sugbroup, return 1 if
272 : cl(bnr)/subgoup/H is cyclic and the signature of the
273 : corresponding field is equal to sig and no finite prime
274 : dividing cond(bnr) is totally split in this field. Return 0
275 : otherwise. */
276 : static long
277 518 : IsGoodSubgroup(GEN H, GEN bnr, GEN map)
278 : {
279 518 : pari_sp av = avma;
280 : GEN S, mod, modH, p1, U, P, PH, bnrH, iH, qH;
281 : long j;
282 :
283 518 : p1 = InitQuotient(H);
284 : /* quotient is non cyclic */
285 518 : if (!cyc_is_cyclic(gel(p1,2))) return gc_long(av,0);
286 :
287 252 : (void)ZM_hnfall_i(shallowconcat(map,H), &U, 0);
288 252 : setlg(U, lg(H));
289 924 : for (j = 1; j < lg(U); j++) setlg(gel(U,j), lg(H));
290 252 : p1 = ZM_hnfmodid(U, bnr_get_cyc(bnr)); /* H as a subgroup of bnr */
291 252 : modH = bnrconductor_raw(bnr, p1);
292 252 : mod = bnr_get_mod(bnr);
293 :
294 : /* is the signature correct? */
295 252 : if (!gequal(gel(modH,2), gel(mod,2))) return gc_long(av, 0);
296 :
297 : /* finite part are the same: OK */
298 182 : if (gequal(gel(modH,1), gel(mod,1))) return gc_long(av, 1);
299 :
300 : /* need to check the splitting of primes dividing mod but not modH */
301 63 : bnrH = Buchray(bnr, modH, nf_INIT);
302 63 : P = divcond(bnr);
303 63 : PH = divcond(bnrH);
304 63 : S = bnrsurjection(bnr, bnrH);
305 : /* H as a subgroup of bnrH */
306 63 : iH = abmap_subgroup_image(S, p1);
307 63 : qH = InitQuotient(iH);
308 203 : for (j = 1; j < lg(P); j++)
309 : {
310 161 : GEN pr = gel(P, j), e;
311 : /* if pr divides modH, it is ramified, so it's good */
312 161 : if (tablesearch(PH, pr, cmp_prime_ideal)) continue;
313 : /* inertia degree of pr in bnr(modH)/H is charorder(cycH, e) */
314 56 : e = ZM_ZC_mul(gel(qH,3), isprincipalray(bnrH, pr));
315 56 : e = ZV_ZV_mod(e, gel(qH,2));
316 56 : if (ZV_equal0(e)) return gc_long(av,0); /* f = 1 */
317 : }
318 42 : return gc_long(av,1);
319 : }
320 :
321 : /* compute list of nontrivial characters trivial on H, modulo complex
322 : * conjugation. If flag is set, impose a nontrivial conductor at infinity */
323 : static GEN
324 392 : AllChars(GEN bnr, GEN dtQ, long flag)
325 : {
326 392 : GEN v, vchi, cyc = bnr_get_cyc(bnr);
327 392 : long n, i, hD = itos(gel(dtQ,1));
328 : hashtable *S;
329 :
330 392 : v = cgetg(hD+1, t_VEC); /* nonconjugate chars */
331 392 : vchi = cyc2elts(gel(dtQ,2));
332 392 : S = hash_create(hD, (ulong(*)(void*))&hash_GEN,
333 : (int(*)(void*,void*))&ZV_equal, 1);
334 3857 : for (i = n = 1; i < hD; i++) /* remove i = hD: trivial char */
335 : { /* lift a character of D in Clk(m) */
336 3465 : GEN F, lchi = LiftChar(dtQ, cyc, zv_to_ZV(gel(vchi,i))), cchi = NULL;
337 :
338 3465 : if (hash_search(S, lchi)) continue;
339 2653 : F = bnrconductor_raw(bnr, lchi);
340 2653 : if (flag && gequal0(gel(F,2))) continue; /* f_oo(chi) trivial ? */
341 :
342 1267 : if (abscmpiu(charorder(cyc,lchi), 2) > 0)
343 : { /* nonreal chi: add its conjugate character to S */
344 812 : cchi = charconj(cyc, lchi);
345 812 : hash_insert(S, cchi, (void*)1);
346 : }
347 1267 : gel(v, n++) = cchi? mkvec3(lchi, F, cchi): mkvec2(lchi, F);
348 : }
349 392 : setlg(v, n); return v;
350 : }
351 :
352 : static GEN InitChar(GEN bnr, GEN CR, long flag, long prec);
353 : static void CharNewPrec(GEN data, long prec);
354 :
355 : /* Given a conductor and a subgroups, return the corresponding complexity and
356 : * precision required using quickpol. Fill data[5] with dataCR */
357 : static long
358 322 : CplxModulus(GEN data, long *newprec)
359 : {
360 322 : long dprec = DEFAULTPREC;
361 322 : pari_sp av = avma;
362 : for (;;)
363 0 : {
364 322 : GEN cpl, pol = AllStark(data, -1, dprec);
365 322 : cpl = RgX_fpnorml2(pol, LOWDEFAULTPREC);
366 322 : dprec = maxss(dprec, nbits2extraprec(gexpo(pol))) + EXTRAPREC64;
367 322 : if (!gequal0(cpl)) { *newprec = dprec; return gexpo(cpl); }
368 0 : set_avma(av);
369 0 : if (DEBUGLEVEL>1) pari_warn(warnprec, "CplxModulus", dprec);
370 0 : CharNewPrec(data, dprec);
371 : }
372 : }
373 :
374 : /* return A \cap B in abelian group defined by cyc. NULL = whole group */
375 : static GEN
376 567 : subgp_intersect(GEN cyc, GEN A, GEN B)
377 : {
378 : GEN H, U;
379 : long k, lH;
380 567 : if (!A) return B;
381 224 : if (!B) return A;
382 224 : H = ZM_hnfall_i(shallowconcat(A,B), &U, 1);
383 224 : setlg(U, lg(A)); lH = lg(H);
384 798 : for (k = 1; k < lg(U); k++) setlg(gel(U,k), lH);
385 224 : return ZM_hnfmodid(ZM_mul(A,U), cyc);
386 : }
387 :
388 : static void CharNewPrec(GEN dataCR, long prec);
389 : /* Let (f,C) be a conductor without infinite part and a congruence group mod f.
390 : * Compute (m,D) such that D is a congruence group of conductor m, f | m,
391 : * divisible by all the infinite places but one, D is a subgroup of index 2 of
392 : * Cm = Ker: Clk(m) -> Clk(f)/C. Consider further the subgroups H of Clk(m)/D
393 : * with cyclic quotient Clk(m)/H such that no place dividing m is totally split
394 : * in the extension KH corresponding to H: we want their intersection to be
395 : * trivial. These H correspond to (the kernels of Galois orbits of) characters
396 : * chi of Clk(m)/D such that chi(log_gen_arch(m_oo)) != 1 and for all pr | m
397 : * we either have
398 : * - chi(log_gen_pr(pr,1)) != 1 [pr | cond(chi) => ramified in KH]
399 : * - or [pr \nmid cond(chi)] chi lifted to Clk(m/pr^oo) is not trivial at pr.
400 : * We want the map from Clk(m)/D given by the vector of such characters to have
401 : * trivial kernel. Return bnr(m), D, Ck(m)/D and Clk(m)/Cm */
402 : static GEN
403 322 : FindModulus(GEN bnr, GEN dtQ, long *newprec)
404 : {
405 322 : const long LIMNORM = 400;
406 322 : long n, i, maxnorm, minnorm, N, pr, rb, iscyc, olde = LONG_MAX;
407 322 : pari_sp av = avma;
408 322 : GEN bnf, nf, f, varch, m, rep = NULL;
409 :
410 322 : bnf = bnr_get_bnf(bnr);
411 322 : nf = bnf_get_nf(bnf);
412 322 : N = nf_get_degree(nf);
413 322 : f = gel(bnr_get_mod(bnr), 1);
414 :
415 : /* if cpl < rb, it is not necessary to try another modulus */
416 322 : rb = expi( powii(mulii(nf_get_disc(nf), ZM_det_triangular(f)),
417 : gmul2n(bnr_get_no(bnr), 3)) );
418 :
419 : /* Initialization of the possible infinite part */
420 322 : varch = cgetg(N+1,t_VEC);
421 1106 : for (i = 1; i <= N; i++)
422 : {
423 784 : GEN a = const_vec(N,gen_1);
424 784 : gel(a, N+1-i) = gen_0;
425 784 : gel(varch, i) = a;
426 : }
427 322 : m = cgetg(3, t_VEC);
428 :
429 : /* Go from minnorm up to maxnorm; if necessary, increase these values.
430 : * If the extension is cyclic then a suitable conductor exists and we go on
431 : * until we find it. Else, stop at norm LIMNORM. */
432 322 : minnorm = 1;
433 322 : maxnorm = 50;
434 322 : iscyc = cyc_is_cyclic(gel(dtQ,2));
435 :
436 322 : if (DEBUGLEVEL>1)
437 0 : err_printf("Looking for a modulus of norm: ");
438 :
439 : for(;;)
440 0 : {
441 322 : GEN listid = ideallist0(nf, maxnorm, 4+8); /* ideals of norm <= maxnorm */
442 322 : pari_sp av1 = avma;
443 1463 : for (n = minnorm; n <= maxnorm; n++, set_avma(av1))
444 : {
445 1463 : GEN idnormn = gel(listid,n);
446 1463 : long nbidnn = lg(idnormn) - 1;
447 1463 : if (DEBUGLEVEL>1) err_printf(" %ld", n);
448 2296 : for (i = 1; i <= nbidnn; i++)
449 : { /* finite part of the conductor */
450 : long s;
451 :
452 1155 : gel(m,1) = idealmul(nf, f, gel(idnormn,i));
453 3213 : for (s = 1; s <= N; s++)
454 : { /* infinite part */
455 : GEN candD, Cm, bnrm;
456 : long lD, c;
457 :
458 2380 : gel(m,2) = gel(varch,s);
459 : /* compute Clk(m), check if m is a conductor */
460 2380 : bnrm = Buchray(bnf, m, nf_INIT);
461 2380 : if (!bnrisconductor(bnrm, NULL)) continue;
462 :
463 : /* compute Im(C) in Clk(m)... */
464 448 : Cm = ComputeKernel(bnrm, bnr, dtQ);
465 : /* ... and its subgroups of index 2 with conductor m */
466 448 : candD = subgrouplist_cond_sub(bnrm, Cm, mkvec(gen_2));
467 448 : lD = lg(candD);
468 455 : for (c = 1; c < lD; c++)
469 : {
470 329 : GEN data, CR, D = gel(candD,c), QD = InitQuotient(D);
471 329 : GEN ord = gel(QD,1), cyc = gel(QD,2), map = gel(QD,3);
472 : long e;
473 :
474 329 : if (!cyc_is_cyclic(cyc)) /* cyclic => suitable, else test */
475 : {
476 77 : GEN lH = subgrouplist(cyc, NULL), IK = NULL;
477 77 : long j, ok = 0;
478 574 : for (j = 1; j < lg(lH); j++)
479 : {
480 567 : GEN H = gel(lH, j), IH = subgp_intersect(cyc, IK, H);
481 : /* if H > IK, no need to test H */
482 567 : if (IK && gidentical(IH, IK)) continue;
483 518 : if (IsGoodSubgroup(H, bnrm, map))
484 : {
485 161 : IK = IH; /* intersection of all good subgroups */
486 161 : if (equalii(ord, ZM_det_triangular(IK))) { ok = 1; break; }
487 : }
488 : }
489 77 : if (!ok) continue;
490 : }
491 322 : CR = InitChar(bnrm, AllChars(bnrm, QD, 1), 0, DEFAULTPREC);
492 322 : data = mkvec4(bnrm, D, subgroup_classes(Cm), CR);
493 322 : if (DEBUGLEVEL>1)
494 0 : err_printf("\nTrying modulus = %Ps and subgroup = %Ps\n",
495 : bnr_get_mod(bnrm), D);
496 322 : e = CplxModulus(data, &pr);
497 322 : if (DEBUGLEVEL>1) err_printf("cpl = 2^%ld\n", e);
498 322 : if (e < olde)
499 : {
500 322 : guncloneNULL(rep); rep = gclone(data);
501 322 : *newprec = pr; olde = e;
502 : }
503 322 : if (olde < rb) goto END; /* OK */
504 0 : if (DEBUGLEVEL>1) err_printf("Trying to find another modulus...");
505 : }
506 : }
507 833 : if (rep) goto END; /* OK */
508 : }
509 : }
510 : /* if necessary compute more ideals */
511 0 : minnorm = maxnorm;
512 0 : maxnorm <<= 1;
513 0 : if (!iscyc && maxnorm > LIMNORM) return NULL;
514 : }
515 322 : END:
516 322 : if (DEBUGLEVEL>1)
517 0 : err_printf("No, we're done!\nModulus = %Ps and subgroup = %Ps\n",
518 0 : bnr_get_mod(gel(rep,1)), gel(rep,2));
519 322 : CharNewPrec(rep, *newprec); return gerepilecopy(av, rep);
520 : }
521 :
522 : /********************************************************************/
523 : /* 2nd part: compute W(X) */
524 : /********************************************************************/
525 :
526 : /* find ilambda s.t. Diff*f*ilambda integral and coprime to f
527 : and ilambda >> 0 at foo, fa = factorization of f */
528 : static GEN
529 812 : get_ilambda(GEN nf, GEN fa, GEN foo)
530 : {
531 812 : GEN x, w, E2, P = gel(fa,1), E = gel(fa,2), D = nf_get_diff(nf);
532 812 : long i, l = lg(P);
533 812 : if (l == 1) return gen_1;
534 679 : w = cgetg(l, t_VEC);
535 679 : E2 = cgetg(l, t_COL);
536 1498 : for (i = 1; i < l; i++)
537 : {
538 819 : GEN pr = gel(P,i), t = pr_get_tau(pr);
539 819 : long e = itou(gel(E,i)), v = idealval(nf, D, pr);
540 819 : if (v) { D = idealdivpowprime(nf, D, pr, utoipos(v)); e += v; }
541 819 : gel(E2,i) = stoi(e+1);
542 819 : if (typ(t) == t_MAT) t = gel(t,1);
543 819 : gel(w,i) = gdiv(nfpow(nf, t, stoi(e)), powiu(pr_get_p(pr),e));
544 : }
545 679 : x = mkmat2(P, E2);
546 679 : return idealchinese(nf, mkvec2(x, foo), w);
547 : }
548 : /* compute the list of W(chi) such that Ld(s,chi) = W(chi) Ld(1 - s, chi*),
549 : * for all chi in LCHI. All chi have the same conductor (= cond(bnr)). */
550 : static GEN
551 1015 : ArtinNumber(GEN bnr, GEN LCHI, long prec)
552 : {
553 1015 : long ic, i, j, nz, nChar = lg(LCHI)-1;
554 : pari_sp av2;
555 : GEN sqrtnc, cond, condZ, cond0, cond1, nf, T, cyc, vN, vB, diff, vt, idh;
556 : GEN zid, gen, z, nchi, indW, W, classe, s0, s, den, ilambda, sarch;
557 : CHI_t **lC;
558 : GROUP_t G;
559 :
560 1015 : lC = (CHI_t**)new_chunk(nChar + 1);
561 1015 : indW = cgetg(nChar + 1, t_VECSMALL);
562 1015 : W = cgetg(nChar + 1, t_VEC);
563 3528 : for (ic = 0, i = 1; i <= nChar; i++)
564 : {
565 2513 : GEN CHI = gel(LCHI,i);
566 2513 : if (chi_get_deg(CHI) <= 2) { gel(W,i) = gen_1; continue; }
567 1631 : ic++; indW[ic] = i;
568 1631 : lC[ic] = (CHI_t*)new_chunk(sizeof(CHI_t));
569 1631 : init_CHI_C(lC[ic], CHI);
570 : }
571 1015 : if (!ic) return W;
572 812 : nChar = ic;
573 :
574 812 : nf = bnr_get_nf(bnr);
575 812 : diff = nf_get_diff(nf);
576 812 : T = nf_get_Tr(nf);
577 812 : cond = bnr_get_mod(bnr);
578 812 : cond0 = gel(cond,1); condZ = gcoeff(cond0,1,1);
579 812 : cond1 = gel(cond,2);
580 :
581 812 : sqrtnc = gsqrt(idealnorm(nf, cond0), prec);
582 812 : ilambda = get_ilambda(nf, bid_get_fact(bnr_get_bid(bnr)), cond1);
583 812 : idh = idealmul(nf, ilambda, idealmul(nf, diff, cond0)); /* integral */
584 812 : ilambda = Q_remove_denom(ilambda, &den);
585 812 : z = den? rootsof1_cx(den, prec): NULL;
586 :
587 : /* compute a system of generators of (Ok/cond)^*, we'll make them
588 : * cond1-positive in the main loop */
589 812 : zid = Idealstar(nf, cond0, nf_GEN);
590 812 : cyc = abgrp_get_cyc(zid);
591 812 : gen = abgrp_get_gen(zid);
592 812 : nz = lg(gen) - 1;
593 812 : sarch = nfarchstar(nf, cond0, vec01_to_indices(cond1));
594 :
595 812 : nchi = cgetg(nChar+1, t_VEC);
596 2443 : for (ic = 1; ic <= nChar; ic++) gel(nchi,ic) = cgetg(nz + 1, t_VECSMALL);
597 1673 : for (i = 1; i <= nz; i++)
598 : {
599 861 : if (is_bigint(gel(cyc,i)))
600 0 : pari_err_OVERFLOW("ArtinNumber [conductor too large]");
601 861 : gel(gen,i) = set_sign_mod_divisor(nf, NULL, gel(gen,i), sarch);
602 861 : classe = isprincipalray(bnr, gel(gen,i));
603 2674 : for (ic = 1; ic <= nChar; ic++) {
604 1813 : GEN n = gel(nchi,ic);
605 1813 : n[i] = CHI_eval_n(lC[ic], classe);
606 : }
607 : }
608 :
609 : /* Sum chi(beta) * exp(2i * Pi * Tr(beta * ilambda) where beta
610 : runs through the classes of (Ok/cond0)^* and beta cond1-positive */
611 812 : vt = gel(T,1); /* ( Tr(w_i) )_i */
612 812 : if (typ(ilambda) == t_COL)
613 588 : vt = ZV_ZM_mul(vt, zk_multable(nf, ilambda));
614 : else
615 224 : vt = ZC_Z_mul(vt, ilambda);
616 : /*vt = den . (Tr(w_i * ilambda))_i */
617 812 : G.cyc = gtovecsmall(cyc);
618 812 : G.r = nz;
619 812 : G.j = zero_zv(nz);
620 812 : vN = zero_Flm_copy(nz, nChar);
621 :
622 812 : av2 = avma;
623 812 : vB = const_vec(nz, gen_1);
624 812 : s0 = z? powgi(z, modii(gel(vt,1), den)): gen_1; /* for beta = 1 */
625 812 : s = const_vec(nChar, s0);
626 :
627 63133 : while ( (i = NextElt(&G)) )
628 : {
629 62321 : GEN b = gel(vB,i);
630 62321 : b = nfmuli(nf, b, gel(gen,i));
631 62321 : b = typ(b) == t_COL? FpC_red(b, condZ): modii(b, condZ);
632 125020 : for (j=1; j<=i; j++) gel(vB,j) = b;
633 :
634 257236 : for (ic = 1; ic <= nChar; ic++)
635 : {
636 194915 : GEN v = gel(vN,ic), n = gel(nchi,ic);
637 194915 : v[i] = Fl_add(v[i], n[i], lC[ic]->ord);
638 195601 : for (j=1; j<i; j++) v[j] = v[i];
639 : }
640 :
641 62321 : gel(vB,i) = b = set_sign_mod_divisor(nf, NULL, b, sarch);
642 62321 : if (!z)
643 0 : s0 = gen_1;
644 : else
645 : {
646 62321 : b = typ(b) == t_COL? ZV_dotproduct(vt, b): mulii(gel(vt,1),b);
647 62321 : s0 = powgi(z, modii(b,den));
648 : }
649 257236 : for (ic = 1; ic <= nChar; ic++)
650 : {
651 194915 : GEN v = gel(vN,ic), val = lC[ic]->val[ v[i] ];
652 194915 : gel(s,ic) = gadd(gel(s,ic), gmul(val, s0));
653 : }
654 :
655 62321 : if (gc_needed(av2, 1))
656 : {
657 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"ArtinNumber");
658 0 : gerepileall(av2, 2, &s, &vB);
659 : }
660 : }
661 :
662 812 : classe = isprincipalray(bnr, idh);
663 812 : z = powIs(- (lg(gel(sarch,1))-1));
664 :
665 2443 : for (ic = 1; ic <= nChar; ic++)
666 : {
667 1631 : s0 = gmul(gel(s,ic), CHI_eval(lC[ic], classe));
668 1631 : gel(W, indW[ic]) = gmul(gdiv(s0, sqrtnc), z);
669 : }
670 812 : return W;
671 : }
672 :
673 : static GEN
674 735 : AllArtinNumbers(GEN CR, long prec)
675 : {
676 735 : pari_sp av = avma;
677 735 : GEN vChar = gel(CR,1), dataCR = gel(CR,2);
678 735 : long j, k, cl = lg(dataCR) - 1, J = lg(vChar)-1;
679 735 : GEN W = cgetg(cl+1,t_VEC), WbyCond, LCHI;
680 :
681 1673 : for (j = 1; j <= J; j++)
682 : {
683 938 : GEN LChar = gel(vChar,j), ldata = vecpermute(dataCR, LChar);
684 938 : GEN dtcr = gel(ldata,1), bnr = ch_bnr(dtcr);
685 938 : long l = lg(LChar);
686 :
687 938 : if (DEBUGLEVEL>1)
688 0 : err_printf("* Root Number: cond. no %ld/%ld (%ld chars)\n", j, J, l-1);
689 938 : LCHI = cgetg(l, t_VEC);
690 3374 : for (k = 1; k < l; k++) gel(LCHI,k) = ch_CHI0(gel(ldata,k));
691 938 : WbyCond = ArtinNumber(bnr, LCHI, prec);
692 3374 : for (k = 1; k < l; k++) gel(W,LChar[k]) = gel(WbyCond,k);
693 : }
694 735 : return gerepilecopy(av, W);
695 : }
696 :
697 : /* compute the constant W of the functional equation of
698 : Lambda(chi). If flag = 1 then chi is assumed to be primitive */
699 : GEN
700 77 : bnrrootnumber(GEN bnr, GEN chi, long flag, long prec)
701 : {
702 77 : pari_sp av = avma;
703 : GEN cyc, W;
704 :
705 77 : if (flag < 0 || flag > 1) pari_err_FLAG("bnrrootnumber");
706 77 : checkbnr(bnr);
707 77 : if (flag)
708 : {
709 0 : cyc = bnr_get_cyc(bnr);
710 0 : if (!char_check(cyc,chi)) pari_err_TYPE("bnrrootnumber [character]", chi);
711 : }
712 : else
713 : {
714 77 : bnr_char_sanitize(&bnr, &chi);
715 77 : cyc = bnr_get_cyc(bnr);
716 : }
717 77 : chi = char_normalize(chi, cyc_normalize(cyc));
718 77 : chi = get_Char(chi, prec);
719 77 : W = ArtinNumber(bnr, mkvec(chi), prec);
720 77 : return gerepilecopy(av, gel(W,1));
721 : }
722 :
723 : /********************************************************************/
724 : /* 3rd part: initialize the characters */
725 : /********************************************************************/
726 :
727 : /* Let chi be a character, A(chi) corresponding to the primes dividing diff
728 : * at s = flag. If s = 0, returns [r, A] where r is the order of vanishing
729 : * at s = 0 corresponding to diff. */
730 : static GEN
731 2184 : AChi(GEN dtcr, long *r, long flag, long prec)
732 : {
733 2184 : GEN A, diff = ch_diff(dtcr), bnrc = ch_bnr(dtcr), chi = ch_CHI0(dtcr);
734 2184 : long i, l = lg(diff);
735 :
736 2184 : A = gen_1; *r = 0;
737 2296 : for (i = 1; i < l; i++)
738 : {
739 112 : GEN B, pr = gel(diff,i), z = CharEval(chi, isprincipalray(bnrc, pr));
740 112 : if (flag)
741 0 : B = gsubsg(1, gdiv(z, pr_norm(pr)));
742 112 : else if (gequal1(z))
743 : {
744 21 : B = glog(pr_norm(pr), prec);
745 21 : (*r)++;
746 : }
747 : else
748 91 : B = gsubsg(1, z);
749 112 : A = gmul(A, B);
750 : }
751 2184 : return A;
752 : }
753 : /* simplified version of Achi: return 1 if L(0,chi) = 0 */
754 : static int
755 1029 : L_vanishes_at_0(GEN D)
756 : {
757 1029 : GEN diff = ch_diff(D), bnrc = ch_bnr(D), chi = ch_CHI0(D);
758 1029 : long i, l = lg(diff);
759 1071 : for (i = 1; i < l; i++)
760 : {
761 56 : GEN pr = gel(diff,i);
762 56 : if (!CharEval_n(chi, isprincipalray(bnrc, pr))) return 1;
763 : }
764 1015 : return 0;
765 : }
766 :
767 : static GEN
768 532 : _data3(GEN arch, long r2)
769 : {
770 532 : GEN z = cgetg(4, t_VECSMALL);
771 532 : long i, r1 = lg(arch) - 1, q = 0;
772 1701 : for (i = 1; i <= r1; i++) if (signe(gel(arch,i))) q++;
773 532 : z[1] = q;
774 532 : z[2] = r1 - q;
775 532 : z[3] = r2; return z;
776 : }
777 : static void
778 1701 : ch_get3(GEN dtcr, long *a, long *b, long *c)
779 1701 : { GEN v = ch_3(dtcr); *a = v[1]; *b = v[2]; *c = v[3]; }
780 : static GEN
781 735 : get_C(GEN nf, long prec)
782 : {
783 735 : long r2 = nf_get_r2(nf), N = nf_get_degree(nf);
784 735 : return gmul2n(sqrtr_abs(divir(nf_get_disc(nf), powru(mppi(prec),N))), -r2);
785 : }
786 : /* sort chars according to conductor */
787 : static GEN
788 385 : sortChars(GEN ch)
789 : {
790 385 : long j, l = lg(ch);
791 385 : GEN F = cgetg(l, t_VEC);
792 1652 : for (j = 1; j < l; j++) gel(F, j) = gmael(ch,j,2);
793 385 : return vec_equiv(F);
794 : }
795 :
796 : /* Given a list [chi, F = cond(chi)] of characters over Cl(bnr), return
797 : * [vChar, dataCR], where vChar contains the equivalence classes of
798 : * characters with the same conductor, and dataCR contains for each character:
799 : * - bnr(F)
800 : * - the constant C(F) [t_REAL]
801 : * - [q, r1 - q, r2, rc] where
802 : * q = number of real places in F
803 : * rc = max{r1 + r2 - q + 1, r2 + q}
804 : * - diff(chi) primes dividing m but not F
805 : * - chi in bnr(m)
806 : * - chi in bnr(F).
807 : * If all is unset, only compute characters s.t. L(chi,0) != 0 */
808 : static GEN
809 385 : InitChar(GEN bnr, GEN ch, long all, long prec)
810 : {
811 385 : GEN bnf = checkbnf(bnr), nf = bnf_get_nf(bnf), mod = bnr_get_mod(bnr);
812 385 : GEN C, dataCR, ncyc, vChar = sortChars(ch);
813 385 : long n, l, r2 = nf_get_r2(nf), prec2 = precdbl(prec) + EXTRAPREC64;
814 385 : long lv = lg(vChar);
815 :
816 385 : C = get_C(nf, prec2);
817 385 : ncyc = cyc_normalize(bnr_get_cyc(bnr));
818 :
819 385 : dataCR = cgetg_copy(ch, &l);
820 917 : for (n = 1; n < lv; n++)
821 : {
822 532 : GEN D, bnrc, v = gel(vChar, n); /* indices of chars of given conductor */
823 532 : long a, i = v[1], lc = lg(v);
824 532 : GEN F = gmael(ch,i,2);
825 :
826 532 : gel(dataCR, i) = D = cgetg(8, t_VEC);
827 532 : ch_C(D) = mulrr(C, gsqrt(ZM_det_triangular(gel(F,1)), prec2));
828 532 : ch_3(D) = _data3(gel(F,2), r2);
829 532 : if (gequal(F, mod))
830 : {
831 371 : ch_bnr(D) = bnrc = bnr;
832 371 : ch_diff(D) = cgetg(1, t_VEC);
833 : }
834 : else
835 : {
836 161 : ch_bnr(D) = bnrc = Buchray(bnf, F, nf_INIT);
837 161 : ch_diff(D) = get_prdiff(divcond(bnr), divcond(bnrc));
838 : }
839 1799 : for (a = 1; a < lc; a++)
840 : {
841 1267 : long i = v[a];
842 1267 : GEN chi = gmael(ch,i,1);
843 :
844 1267 : if (a > 1) gel(dataCR, i) = D = leafcopy(D);
845 1267 : chi = char_normalize(chi,ncyc);
846 1267 : ch_CHI(D) = get_Char(chi, prec2);
847 1267 : if (bnrc == bnr)
848 1050 : ch_CHI0(D) = ch_CHI(D);
849 : else
850 : {
851 217 : chi = bnrchar_primitive(bnr, chi, bnrc);
852 217 : ch_CHI0(D) = get_Char(chi, prec2);
853 : }
854 : /* set last */
855 1267 : ch_small(D) = mkvecsmall2(all || !L_vanishes_at_0(D),
856 1267 : eulerphiu(itou(gel(chi,1))));
857 : }
858 : }
859 385 : return mkvec2(vChar, dataCR);
860 : }
861 :
862 : /* recompute dataCR with the new precision, modify bnr components in place */
863 : static void
864 350 : CharNewPrec(GEN data, long prec)
865 : {
866 350 : long j, l, prec2 = precdbl(prec) + EXTRAPREC64;
867 350 : GEN C, nf, dataCR = gmael(data,4,2), D = gel(dataCR,1);
868 :
869 350 : if (ch_prec(D) >= prec2) return;
870 350 : nf = bnr_get_nf(ch_bnr(D));
871 350 : if (nf_get_prec(nf) < prec) nf = nfnewprec_shallow(nf, prec); /* not prec2 */
872 350 : C = get_C(nf, prec2); l = lg(dataCR);
873 1519 : for (j = 1; j < l; j++)
874 : {
875 : GEN f0;
876 1169 : D = gel(dataCR,j); f0 = gel(bnr_get_mod(ch_bnr(D)), 1);
877 1169 : ch_C(D) = mulrr(C, gsqrt(ZM_det_triangular(f0), prec2));
878 1169 : gmael(ch_bnr(D), 1, 7) = nf;
879 1169 : ch_CHI(D) = get_Char(gel(ch_CHI(D),1), prec2);
880 1169 : ch_CHI0(D)= get_Char(gel(ch_CHI0(D),1), prec2);
881 : }
882 : }
883 :
884 : /********************************************************************/
885 : /* 4th part: compute the coefficients an(chi) */
886 : /* */
887 : /* matan entries are arrays of ints containing the coefficients of */
888 : /* an(chi) as a polmod modulo polcyclo(order(chi)) */
889 : /********************************************************************/
890 :
891 : static void
892 1368550 : _0toCoeff(int *rep, long deg)
893 : {
894 : long i;
895 6841980 : for (i=0; i<deg; i++) rep[i] = 0;
896 1368550 : }
897 :
898 : /* transform a polmod into Coeff */
899 : static void
900 394440 : Polmod2Coeff(int *rep, GEN polmod, long deg)
901 : {
902 : long i;
903 394440 : if (typ(polmod) == t_POLMOD)
904 : {
905 282396 : GEN pol = gel(polmod,2);
906 282396 : long d = degpol(pol);
907 :
908 282396 : pol += 2;
909 1270330 : for (i=0; i<=d; i++) rep[i] = itos(gel(pol,i));
910 694388 : for ( ; i<deg; i++) rep[i] = 0;
911 : }
912 : else
913 : {
914 112044 : rep[0] = itos(polmod);
915 125670 : for (i=1; i<deg; i++) rep[i] = 0;
916 : }
917 394440 : }
918 :
919 : /* initialize a deg * n matrix of ints */
920 : static int**
921 4186 : InitMatAn(long n, long deg, long flag)
922 : {
923 : long i, j;
924 4186 : int *a, **A = (int**)pari_malloc((n+1)*sizeof(int*));
925 4186 : A[0] = NULL;
926 5892609 : for (i = 1; i <= n; i++)
927 : {
928 5888423 : a = (int*)pari_malloc(deg*sizeof(int));
929 5888423 : A[i] = a; a[0] = (i == 1 || flag);
930 19110815 : for (j = 1; j < deg; j++) a[j] = 0;
931 : }
932 4186 : return A;
933 : }
934 :
935 : static void
936 6608 : FreeMat(int **A, long n)
937 : {
938 : long i;
939 5908387 : for (i = 0; i <= n; i++)
940 5901779 : if (A[i]) pari_free((void*)A[i]);
941 6608 : pari_free((void*)A);
942 6608 : }
943 :
944 : /* initialize Coeff reduction */
945 : static int**
946 2422 : InitReduction(long d, long deg)
947 : {
948 : long j;
949 2422 : pari_sp av = avma;
950 : int **A;
951 : GEN polmod, pol;
952 :
953 2422 : A = (int**)pari_malloc(deg*sizeof(int*));
954 2422 : pol = polcyclo(d, 0);
955 11592 : for (j = 0; j < deg; j++)
956 : {
957 9170 : A[j] = (int*)pari_malloc(deg*sizeof(int));
958 9170 : polmod = gmodulo(pol_xn(deg+j, 0), pol);
959 9170 : Polmod2Coeff(A[j], polmod, deg);
960 : }
961 :
962 2422 : set_avma(av); return A;
963 : }
964 :
965 : #if 0
966 : void
967 : pan(int **an, long n, long deg)
968 : {
969 : long i,j;
970 : for (i = 1; i <= n; i++)
971 : {
972 : err_printf("n = %ld: ",i);
973 : for (j = 0; j < deg; j++) err_printf("%d ",an[i][j]);
974 : err_printf("\n");
975 : }
976 : }
977 : #endif
978 :
979 : /* returns 0 if c is zero, 1 otherwise. */
980 : static int
981 7574308 : IsZero(int* c, long deg)
982 : {
983 : long i;
984 25160087 : for (i = 0; i < deg; i++)
985 19789393 : if (c[i]) return 0;
986 5370694 : return 1;
987 : }
988 :
989 : /* set c0 <-- c0 * c1 */
990 : static void
991 2118883 : MulCoeff(int *c0, int* c1, int** reduc, long deg)
992 : {
993 : long i,j;
994 : int c, *T;
995 :
996 2118883 : if (IsZero(c0,deg)) return;
997 :
998 1128589 : T = (int*)new_chunk(2*deg);
999 13916839 : for (i = 0; i < 2*deg; i++)
1000 : {
1001 12788250 : c = 0;
1002 161443073 : for (j = 0; j <= i; j++)
1003 148654823 : if (j < deg && j > i - deg) c += c0[j] * c1[i-j];
1004 12788250 : T[i] = c;
1005 : }
1006 7522714 : for (i = 0; i < deg; i++)
1007 : {
1008 6394125 : c = T[i];
1009 77524474 : for (j = 0; j < deg; j++) c += reduc[j][i] * T[deg+j];
1010 6394125 : c0[i] = c;
1011 : }
1012 : }
1013 :
1014 : /* c0 <- c0 + c1 * c2 */
1015 : static void
1016 5455425 : AddMulCoeff(int *c0, int *c1, int* c2, int** reduc, long deg)
1017 : {
1018 : long i, j;
1019 : pari_sp av;
1020 : int c, *t;
1021 :
1022 5455425 : if (IsZero(c2,deg)) return;
1023 1075025 : if (!c1) /* c1 == 1 */
1024 : {
1025 680477 : for (i = 0; i < deg; i++) c0[i] += c2[i];
1026 285138 : return;
1027 : }
1028 789887 : av = avma;
1029 789887 : t = (int*)new_chunk(2*deg); /* = c1 * c2, not reduced */
1030 5809769 : for (i = 0; i < 2*deg; i++)
1031 : {
1032 5019882 : c = 0;
1033 36510313 : for (j = 0; j <= i; j++)
1034 31490431 : if (j < deg && j > i - deg) c += c1[j] * c2[i-j];
1035 5019882 : t[i] = c;
1036 : }
1037 3299828 : for (i = 0; i < deg; i++)
1038 : {
1039 2509941 : c = t[i];
1040 17000186 : for (j = 0; j < deg; j++) c += reduc[j][i] * t[deg+j];
1041 2509941 : c0[i] += c;
1042 : }
1043 789887 : set_avma(av);
1044 : }
1045 :
1046 : /* evaluate the Coeff. No Garbage collector */
1047 : static GEN
1048 3764622 : EvalCoeff(GEN z, int* c, long deg)
1049 : {
1050 : long i,j;
1051 : GEN e, r;
1052 :
1053 3764622 : if (!c) return gen_0;
1054 : #if 0
1055 : /* standard Horner */
1056 : e = stoi(c[deg - 1]);
1057 : for (i = deg - 2; i >= 0; i--)
1058 : e = gadd(stoi(c[i]), gmul(z, e));
1059 : #else
1060 : /* specific attention to sparse polynomials */
1061 3764622 : e = NULL;
1062 5602386 : for (i = deg-1; i >=0; i=j-1)
1063 : {
1064 13383401 : for (j=i; c[j] == 0; j--)
1065 11545637 : if (j==0)
1066 : {
1067 3060993 : if (!e) return NULL;
1068 368187 : if (i!=j) z = gpowgs(z,i-j+1);
1069 368187 : return gmul(e,z);
1070 : }
1071 1837764 : if (e)
1072 : {
1073 765948 : r = (i==j)? z: gpowgs(z,i-j+1);
1074 765948 : e = gadd(gmul(e,r), stoi(c[j]));
1075 : }
1076 : else
1077 1071816 : e = stoi(c[j]);
1078 : }
1079 : #endif
1080 703629 : return e;
1081 : }
1082 :
1083 : /* a2 <- copy the n x m array */
1084 : static void
1085 310912 : CopyCoeff(int** a, int** a2, long n, long m)
1086 : {
1087 : long i,j;
1088 :
1089 4574071 : for (i = 1; i <= n; i++)
1090 : {
1091 4263159 : int *b = a[i], *b2 = a2[i];
1092 15526345 : for (j = 0; j < m; j++) b2[j] = b[j];
1093 : }
1094 310912 : }
1095 :
1096 : static void
1097 310912 : an_AddMul(int **an,int **an2, long np, long n, long deg, GEN chi, int **reduc)
1098 : {
1099 310912 : GEN chi2 = chi;
1100 : long q, qk, k;
1101 310912 : int *c, *c2 = (int*)new_chunk(deg);
1102 :
1103 310912 : CopyCoeff(an, an2, n/np, deg);
1104 310912 : for (q=np;;)
1105 : {
1106 344029 : if (gequal1(chi2)) c = NULL; else { Polmod2Coeff(c2, chi2, deg); c = c2; }
1107 5799454 : for(k = 1, qk = q; qk <= n; k++, qk += q)
1108 5455425 : AddMulCoeff(an[qk], c, an2[k], reduc, deg);
1109 344029 : if (! (q = umuluu_le(q,np, n)) ) break;
1110 :
1111 33117 : chi2 = gmul(chi2, chi);
1112 : }
1113 310912 : }
1114 :
1115 : /* correct the coefficients an(chi) according with diff(chi) in place */
1116 : static void
1117 2422 : CorrectCoeff(GEN dtcr, int** an, int** reduc, long n, long deg)
1118 : {
1119 2422 : pari_sp av = avma;
1120 : long lg, j;
1121 : pari_sp av1;
1122 : int **an2;
1123 : GEN bnrc, diff;
1124 : CHI_t C;
1125 :
1126 2422 : diff = ch_diff(dtcr); lg = lg(diff) - 1;
1127 2422 : if (!lg) return;
1128 :
1129 175 : if (DEBUGLEVEL>2) err_printf("diff(CHI) = %Ps", diff);
1130 175 : bnrc = ch_bnr(dtcr);
1131 175 : init_CHI_alg(&C, ch_CHI0(dtcr));
1132 :
1133 175 : an2 = InitMatAn(n, deg, 0);
1134 175 : av1 = avma;
1135 378 : for (j = 1; j <= lg; j++)
1136 : {
1137 203 : GEN pr = gel(diff,j);
1138 203 : long Np = upr_norm(pr);
1139 203 : GEN chi = CHI_eval(&C, isprincipalray(bnrc, pr));
1140 203 : an_AddMul(an,an2,Np,n,deg,chi,reduc);
1141 203 : set_avma(av1);
1142 : }
1143 175 : FreeMat(an2, n); set_avma(av);
1144 : }
1145 :
1146 : /* compute the coefficients an in the general case */
1147 : static int**
1148 1589 : ComputeCoeff(GEN dtcr, LISTray *R, long n, long deg)
1149 : {
1150 1589 : pari_sp av = avma, av2;
1151 : long i, l;
1152 : int **an, **reduc, **an2;
1153 : GEN L;
1154 : CHI_t C;
1155 :
1156 1589 : init_CHI_alg(&C, ch_CHI(dtcr));
1157 1589 : an = InitMatAn(n, deg, 0);
1158 1589 : an2 = InitMatAn(n, deg, 0);
1159 1589 : reduc = InitReduction(C.ord, deg);
1160 1589 : av2 = avma;
1161 :
1162 1589 : L = R->L1; l = lg(L);
1163 312298 : for (i=1; i<l; i++, set_avma(av2))
1164 : {
1165 310709 : long np = L[i];
1166 310709 : GEN chi = CHI_eval(&C, gel(R->L1ray,i));
1167 310709 : an_AddMul(an,an2,np,n,deg,chi,reduc);
1168 : }
1169 1589 : FreeMat(an2, n);
1170 :
1171 1589 : CorrectCoeff(dtcr, an, reduc, n, deg);
1172 1589 : FreeMat(reduc, deg-1);
1173 1589 : set_avma(av); return an;
1174 : }
1175 :
1176 : /********************************************************************/
1177 : /* 5th part: compute L-functions at s=1 */
1178 : /********************************************************************/
1179 : static void
1180 504 : deg11(LISTray *R, long p, GEN bnr, GEN pr) {
1181 504 : vecsmalltrunc_append(R->L1, p);
1182 504 : vectrunc_append(R->L1ray, isprincipalray(bnr, pr));
1183 504 : }
1184 : static void
1185 32142 : deg12(LISTray *R, long p, GEN bnr, GEN pr) {
1186 32142 : vecsmalltrunc_append(R->L11, p);
1187 32142 : vectrunc_append(R->L11ray, isprincipalray(bnr, pr));
1188 32142 : }
1189 : static void
1190 42 : deg0(LISTray *R, long p) { vecsmalltrunc_append(R->L0, p); }
1191 : static void
1192 34039 : deg2(LISTray *R, long p) { vecsmalltrunc_append(R->L2, p); }
1193 :
1194 : static void
1195 224 : InitPrimesQuad(GEN bnr, ulong N0, LISTray *R)
1196 : {
1197 224 : pari_sp av = avma;
1198 224 : GEN bnf = bnr_get_bnf(bnr), F = gel(bnr_get_mod(bnr), 1);
1199 224 : GEN v, N, nf = bnf_get_nf(bnf), dk = nf_get_disc(nf);
1200 224 : long l = 1 + primepi_upper_bound(N0);
1201 224 : ulong i, p, FZ = itou(gcoeff(F,1,1)), FZ2 = itou(gcoeff(F,2,2));
1202 : forprime_t T;
1203 :
1204 224 : FZ2 = ugcd(FZ, FZ2); /* content(F) */
1205 224 : R->L0 = vecsmalltrunc_init(l);
1206 224 : R->L2 = vecsmalltrunc_init(l); R->condZ = FZ;
1207 224 : R->L1 = vecsmalltrunc_init(l); R->L1ray = vectrunc_init(l);
1208 224 : R->L11= vecsmalltrunc_init(l); R->L11ray= vectrunc_init(l);
1209 224 : N = utoipos(2);
1210 224 : u_forprime_init(&T, 2, N0);
1211 66951 : while ( (p = u_forprime_next(&T)) )
1212 : {
1213 66727 : N[2] = p;
1214 66727 : switch (kroiu(dk, p))
1215 : {
1216 34060 : case -1: /* inert */
1217 34060 : if (FZ % p == 0) deg0(R,p); else deg2(R,p);
1218 34060 : break;
1219 32359 : case 1: /* split */
1220 32359 : if (FZ2 % p == 0) deg0(R,p);
1221 : else
1222 : {
1223 32359 : GEN Lpr = idealprimedec(nf, N);
1224 32359 : if (FZ % p) deg12(R, p, bnr, gel(Lpr,1));
1225 : else
1226 : {
1227 217 : long t = ZC_prdvd(gel(F,2), gel(Lpr,1))? 2: 1;
1228 217 : deg11(R, p, bnr, gel(Lpr,t));
1229 : }
1230 : }
1231 32359 : break;
1232 308 : default: /* ramified */
1233 308 : if (FZ % p == 0)
1234 21 : deg0(R,p);
1235 : else
1236 287 : deg11(R, p, bnr, idealprimedec_galois(nf, N));
1237 308 : break;
1238 : }
1239 : }
1240 : /* precompute isprincipalray(x), x in Z */
1241 224 : v = coprimes_zv(FZ);
1242 224 : R->rayZ = cgetg(FZ, t_VEC);
1243 3521 : for (i = 1; i < FZ; i++)
1244 : {
1245 3297 : N[2] = i;
1246 3297 : gel(R->rayZ,i) = v[i]? isprincipalray(bnr, N): gen_0;
1247 : }
1248 224 : gerepileall(av, 7, &(R->L0), &(R->L2), &(R->rayZ),
1249 : &(R->L1), &(R->L1ray), &(R->L11), &(R->L11ray));
1250 224 : }
1251 :
1252 : static void
1253 511 : InitPrimes(GEN bnr, ulong N0, LISTray *R)
1254 : {
1255 511 : GEN bnf = bnr_get_bnf(bnr), cond = gel(bnr_get_mod(bnr), 1);
1256 511 : long p, l, condZ, N = lg(cond)-1;
1257 511 : GEN DL, prime, BOUND, nf = bnf_get_nf(bnf);
1258 : forprime_t T;
1259 :
1260 511 : R->condZ = condZ = itos(gcoeff(cond,1,1));
1261 511 : l = primepi_upper_bound(N0) * N;
1262 511 : DL = cgetg(N+1, t_VEC);
1263 511 : R->L1 = vecsmalltrunc_init(l);
1264 511 : R->L1ray = vectrunc_init(l);
1265 511 : u_forprime_init(&T, 2, N0);
1266 511 : prime = utoipos(2);
1267 511 : BOUND = utoi(N0);
1268 113946 : while ( (p = u_forprime_next(&T)) )
1269 : {
1270 113435 : pari_sp av = avma;
1271 : long j, k, lP;
1272 : GEN P;
1273 113435 : prime[2] = p;
1274 113435 : if (DEBUGLEVEL>1 && (p & 2047) == 1) err_printf("%ld ", p);
1275 113435 : P = idealprimedec_limit_norm(nf, prime, BOUND); lP = lg(P);
1276 226310 : for (j = 1; j < lP; j++)
1277 : {
1278 112875 : GEN pr = gel(P,j), dl = NULL;
1279 112875 : if (condZ % p || !idealval(nf, cond, pr))
1280 : {
1281 112406 : dl = gclone( isprincipalray(bnr, pr) );
1282 112406 : vecsmalltrunc_append(R->L1, upowuu(p, pr_get_f(pr)));
1283 : }
1284 112875 : gel(DL,j) = dl;
1285 : }
1286 113435 : set_avma(av);
1287 226310 : for (k = 1; k < j; k++)
1288 : {
1289 112875 : if (!DL[k]) continue;
1290 112406 : vectrunc_append(R->L1ray, ZC_copy(gel(DL,k)));
1291 112406 : gunclone(gel(DL,k));
1292 : }
1293 : }
1294 511 : }
1295 :
1296 : static GEN /* cf polcoef */
1297 406524 : _sercoeff(GEN x, long n)
1298 : {
1299 406524 : long i = n - valser(x);
1300 406524 : return (i < 0)? gen_0: gel(x,i+2);
1301 : }
1302 :
1303 : static void
1304 406524 : affect_coeff(GEN q, long n, GEN y, long t)
1305 : {
1306 406524 : GEN x = _sercoeff(q,-n);
1307 406524 : if (x == gen_0) gel(y,n) = NULL;
1308 206264 : else { affgr(x, gel(y,n)); shiftr_inplace(gel(y,n), t); }
1309 406524 : }
1310 : /* (x-i)(x-(i+1)) */
1311 : static GEN
1312 103854 : d2(long i) { return deg2pol_shallow(gen_1, utoineg(2*i+1), muluu(i,i+1), 0); }
1313 : /* x-i */
1314 : static GEN
1315 311618 : d1(long i) { return deg1pol_shallow(gen_1, stoi(-i), 0); }
1316 :
1317 : typedef struct {
1318 : GEN c1, aij, bij, cS, cT, powracpi;
1319 : long i0, a,b,c, r, rc1, rc2;
1320 : } ST_t;
1321 :
1322 : /* compute the principal part at the integers s = 0, -1, -2, ..., -i0
1323 : * of Gamma((s+1)/2)^a Gamma(s/2)^b Gamma(s)^c / (s - z) with z = 0 and 1 */
1324 : static void
1325 308 : ppgamma(ST_t *T, long prec)
1326 : {
1327 : GEN G, G1, G2, A, E, O, x, sqpi, aij, bij;
1328 308 : long c = T->c, r = T->r, i0 = T->i0, i, j, s, t, dx;
1329 : pari_sp av;
1330 :
1331 308 : T->aij = aij = cgetg(i0+1, t_VEC);
1332 308 : T->bij = bij = cgetg(i0+1, t_VEC);
1333 104162 : for (i = 1; i <= i0; i++)
1334 : {
1335 : GEN p1, p2;
1336 103854 : gel(aij,i) = p1 = cgetg(r+1, t_VEC);
1337 103854 : gel(bij,i) = p2 = cgetg(r+1, t_VEC);
1338 307116 : for (j=1; j<=r; j++) { gel(p1,j) = cgetr(prec); gel(p2,j) = cgetr(prec); }
1339 : }
1340 308 : av = avma; x = pol_x(0);
1341 308 : sqpi = sqrtr_abs(mppi(prec)); /* Gamma(1/2) */
1342 :
1343 308 : G1 = gexp(integser(psi1series(r-1, 0, prec)), prec); /* Gamma(1 + x) */
1344 308 : G = shallowcopy(G1); setvalser(G,-1); /* Gamma(x) */
1345 :
1346 : /* expansion of log(Gamma(u) / Gamma(1/2)) at u = 1/2 */
1347 308 : G2 = cgetg(r+2, t_SER);
1348 308 : G2[1] = evalsigne(1) | _evalvalser(1) | evalvarn(0);
1349 308 : gel(G2,2) = gneg(gadd(gmul2n(mplog2(prec), 1), mpeuler(prec)));
1350 644 : for (i = 1; i < r; i++) gel(G2,i+2) = mulri(gel(G1,i+2), int2um1(i));
1351 308 : G2 = gmul(sqpi, gexp(G2, prec)); /* Gamma(1/2 + x) */
1352 :
1353 : /* We simplify to get one of the following two expressions
1354 : * if (b > a) : sqrt(Pi)^a 2^{a-au} Gamma(u)^{a+c} Gamma( u/2 )^{|b-a|}
1355 : * if (b <= a): sqrt(Pi)^b 2^{b-bu} Gamma(u)^{b+c} Gamma((u+1)/2)^{|b-a|} */
1356 308 : if (T->b > T->a)
1357 : {
1358 56 : t = T->a; s = T->b; dx = 1;
1359 56 : E = ser_unscale(G, ghalf);
1360 56 : O = gmul2n(gdiv(ser_unscale(G2, ghalf), d1(1)), 1); /* Gamma((x-1)/2) */
1361 : }
1362 : else
1363 : {
1364 252 : t = T->b; s = T->a; dx = 0;
1365 252 : E = ser_unscale(G2, ghalf);
1366 252 : O = ser_unscale(G, ghalf);
1367 : }
1368 : /* (sqrt(Pi) 2^{1-x})^t Gamma(x)^{t+c} */
1369 308 : A = gmul(gmul(powru(gmul2n(sqpi,1), t), gpowgs(G, t+c)),
1370 : gpow(gen_2, RgX_to_ser(gmulgs(x,-t), r+2), prec));
1371 : /* A * Gamma((x - dx + 1)/2)^{s-t} */
1372 308 : E = gmul(A, gpowgs(E, s-t));
1373 : /* A * Gamma((x - dx)/2)^{s-t} */
1374 308 : O = gdiv(gmul(A, gpowgs(O, s-t)), gpowgs(gsubgs(x, 1), t+c));
1375 52235 : for (i = 0; i < i0/2; i++)
1376 : {
1377 51927 : GEN p1, q1, A1 = gel(aij,2*i+1), B1 = gel(bij,2*i+1);
1378 51927 : GEN p2, q2, A2 = gel(aij,2*i+2), B2 = gel(bij,2*i+2);
1379 51927 : long t1 = i * (s+t), t2 = t1 + t;
1380 :
1381 51927 : p1 = gdiv(E, d1(2*i));
1382 51927 : q1 = gdiv(E, d1(2*i+1));
1383 51927 : p2 = gdiv(O, d1(2*i+1));
1384 51927 : q2 = gdiv(O, d1(2*i+2));
1385 153558 : for (j = 1; j <= r; j++)
1386 : {
1387 101631 : affect_coeff(p1, j, A1, t1); affect_coeff(q1, j, B1, t1);
1388 101631 : affect_coeff(p2, j, A2, t2); affect_coeff(q2, j, B2, t2);
1389 : }
1390 51927 : E = gdiv(E, gmul(gpowgs(d1(2*i+1+dx), s-t), gpowgs(d2(2*i+1), t+c)));
1391 51927 : O = gdiv(O, gmul(gpowgs(d1(2*i+2+dx), s-t), gpowgs(d2(2*i+2), t+c)));
1392 : }
1393 308 : set_avma(av);
1394 308 : }
1395 :
1396 : /* chi != 1. Return L(1, chi) if fl & 1, else [r, c] where L(s, chi) ~ c s^r
1397 : * at s = 0. */
1398 : static GEN
1399 1393 : GetValue(GEN dtcr, GEN W, GEN S, GEN T, long fl, long prec)
1400 : {
1401 : GEN cf, z;
1402 : long q, b, c, r;
1403 1393 : int isreal = (ch_deg(dtcr) <= 2);
1404 :
1405 1393 : ch_get3(dtcr, &q, &b, &c);
1406 1393 : if (fl & 1)
1407 : { /* S(chi) + W(chi).T(chi)) / (C(chi) sqrt(Pi)^{r1 - q}) */
1408 196 : cf = gmul(ch_C(dtcr), powruhalf(mppi(prec), b));
1409 :
1410 196 : z = gadd(S, gmul(W, T));
1411 196 : if (isreal) z = real_i(z);
1412 196 : z = gdiv(z, cf);
1413 196 : if (fl & 2) z = gmul(z, AChi(dtcr, &r, 1, prec));
1414 : }
1415 : else
1416 : { /* (W(chi).S(conj(chi)) + T(chi)) / (sqrt(Pi)^q 2^{r1 - q}) */
1417 1197 : cf = gmul2n(powruhalf(mppi(prec), q), b);
1418 :
1419 1197 : z = gadd(gmul(W, conj_i(S)), conj_i(T));
1420 1197 : if (isreal) z = real_i(z);
1421 1197 : z = gdiv(z, cf); r = 0;
1422 1197 : if (fl & 2) z = gmul(z, AChi(dtcr, &r, 0, prec));
1423 1197 : z = mkvec2(utoi(b + c + r), z);
1424 : }
1425 1393 : return z;
1426 : }
1427 :
1428 : /* return the order and the first nonzero term of L(s, chi0)
1429 : at s = 0. If flag != 0, adjust the value to get L_S(s, chi0). */
1430 : static GEN
1431 35 : GetValue1(GEN bnr, long flag, long prec)
1432 : {
1433 35 : GEN bnf = bnr_get_bnf(bnr), nf = bnf_get_nf(bnf);
1434 35 : GEN h = bnf_get_no(bnf), R = bnf_get_reg(bnf);
1435 35 : GEN c = gdivgs(mpmul(h, R), -bnf_get_tuN(bnf));
1436 35 : long r = lg(nf_get_roots(nf)) - 2; /* r1 + r2 - 1 */;
1437 35 : if (flag)
1438 : {
1439 0 : GEN diff = divcond(bnr);
1440 : long i, l;
1441 0 : l = lg(diff) - 1; r += l;
1442 0 : for (i = 1; i <= l; i++) c = gmul(c, glog(pr_norm(gel(diff,i)), prec));
1443 : }
1444 35 : return mkvec2(utoi(r), c);
1445 : }
1446 :
1447 : /********************************************************************/
1448 : /* 6th part: recover the coefficients */
1449 : /********************************************************************/
1450 : static long
1451 2508 : TestOne(GEN plg, RC_data *d)
1452 : {
1453 2508 : long j, v = d->v;
1454 2508 : GEN z = gsub(d->beta, gel(plg,v));
1455 2508 : if (expo(z) >= d->G) return 0;
1456 6705 : for (j = 1; j < lg(plg); j++)
1457 4683 : if (j != v && mpcmp(d->B, mpabs_shallow(gel(plg,j))) < 0) return 0;
1458 2022 : return 1;
1459 : }
1460 :
1461 : static GEN
1462 461 : chk_reccoeff_init(FP_chk_fun *chk, GEN r, GEN mat)
1463 : {
1464 461 : RC_data *d = (RC_data*)chk->data;
1465 461 : (void)r; d->U = mat; return d->nB;
1466 : }
1467 :
1468 : static GEN
1469 425 : chk_reccoeff(void *data, GEN x)
1470 : {
1471 425 : RC_data *d = (RC_data*)data;
1472 425 : GEN v = gmul(d->U, x), z = gel(v,1);
1473 :
1474 425 : if (!gequal1(z)) return NULL;
1475 418 : *++v = evaltyp(t_COL) | evallg( lg(d->M) );
1476 418 : if (TestOne(gmul(d->M, v), d)) return v;
1477 0 : return NULL;
1478 : }
1479 :
1480 : /* Using Cohen's method */
1481 : static GEN
1482 461 : RecCoeff3(GEN nf, RC_data *d, long prec)
1483 : {
1484 : GEN A, M, nB, cand, p1, B2, C2, tB, beta2, nf2, Bd;
1485 461 : GEN beta = d->beta, B = d->B;
1486 461 : long N = d->N, v = d->v, e, BIG;
1487 461 : long i, j, k, ct = 0, prec2;
1488 461 : FP_chk_fun chk = { &chk_reccoeff, &chk_reccoeff_init, NULL, NULL, 0 };
1489 461 : chk.data = (void*)d;
1490 :
1491 461 : d->G = minss(-10, -prec2nbits(prec) >> 4);
1492 461 : BIG = maxss(32, -2*d->G);
1493 461 : tB = sqrtnr(real2n(BIG-N,DEFAULTPREC), N-1);
1494 461 : Bd = grndtoi(gmin_shallow(B, tB), &e);
1495 461 : if (e > 0) return NULL; /* failure */
1496 461 : Bd = addiu(Bd, 1);
1497 461 : prec2 = nbits2prec( expi(Bd) + 192 );
1498 461 : prec2 = maxss(precdbl(prec), prec2);
1499 461 : B2 = sqri(Bd);
1500 461 : C2 = shifti(B2, BIG<<1);
1501 :
1502 461 : LABrcf: ct++;
1503 461 : beta2 = gprec_w(beta, prec2);
1504 461 : nf2 = nfnewprec_shallow(nf, prec2);
1505 461 : d->M = M = nf_get_M(nf2);
1506 :
1507 461 : A = cgetg(N+2, t_MAT);
1508 1935 : for (i = 1; i <= N+1; i++) gel(A,i) = cgetg(N+2, t_COL);
1509 :
1510 461 : gcoeff(A, 1, 1) = gadd(gmul(C2, gsqr(beta2)), B2);
1511 1474 : for (j = 2; j <= N+1; j++)
1512 : {
1513 1013 : p1 = gmul(C2, gmul(gneg_i(beta2), gcoeff(M, v, j-1)));
1514 1013 : gcoeff(A, 1, j) = gcoeff(A, j, 1) = p1;
1515 : }
1516 1474 : for (i = 2; i <= N+1; i++)
1517 2669 : for (j = i; j <= N+1; j++)
1518 : {
1519 1656 : p1 = gen_0;
1520 5514 : for (k = 1; k <= N; k++)
1521 : {
1522 3858 : GEN p2 = gmul(gcoeff(M, k, j-1), gcoeff(M, k, i-1));
1523 3858 : if (k == v) p2 = gmul(C2, p2);
1524 3858 : p1 = gadd(p1,p2);
1525 : }
1526 1656 : gcoeff(A, i, j) = gcoeff(A, j, i) = p1;
1527 : }
1528 :
1529 461 : nB = mului(N+1, B2);
1530 461 : d->nB = nB;
1531 461 : cand = fincke_pohst(A, nB, -1, prec2, &chk);
1532 :
1533 461 : if (!cand)
1534 : {
1535 0 : if (ct > 3) return NULL;
1536 0 : prec2 = precdbl(prec2);
1537 0 : if (DEBUGLEVEL>1) pari_warn(warnprec,"RecCoeff", prec2);
1538 0 : goto LABrcf;
1539 : }
1540 :
1541 461 : cand = gel(cand,1);
1542 461 : if (lg(cand) == 2) return gel(cand,1);
1543 :
1544 252 : if (DEBUGLEVEL>1) err_printf("RecCoeff3: no solution found!\n");
1545 252 : return NULL;
1546 : }
1547 :
1548 : /* Using linear dependence relations */
1549 : static GEN
1550 2065 : RecCoeff2(GEN nf, RC_data *d, long prec)
1551 : {
1552 : pari_sp av;
1553 2065 : GEN vec, M = nf_get_M(nf), beta = d->beta;
1554 2065 : long bit, min, max, lM = lg(M);
1555 :
1556 2065 : d->G = minss(-20, -prec2nbits(prec) >> 4);
1557 :
1558 2065 : vec = vec_prepend(row(M, d->v), gneg(beta));
1559 2065 : min = (long)prec2nbits_mul(prec, 0.75);
1560 2065 : max = (long)prec2nbits_mul(prec, 0.98);
1561 2065 : av = avma;
1562 2631 : for (bit = max; bit >= min; bit-=32, set_avma(av))
1563 : {
1564 : long e;
1565 2170 : GEN v = lindep_bit(vec, bit), z = gel(v,1);
1566 2170 : if (!signe(z)) continue;
1567 2090 : *++v = evaltyp(t_COL) | evallg(lM);
1568 2090 : v = grndtoi(gdiv(v, z), &e);
1569 2090 : if (e > 0) break;
1570 2090 : if (TestOne(RgM_RgC_mul(M, v), d)) return v;
1571 : }
1572 : /* failure */
1573 461 : return RecCoeff3(nf,d,prec);
1574 : }
1575 :
1576 : /* Attempts to find a polynomial with coefficients in nf such that
1577 : its coefficients are close to those of pol at the place v and
1578 : less than B at all the other places */
1579 : static GEN
1580 595 : RecCoeff(GEN nf, GEN pol, long v, long prec)
1581 : {
1582 595 : long j, md, cl = degpol(pol);
1583 595 : pari_sp av = avma;
1584 : RC_data d;
1585 :
1586 : /* if precision(pol) is too low, abort */
1587 3955 : for (j = 2; j <= cl+1; j++)
1588 : {
1589 3381 : GEN t = gel(pol, j);
1590 3381 : if (prec2nbits(precision(t)) - gexpo(t) < 34) return NULL;
1591 : }
1592 :
1593 574 : md = cl/2;
1594 574 : pol = leafcopy(pol);
1595 :
1596 574 : d.N = nf_get_degree(nf);
1597 574 : d.v = v;
1598 :
1599 2387 : for (j = 1; j <= cl; j++)
1600 : { /* start with the coefficients in the middle,
1601 : since they are the harder to recognize! */
1602 2065 : long cf = md + (j%2? j/2: -j/2);
1603 2065 : GEN t, bound = shifti(binomial(utoipos(cl), cf), cl-cf);
1604 :
1605 2065 : if (DEBUGLEVEL>1) err_printf("RecCoeff (cf = %ld, B = %Ps)\n", cf, bound);
1606 2065 : d.beta = real_i( gel(pol,cf+2) );
1607 2065 : d.B = bound;
1608 2065 : if (! (t = RecCoeff2(nf, &d, prec)) ) return NULL;
1609 1813 : gel(pol, cf+2) = coltoalg(nf,t);
1610 : }
1611 322 : gel(pol,cl+2) = gen_1;
1612 322 : return gerepilecopy(av, pol);
1613 : }
1614 :
1615 : /* an[q * i] *= chi for all (i,p)=1 */
1616 : static void
1617 149140 : an_mul(int **an, long p, long q, long n, long deg, GEN chi, int **reduc)
1618 : {
1619 : pari_sp av;
1620 : long c,i;
1621 : int *T;
1622 :
1623 149140 : if (gequal1(chi)) return;
1624 138646 : av = avma;
1625 138646 : T = (int*)new_chunk(deg); Polmod2Coeff(T,chi, deg);
1626 3009848 : for (c = 1, i = q; i <= n; i += q, c++)
1627 2871202 : if (c == p) c = 0; else MulCoeff(an[i], T, reduc, deg);
1628 138646 : set_avma(av);
1629 : }
1630 : /* an[q * i] = 0 for all (i,p)=1 */
1631 : static void
1632 133310 : an_set0_coprime(int **an, long p, long q, long n, long deg)
1633 : {
1634 : long c,i;
1635 1679399 : for (c = 1, i = q; i <= n; i += q, c++)
1636 1546089 : if (c == p) c = 0; else _0toCoeff(an[i], deg);
1637 133310 : }
1638 : /* an[q * i] = 0 for all i */
1639 : static void
1640 140 : an_set0(int **an, long p, long n, long deg)
1641 : {
1642 : long i;
1643 64663 : for (i = p; i <= n; i += p) _0toCoeff(an[i], deg);
1644 140 : }
1645 :
1646 : /* compute the coefficients an for the quadratic case */
1647 : static int**
1648 833 : computean(GEN dtcr, LISTray *R, long n, long deg)
1649 : {
1650 833 : pari_sp av = avma, av2;
1651 : long i, p, q, condZ, l;
1652 : int **an, **reduc;
1653 : GEN L, chi, chi1;
1654 : CHI_t C;
1655 :
1656 833 : init_CHI_alg(&C, ch_CHI(dtcr));
1657 833 : condZ= R->condZ;
1658 :
1659 833 : an = InitMatAn(n, deg, 1);
1660 833 : reduc = InitReduction(C.ord, deg);
1661 833 : av2 = avma;
1662 :
1663 : /* all pr | p divide cond */
1664 833 : L = R->L0; l = lg(L);
1665 973 : for (i=1; i<l; i++) an_set0(an,L[i],n,deg);
1666 :
1667 : /* 1 prime of degree 2 */
1668 833 : L = R->L2; l = lg(L);
1669 132494 : for (i=1; i<l; i++, set_avma(av2))
1670 : {
1671 131661 : p = L[i];
1672 131661 : if (condZ == 1) chi = C.val[0]; /* 1 */
1673 131507 : else chi = CHI_eval(&C, gel(R->rayZ, p%condZ));
1674 131661 : chi1 = chi;
1675 131661 : for (q=p;;)
1676 : {
1677 133310 : an_set0_coprime(an, p,q,n,deg); /* v_p(q) odd */
1678 133310 : if (! (q = umuluu_le(q,p, n)) ) break;
1679 :
1680 5560 : an_mul(an,p,q,n,deg,chi,reduc);
1681 5560 : if (! (q = umuluu_le(q,p, n)) ) break;
1682 1649 : chi = gmul(chi, chi1);
1683 : }
1684 : }
1685 :
1686 : /* 1 prime of degree 1 */
1687 833 : L = R->L1; l = lg(L);
1688 2856 : for (i=1; i<l; i++, set_avma(av2))
1689 : {
1690 2023 : p = L[i];
1691 2023 : chi = CHI_eval(&C, gel(R->L1ray,i));
1692 2023 : chi1 = chi;
1693 2023 : for(q=p;;)
1694 : {
1695 9168 : an_mul(an,p,q,n,deg,chi,reduc);
1696 9168 : if (! (q = umuluu_le(q,p, n)) ) break;
1697 7145 : chi = gmul(chi, chi1);
1698 : }
1699 : }
1700 :
1701 : /* 2 primes of degree 1 */
1702 833 : L = R->L11; l = lg(L);
1703 126472 : for (i=1; i<l; i++, set_avma(av2))
1704 : {
1705 : GEN ray1, ray2, chi11, chi12, chi2;
1706 :
1707 125639 : p = L[i]; ray1 = gel(R->L11ray,i); /* use pr1 pr2 = (p) */
1708 125639 : if (condZ == 1)
1709 112 : ray2 = ZC_neg(ray1);
1710 : else
1711 125527 : ray2 = ZC_sub(gel(R->rayZ, p%condZ), ray1);
1712 125639 : chi11 = CHI_eval(&C, ray1);
1713 125639 : chi12 = CHI_eval(&C, ray2);
1714 :
1715 125639 : chi1 = gadd(chi11, chi12);
1716 125639 : chi2 = chi12;
1717 125639 : for(q=p;;)
1718 : {
1719 134412 : an_mul(an,p,q,n,deg,chi1,reduc);
1720 134412 : if (! (q = umuluu_le(q,p, n)) ) break;
1721 8773 : chi2 = gmul(chi2, chi12);
1722 8773 : chi1 = gadd(chi2, gmul(chi1, chi11));
1723 : }
1724 : }
1725 :
1726 833 : CorrectCoeff(dtcr, an, reduc, n, deg);
1727 833 : FreeMat(reduc, deg-1);
1728 833 : set_avma(av); return an;
1729 : }
1730 :
1731 : /* return the vector of A^i/i for i = 1...n */
1732 : static GEN
1733 259 : mpvecpowdiv(GEN A, long n)
1734 : {
1735 259 : pari_sp av = avma;
1736 : long i;
1737 259 : GEN v = powersr(A, n);
1738 259 : GEN w = cgetg(n+1, t_VEC);
1739 259 : gel(w,1) = rcopy(gel(v,2));
1740 485716 : for (i=2; i<=n; i++) gel(w,i) = divru(gel(v,i+1), i);
1741 259 : return gerepileupto(av, w);
1742 : }
1743 :
1744 : static void GetST0(GEN bnr, GEN *pS, GEN *pT, GEN CR, long prec);
1745 : /* allocate memory for GetST answer */
1746 : static void
1747 448 : ST_alloc(GEN *pS, GEN *pT, long l, long prec)
1748 : {
1749 : long j;
1750 448 : *pS = cgetg(l, t_VEC);
1751 448 : *pT = cgetg(l, t_VEC);
1752 2051 : for (j = 1; j < l; j++)
1753 : {
1754 1603 : gel(*pS,j) = cgetc(prec);
1755 1603 : gel(*pT,j) = cgetc(prec);
1756 : }
1757 448 : }
1758 :
1759 : /* compute S and T for the quadratic case. The following cases are:
1760 : * 1) bnr complex;
1761 : * 2) bnr real and no infinite place divide cond_chi (TODO);
1762 : * 3) bnr real and one infinite place divide cond_chi;
1763 : * 4) bnr real and both infinite places divide cond_chi (TODO) */
1764 : static void
1765 259 : QuadGetST(GEN bnr, GEN *pS, GEN *pT, GEN CR, long prec)
1766 : {
1767 : pari_sp av, av1, av2;
1768 : long ncond, n, j, k, n0;
1769 259 : GEN vChar = gel(CR,1), dataCR = gel(CR,2), S, T, an, cs, N0, C;
1770 : LISTray LIST;
1771 :
1772 : /* initializations */
1773 259 : ST_alloc(pS, pT, lg(dataCR), prec); T = *pT; S = *pS;
1774 259 : av = avma;
1775 259 : ncond = lg(vChar)-1;
1776 259 : C = cgetg(ncond+1, t_VEC);
1777 259 : N0 = cgetg(ncond+1, t_VECSMALL);
1778 259 : cs = cgetg(ncond+1, t_VECSMALL);
1779 259 : n0 = 0;
1780 518 : for (j = 1; j <= ncond; j++)
1781 : {
1782 294 : GEN dtcr = gel(dataCR, mael(vChar,j,1)), c = ch_C(dtcr);
1783 : long r1, r2;
1784 :
1785 294 : gel(C,j) = c;
1786 294 : nf_get_sign(bnr_get_nf(ch_bnr(dtcr)), &r1, &r2);
1787 294 : if (r1 == 2) /* real quadratic */
1788 : {
1789 280 : cs[j] = 2 + ch_q(dtcr);
1790 280 : if (cs[j] == 2 || cs[j] == 4)
1791 : { /* NOT IMPLEMENTED YET */
1792 35 : GetST0(bnr, pS, pT, CR, prec);
1793 35 : return;
1794 : }
1795 : /* FIXME: is this value of N0 correct for the general case ? */
1796 245 : N0[j] = (long)prec2nbits_mul(prec, 0.35 * gtodouble(c));
1797 : }
1798 : else /* complex quadratic */
1799 : {
1800 14 : cs[j] = 1;
1801 14 : N0[j] = (long)prec2nbits_mul(prec, 0.7 * gtodouble(c));
1802 : }
1803 259 : if (n0 < N0[j]) n0 = N0[j];
1804 : }
1805 224 : if (DEBUGLEVEL>1) err_printf("N0 = %ld\n", n0);
1806 224 : InitPrimesQuad(bnr, n0, &LIST);
1807 :
1808 224 : av1 = avma;
1809 : /* loop over conductors */
1810 483 : for (j = 1; j <= ncond; j++)
1811 : {
1812 259 : GEN c0 = gel(C,j), c1 = divur(1, c0), c2 = divur(2, c0);
1813 259 : GEN ec1 = mpexp(c1), ec2 = mpexp(c2), LChar = gel(vChar,j);
1814 : GEN vf0, vf1, cf0, cf1;
1815 259 : const long nChar = lg(LChar)-1, NN = N0[j];
1816 :
1817 259 : if (DEBUGLEVEL>1)
1818 0 : err_printf("* conductor no %ld/%ld (N = %ld)\n\tInit: ", j,ncond,NN);
1819 259 : if (realprec(ec1) > prec) ec1 = rtor(ec1, prec);
1820 259 : if (realprec(ec2) > prec) ec2 = rtor(ec2, prec);
1821 259 : switch(cs[j])
1822 : {
1823 14 : case 1:
1824 14 : cf0 = gen_1;
1825 14 : cf1 = c0;
1826 14 : vf0 = mpveceint1(rtor(c1, prec), ec1, NN);
1827 14 : vf1 = mpvecpowdiv(invr(ec1), NN); break;
1828 :
1829 245 : case 3:
1830 245 : cf0 = sqrtr(mppi(prec));
1831 245 : cf1 = gmul2n(cf0, 1);
1832 245 : cf0 = gmul(cf0, c0);
1833 245 : vf0 = mpvecpowdiv(invr(ec2), NN);
1834 245 : vf1 = mpveceint1(rtor(c2, prec), ec2, NN); break;
1835 :
1836 0 : default:
1837 0 : cf0 = cf1 = NULL; /* FIXME: not implemented */
1838 0 : vf0 = vf1 = NULL;
1839 : }
1840 1099 : for (k = 1; k <= nChar; k++)
1841 : {
1842 840 : long u = LChar[k], d, c;
1843 840 : GEN dtcr = gel(dataCR, u), z, s, t;
1844 : int **matan;
1845 :
1846 840 : if (!ch_comp(dtcr)) continue;
1847 833 : if (DEBUGLEVEL>1) err_printf("\tchar no: %ld (%ld/%ld)\n", u,k,nChar);
1848 833 : d = ch_phideg(dtcr);
1849 833 : z = gel(ch_CHI(dtcr), 2); s = t = gen_0; av2 = avma;
1850 833 : matan = computean(gel(dataCR,u), &LIST, NN, d);
1851 1722878 : for (n = 1, c = 0; n <= NN; n++)
1852 1722045 : if ((an = EvalCoeff(z, matan[n], d)))
1853 : {
1854 455032 : s = gadd(s, gmul(an, gel(vf0,n)));
1855 455032 : t = gadd(t, gmul(an, gel(vf1,n)));
1856 455032 : if (++c == 256) { gerepileall(av2,2, &s,&t); c = 0; }
1857 : }
1858 833 : gaffect(gmul(cf0, s), gel(S,u));
1859 833 : gaffect(gmul(cf1, conj_i(t)), gel(T,u));
1860 833 : FreeMat(matan,NN); set_avma(av2);
1861 : }
1862 259 : if (DEBUGLEVEL>1) err_printf("\n");
1863 259 : set_avma(av1);
1864 : }
1865 224 : set_avma(av);
1866 : }
1867 :
1868 : /* s += t*u. All 3 of them t_REAL, except we allow s or u = NULL (for 0) */
1869 : static GEN
1870 49039438 : _addmulrr(GEN s, GEN t, GEN u)
1871 : {
1872 49039438 : if (u)
1873 : {
1874 48778009 : GEN v = mulrr(t, u);
1875 48778009 : return s? addrr(s, v): v;
1876 : }
1877 261429 : return s;
1878 : }
1879 : /* s += t. Both real, except we allow s or t = NULL (for exact 0) */
1880 : static GEN
1881 99843892 : _addrr(GEN s, GEN t)
1882 99843892 : { return t? (s? addrr(s, t): t) : s; }
1883 :
1884 : /* S & T for the general case. This is time-critical: optimize */
1885 : static void
1886 500087 : get_cS_cT(ST_t *T, long n)
1887 : {
1888 : pari_sp av;
1889 : GEN csurn, nsurc, lncsurn, A, B, s, t, Z, aij, bij;
1890 : long i, j, r, i0;
1891 :
1892 500087 : if (gel(T->cS,n)) return;
1893 :
1894 237664 : av = avma;
1895 237664 : aij = T->aij; i0= T->i0;
1896 237664 : bij = T->bij; r = T->r;
1897 237664 : Z = cgetg(r+1, t_VEC);
1898 237664 : gel(Z,1) = NULL; /* unused */
1899 :
1900 237664 : csurn = divru(T->c1, n);
1901 237664 : nsurc = invr(csurn);
1902 237664 : lncsurn = logr_abs(csurn);
1903 :
1904 237664 : if (r > 1)
1905 : {
1906 237489 : gel(Z,2) = lncsurn; /* r >= 2 */
1907 241584 : for (i = 3; i <= r; i++)
1908 4095 : gel(Z,i) = divru(mulrr(gel(Z,i-1), lncsurn), i-1);
1909 : /* Z[i] = ln^(i-1)(c1/n) / (i-1)! */
1910 : }
1911 :
1912 : /* i = i0 */
1913 237664 : A = gel(aij,i0); t = _addrr(NULL, gel(A,1));
1914 237664 : B = gel(bij,i0); s = _addrr(NULL, gel(B,1));
1915 479248 : for (j = 2; j <= r; j++)
1916 : {
1917 241584 : s = _addmulrr(s, gel(Z,j),gel(B,j));
1918 241584 : t = _addmulrr(t, gel(Z,j),gel(A,j));
1919 : }
1920 49565450 : for (i = i0 - 1; i > 1; i--)
1921 : {
1922 49327786 : A = gel(aij,i); if (t) t = mulrr(t, nsurc);
1923 49327786 : B = gel(bij,i); if (s) s = mulrr(s, nsurc);
1924 73364337 : for (j = odd(i)? T->rc2: T->rc1; j > 1; j--)
1925 : {
1926 24036551 : s = _addmulrr(s, gel(Z,j),gel(B,j));
1927 24036551 : t = _addmulrr(t, gel(Z,j),gel(A,j));
1928 : }
1929 49327786 : s = _addrr(s, gel(B,1));
1930 49327786 : t = _addrr(t, gel(A,1));
1931 : }
1932 : /* i = 1 */
1933 237664 : A = gel(aij,1); if (t) t = mulrr(t, nsurc);
1934 237664 : B = gel(bij,1); if (s) s = mulrr(s, nsurc);
1935 237664 : s = _addrr(s, gel(B,1));
1936 237664 : t = _addrr(t, gel(A,1));
1937 479248 : for (j = 2; j <= r; j++)
1938 : {
1939 241584 : s = _addmulrr(s, gel(Z,j),gel(B,j));
1940 241584 : t = _addmulrr(t, gel(Z,j),gel(A,j));
1941 : }
1942 237664 : s = _addrr(s, T->b? mulrr(csurn, gel(T->powracpi,T->b+1)): csurn);
1943 237664 : if (!s) s = gen_0;
1944 237664 : if (!t) t = gen_0;
1945 237664 : gel(T->cS,n) = gclone(s);
1946 237664 : gel(T->cT,n) = gclone(t); set_avma(av);
1947 : }
1948 :
1949 : static void
1950 497 : clear_cScT(ST_t *T, long N)
1951 : {
1952 497 : GEN cS = T->cS, cT = T->cT;
1953 : long i;
1954 1464166 : for (i=1; i<=N; i++)
1955 1463669 : if (cS[i]) {
1956 237664 : gunclone(gel(cS,i));
1957 237664 : gunclone(gel(cT,i)); gel(cS,i) = gel(cT,i) = NULL;
1958 : }
1959 497 : }
1960 :
1961 : static void
1962 308 : init_cScT(ST_t *T, GEN dtcr, long N, long prec)
1963 : {
1964 308 : ch_get3(dtcr, &T->a, &T->b, &T->c);
1965 308 : T->rc1 = T->a + T->c;
1966 308 : T->rc2 = T->b + T->c;
1967 308 : T->r = maxss(T->rc2+1, T->rc1); /* >= 2 */
1968 308 : ppgamma(T, prec);
1969 308 : clear_cScT(T, N);
1970 308 : }
1971 :
1972 : /* return a t_REAL */
1973 : static GEN
1974 511 : zeta_get_limx(long r1, long r2, long bit)
1975 : {
1976 511 : pari_sp av = avma;
1977 : GEN p1, p2, c0, c1, A0;
1978 511 : long r = r1 + r2, N = r + r2;
1979 :
1980 : /* c1 = N 2^(-2r2 / N) */
1981 511 : c1 = mulrs(powrfrac(real2n(1, DEFAULTPREC), -2*r2, N), N);
1982 :
1983 511 : p1 = powru(Pi2n(1, DEFAULTPREC), r - 1);
1984 511 : p2 = mulir(powuu(N,r), p1); shiftr_inplace(p2, -r2);
1985 511 : c0 = sqrtr( divrr(p2, powru(c1, r+1)) );
1986 :
1987 511 : A0 = logr_abs( gmul2n(c0, bit) ); p2 = divrr(A0, c1);
1988 511 : p1 = divrr(mulur(N*(r+1), logr_abs(p2)), addsr(2*(r+1), gmul2n(A0,2)));
1989 511 : return gerepileuptoleaf(av, divrr(addrs(p1, 1), powruhalf(p2, N)));
1990 : }
1991 : /* N_0 = floor( C_K / limx ). Large */
1992 : static long
1993 630 : zeta_get_N0(GEN C, GEN limx)
1994 : {
1995 : long e;
1996 630 : pari_sp av = avma;
1997 630 : GEN z = gcvtoi(gdiv(C, limx), &e); /* avoid truncation error */
1998 630 : if (e >= 0 || is_bigint(z))
1999 0 : pari_err_OVERFLOW("zeta_get_N0 [need too many primes]");
2000 630 : return gc_long(av, itos(z));
2001 : }
2002 :
2003 : static GEN
2004 1897 : eval_i(long r1, long r2, GEN limx, long i)
2005 : {
2006 1897 : GEN t = powru(limx, i);
2007 1897 : if (!r1) t = mulrr(t, powru(mpfactr(i , DEFAULTPREC), r2));
2008 1897 : else if (!r2) t = mulrr(t, powru(mpfactr(i/2, DEFAULTPREC), r1));
2009 : else {
2010 0 : GEN u1 = mpfactr(i/2, DEFAULTPREC);
2011 0 : GEN u2 = mpfactr(i, DEFAULTPREC);
2012 0 : if (r1 == r2) t = mulrr(t, powru(mulrr(u1,u2), r1));
2013 0 : else t = mulrr(t, mulrr(powru(u1,r1), powru(u2,r2)));
2014 : }
2015 1897 : return t;
2016 : }
2017 :
2018 : /* "small" even i such that limx^i ( (i\2)! )^r1 ( i! )^r2 > B. */
2019 : static long
2020 189 : get_i0(long r1, long r2, GEN B, GEN limx)
2021 : {
2022 189 : long imin = 1, imax = 1400;
2023 196 : while (mpcmp(eval_i(r1,r2,limx, imax), B) < 0) { imin = imax; imax *= 2; }
2024 1890 : while(imax - imin >= 4)
2025 : {
2026 1701 : long m = (imax + imin) >> 1;
2027 1701 : if (mpcmp(eval_i(r1,r2,limx, m), B) >= 0) imax = m; else imin = m;
2028 : }
2029 189 : return imax & ~1; /* make it even */
2030 : }
2031 : /* limx = zeta_get_limx(r1, r2, bit), a t_REAL */
2032 : static long
2033 189 : zeta_get_i0(long r1, long r2, long bit, GEN limx)
2034 : {
2035 189 : pari_sp av = avma;
2036 189 : GEN B = gmul(sqrtr( divrr(powrs(mppi(DEFAULTPREC), r2-3), limx) ),
2037 : gmul2n(powuu(5, r1), bit + r2));
2038 189 : return gc_long(av, get_i0(r1, r2, B, limx));
2039 : }
2040 :
2041 : static void
2042 189 : GetST0(GEN bnr, GEN *pS, GEN *pT, GEN CR, long prec)
2043 : {
2044 : pari_sp av, av1, av2;
2045 : long n, j, k, jc, n0, prec2, i0, r1, r2, ncond;
2046 189 : GEN nf = bnr_get_nf(bnr);
2047 189 : GEN vChar = gel(CR,1), dataCR = gel(CR,2), N0, C, an, limx, S, T;
2048 : LISTray LIST;
2049 : ST_t cScT;
2050 :
2051 189 : ST_alloc(pS, pT, lg(dataCR), prec); T = *pT; S = *pS;
2052 189 : av = avma;
2053 189 : nf_get_sign(nf,&r1,&r2);
2054 189 : ncond = lg(vChar)-1;
2055 189 : C = cgetg(ncond+1, t_VEC);
2056 189 : N0 = cgetg(ncond+1, t_VECSMALL);
2057 189 : n0 = 0;
2058 189 : limx = zeta_get_limx(r1, r2, prec2nbits(prec));
2059 497 : for (j = 1; j <= ncond; j++)
2060 : {
2061 308 : GEN dtcr = gel(dataCR, mael(vChar,j,1)), c = ch_C(dtcr);
2062 308 : gel(C,j) = c;
2063 308 : N0[j] = zeta_get_N0(c, limx);
2064 308 : if (n0 < N0[j]) n0 = N0[j];
2065 : }
2066 189 : cScT.i0 = i0 = zeta_get_i0(r1, r2, prec2nbits(prec), limx);
2067 189 : if (DEBUGLEVEL>1) err_printf("i0 = %ld, N0 = %ld\n",i0, n0);
2068 189 : InitPrimes(bnr, n0, &LIST);
2069 189 : prec2 = precdbl(prec) + EXTRAPREC64;
2070 189 : cScT.powracpi = powersr(sqrtr(mppi(prec2)), r1);
2071 189 : cScT.cS = cgetg(n0+1, t_VEC);
2072 189 : cScT.cT = cgetg(n0+1, t_VEC);
2073 707013 : for (j=1; j<=n0; j++) gel(cScT.cS,j) = gel(cScT.cT,j) = NULL;
2074 :
2075 189 : av1 = avma;
2076 497 : for (jc = 1; jc <= ncond; jc++)
2077 : {
2078 308 : GEN LChar = gel(vChar,jc);
2079 308 : long nChar = lg(LChar)-1, N = N0[jc];
2080 :
2081 : /* Can discard completely a conductor if all chars satisfy L(0,chi) = 0
2082 : * Not sure whether this is possible. */
2083 308 : if (DEBUGLEVEL>1)
2084 0 : err_printf("* conductor no %ld/%ld (N = %ld)\n\tInit: ", jc,ncond,N);
2085 :
2086 308 : cScT.c1 = gel(C,jc);
2087 308 : init_cScT(&cScT, gel(dataCR, LChar[1]), N, prec2);
2088 308 : av2 = avma;
2089 875 : for (k = 1; k <= nChar; k++)
2090 : {
2091 567 : long d, c, u = LChar[k];
2092 567 : GEN dtcr = gel(dataCR, u), z, s, t;
2093 : int **matan;
2094 :
2095 567 : if (!ch_comp(dtcr)) continue;
2096 560 : if (DEBUGLEVEL>1) err_printf("\tchar no: %ld (%ld/%ld)\n", u,k,nChar);
2097 560 : z = gel(ch_CHI(dtcr), 2);
2098 560 : d = ch_phideg(dtcr); s = t = gen_0;
2099 560 : matan = ComputeCoeff(dtcr, &LIST, N, d);
2100 1741087 : for (n = 1, c = 0; n <= N; n++)
2101 1740527 : if ((an = EvalCoeff(z, matan[n], d)))
2102 : {
2103 500087 : get_cS_cT(&cScT, n);
2104 500087 : s = gadd(s, gmul(an, gel(cScT.cS,n)));
2105 500087 : t = gadd(t, gmul(an, gel(cScT.cT,n)));
2106 500087 : if (++c == 256) { gerepileall(av2,2, &s,&t); c = 0; }
2107 : }
2108 560 : gaffect(s, gel(S,u));
2109 560 : gaffect(conj_i(t), gel(T,u));
2110 560 : FreeMat(matan, N); set_avma(av2);
2111 : }
2112 308 : if (DEBUGLEVEL>1) err_printf("\n");
2113 308 : set_avma(av1);
2114 : }
2115 189 : clear_cScT(&cScT, n0);
2116 189 : set_avma(av);
2117 189 : }
2118 :
2119 : static void
2120 413 : GetST(GEN bnr, GEN *pS, GEN *pT, GEN CR, long prec)
2121 : {
2122 413 : if (nf_get_degree(bnr_get_nf(bnr)) == 2)
2123 259 : QuadGetST(bnr, pS, pT, CR, prec);
2124 : else
2125 154 : GetST0(bnr, pS, pT, CR, prec);
2126 413 : }
2127 :
2128 : /*******************************************************************/
2129 : /* */
2130 : /* Class fields of real quadratic fields using Stark units */
2131 : /* */
2132 : /*******************************************************************/
2133 : /* compute the Hilbert class field using genus class field theory when
2134 : the exponent of the class group is exactly 2 (trivial group not covered) */
2135 : /* Cf Herz, Construction of class fields, LNM 21, Theorem 1 (VII-6) */
2136 : static GEN
2137 14 : GenusFieldQuadReal(GEN disc)
2138 : {
2139 14 : GEN T = NULL, p0 = NULL, P = gel(Z_factor(disc), 1);
2140 14 : long i, i0 = 0, l = lg(P);
2141 :
2142 42 : for (i = 1; i < l; i++)
2143 : {
2144 35 : GEN p = gel(P,i);
2145 35 : if (mod4(p) == 3) { p0 = p; i0 = i; break; }
2146 : }
2147 14 : l--; /* remove last prime */
2148 14 : if (i0 == l) l--; /* ... remove p0 and last prime */
2149 49 : for (i = 1; i < l; i++)
2150 : {
2151 35 : GEN p = gel(P,i), d, t;
2152 35 : if (i == i0) continue;
2153 28 : if (absequaliu(p, 2))
2154 14 : switch (mod32(disc))
2155 : {
2156 14 : case 8: d = gen_2; break;
2157 0 : case 24: d = shifti(p0, 1); break;
2158 0 : default: d = p0; break;
2159 : }
2160 : else
2161 14 : d = (mod4(p) == 1)? p: mulii(p0, p);
2162 28 : t = mkpoln(3, gen_1, gen_0, negi(d)); /* x^2 - d */
2163 28 : T = T? ZX_composedsum(T, t): t;
2164 : }
2165 14 : return polredbest(T, 0);
2166 : }
2167 : static GEN
2168 406 : GenusFieldQuadImag(GEN disc)
2169 : {
2170 406 : GEN T = NULL, P = gel(absZ_factor(disc), 1);
2171 406 : long i, n = lg(P)-1;
2172 1183 : for (i = 1; i < n; i++) /* remove last prime */
2173 : {
2174 777 : GEN p = gel(P,i), d, t;
2175 777 : if (absequaliu(p, 2))
2176 231 : switch (mod32(disc))
2177 : {
2178 56 : case 24: d = gen_2; break; /* disc = 8 mod 32 */
2179 42 : case 8: d = gen_m2; break; /* disc =-8 mod 32 */
2180 133 : default: d = gen_m1; break;
2181 : }
2182 : else
2183 546 : d = (mod4(p) == 1)? p: negi(p);
2184 777 : t = mkpoln(3, gen_1, gen_0, negi(d)); /* x^2 - d */
2185 777 : T = T? ZX_composedsum(T, t): t;
2186 : }
2187 406 : return polredbest(T, 0);
2188 : }
2189 :
2190 : /* if flag != 0, computes a fast and crude approximation of the result */
2191 : static GEN
2192 644 : AllStark(GEN data, long flag, long newprec)
2193 : {
2194 644 : const long BND = 300;
2195 644 : long cl, i, j, cpt = 0, N, h, v, n, r1, r2, den;
2196 : pari_sp av, av2;
2197 : int **matan;
2198 644 : GEN bnr = gel(data,1), nf = bnr_get_nf(bnr), p1, p2, S, T;
2199 644 : GEN CR = gel(data,4), dataCR = gel(CR,2);
2200 : GEN polrelnum, polrel, Lp, W, vzeta, C, cond1, L1, an;
2201 : LISTray LIST;
2202 : pari_timer ti;
2203 :
2204 644 : nf_get_sign(nf, &r1,&r2);
2205 644 : N = nf_get_degree(nf);
2206 644 : cond1 = gel(bnr_get_mod(bnr), 2);
2207 :
2208 644 : v = 1;
2209 1344 : while (gequal1(gel(cond1,v))) v++;
2210 644 : cl = lg(dataCR)-1;
2211 644 : h = itos(ZM_det_triangular(gel(data,2))) >> 1;
2212 :
2213 672 : LABDOUB:
2214 672 : if (DEBUGLEVEL) timer_start(&ti);
2215 672 : av = avma;
2216 672 : W = AllArtinNumbers(CR, newprec);
2217 672 : if (DEBUGLEVEL) timer_printf(&ti,"Compute W");
2218 672 : Lp = cgetg(cl + 1, t_VEC);
2219 672 : if (!flag)
2220 : {
2221 350 : GetST(bnr, &S, &T, CR, newprec);
2222 350 : if (DEBUGLEVEL) timer_printf(&ti, "S&T");
2223 1519 : for (i = 1; i <= cl; i++)
2224 : {
2225 1169 : GEN chi = gel(dataCR, i), v = gen_0;
2226 1169 : if (ch_comp(chi))
2227 1155 : v = gel(GetValue(chi, gel(W,i), gel(S,i), gel(T,i), 2, newprec), 2);
2228 1169 : gel(Lp, i) = v;
2229 : }
2230 : }
2231 : else
2232 : { /* compute a crude approximation of the result */
2233 322 : C = cgetg(cl + 1, t_VEC);
2234 1351 : for (i = 1; i <= cl; i++) gel(C,i) = ch_C(gel(dataCR, i));
2235 322 : n = zeta_get_N0(vecmax(C), zeta_get_limx(r1, r2, prec2nbits(newprec)));
2236 322 : if (n > BND) n = BND;
2237 322 : if (DEBUGLEVEL) err_printf("N0 in QuickPol: %ld \n", n);
2238 322 : InitPrimes(bnr, n, &LIST);
2239 :
2240 322 : L1 = cgetg(cl+1, t_VEC);
2241 : /* use L(1) = sum (an / n) */
2242 1351 : for (i = 1; i <= cl; i++)
2243 : {
2244 1029 : GEN dtcr = gel(dataCR,i);
2245 1029 : long d = ch_phideg(dtcr);
2246 1029 : matan = ComputeCoeff(dtcr, &LIST, n, d);
2247 1029 : av2 = avma;
2248 1029 : p1 = real_0(newprec); p2 = gel(ch_CHI(dtcr), 2);
2249 303079 : for (j = 1; j <= n; j++)
2250 302050 : if ( (an = EvalCoeff(p2, matan[j], d)) )
2251 116697 : p1 = gadd(p1, gdivgu(an, j));
2252 1029 : gel(L1,i) = gerepileupto(av2, p1);
2253 1029 : FreeMat(matan, n);
2254 : }
2255 322 : p1 = gmul2n(powruhalf(mppi(newprec), N-2), 1);
2256 :
2257 1351 : for (i = 1; i <= cl; i++)
2258 : {
2259 : long r;
2260 1029 : GEN WW, A = AChi(gel(dataCR,i), &r, 0, newprec);
2261 1029 : WW = gmul(gel(C,i), gmul(A, gel(W,i)));
2262 1029 : gel(Lp,i) = gdiv(gmul(WW, conj_i(gel(L1,i))), p1);
2263 : }
2264 : }
2265 :
2266 672 : p1 = gel(data,3);
2267 672 : den = flag ? h: 2*h;
2268 672 : vzeta = cgetg(h + 1, t_VEC);
2269 4333 : for (i = 1; i <= h; i++)
2270 : {
2271 3661 : GEN z = gen_0, sig = gel(p1,i);
2272 18837 : for (j = 1; j <= cl; j++)
2273 : {
2274 15176 : GEN dtcr = gel(dataCR,j), CHI = ch_CHI(dtcr);
2275 15176 : GEN t = mulreal(gel(Lp,j), CharEval(CHI, sig));
2276 15176 : if (chi_get_deg(CHI) != 2) t = gmul2n(t, 1); /* character not real */
2277 15176 : z = gadd(z, t);
2278 : }
2279 3661 : gel(vzeta,i) = gmul2n(gcosh(gdivgu(z,den), newprec), 1);
2280 : }
2281 672 : polrelnum = roots_to_pol(vzeta, 0);
2282 672 : if (DEBUGLEVEL)
2283 : {
2284 0 : if (DEBUGLEVEL>1) {
2285 0 : err_printf("polrelnum = %Ps\n", polrelnum);
2286 0 : err_printf("zetavalues = %Ps\n", vzeta);
2287 0 : if (!flag)
2288 0 : err_printf("Checking the square-root of the Stark unit...\n");
2289 : }
2290 0 : timer_printf(&ti, "Compute %s", flag? "quickpol": "polrelnum");
2291 : }
2292 672 : if (flag) return gerepilecopy(av, polrelnum);
2293 :
2294 : /* try to recognize this polynomial */
2295 350 : polrel = RecCoeff(nf, polrelnum, v, newprec);
2296 350 : if (!polrel)
2297 : {
2298 1785 : for (j = 1; j <= h; j++)
2299 1540 : gel(vzeta,j) = gsubgs(gsqr(gel(vzeta,j)), 2);
2300 245 : polrelnum = roots_to_pol(vzeta, 0);
2301 245 : if (DEBUGLEVEL)
2302 : {
2303 0 : if (DEBUGLEVEL>1) {
2304 0 : err_printf("It's not a square...\n");
2305 0 : err_printf("polrelnum = %Ps\n", polrelnum);
2306 : }
2307 0 : timer_printf(&ti, "Compute polrelnum");
2308 : }
2309 245 : polrel = RecCoeff(nf, polrelnum, v, newprec);
2310 : }
2311 350 : if (!polrel) /* FAILED */
2312 : {
2313 28 : const long EXTRA_BITS = 64;
2314 : long incr_pr;
2315 28 : if (++cpt >= 3) pari_err_PREC( "stark (computation impossible)");
2316 : /* estimate needed precision */
2317 28 : incr_pr = prec2nbits(gprecision(polrelnum))- gexpo(polrelnum);
2318 28 : if (incr_pr < 0) incr_pr = -incr_pr + EXTRA_BITS;
2319 28 : newprec += nbits2extraprec(maxss(3*EXTRA_BITS, cpt*incr_pr));
2320 28 : if (DEBUGLEVEL) pari_warn(warnprec, "AllStark", newprec);
2321 28 : CharNewPrec(data, newprec);
2322 28 : nf = bnr_get_nf(ch_bnr(gel(dataCR,1)));
2323 28 : goto LABDOUB;
2324 : }
2325 :
2326 322 : if (DEBUGLEVEL) {
2327 0 : if (DEBUGLEVEL>1) err_printf("polrel = %Ps\n", polrel);
2328 0 : timer_printf(&ti, "Recpolnum");
2329 : }
2330 322 : return gerepilecopy(av, polrel);
2331 : }
2332 :
2333 : /********************************************************************/
2334 : /* Main functions */
2335 : /********************************************************************/
2336 : static GEN
2337 0 : bnrstark_cyclic(GEN bnr, GEN dtQ, long prec)
2338 : {
2339 0 : GEN v, vH, cyc = gel(dtQ,2), U = gel(dtQ,3), M = ZM_inv(U, NULL);
2340 0 : long i, j, l = lg(M);
2341 :
2342 : /* M = indep. generators of Cl_f/subgp, restrict to cyclic components */
2343 0 : vH = cgetg(l, t_VEC);
2344 0 : for (i = j = 1; i < l; i++)
2345 : {
2346 0 : if (is_pm1(gel(cyc,i))) break;
2347 0 : gel(vH, j++) = ZM_hnfmodid(vecsplice(M,i), cyc);
2348 : }
2349 0 : setlg(vH, j); v = cgetg(l, t_VEC);
2350 0 : for (i = 1; i < j; i++) gel(v,i) = bnrstark(bnr, gel(vH,i), prec);
2351 0 : return v;
2352 : }
2353 : GEN
2354 203 : bnrstark(GEN bnr, GEN subgrp, long prec)
2355 : {
2356 : long newprec;
2357 203 : pari_sp av = avma;
2358 : GEN nf, data, dtQ;
2359 :
2360 : /* check the bnr */
2361 203 : checkbnr(bnr); nf = bnr_get_nf(bnr);
2362 203 : if (nf_get_degree(nf) == 1) return galoissubcyclo(bnr, subgrp, 0, 0);
2363 203 : if (!nf_get_varn(nf))
2364 0 : pari_err_PRIORITY("bnrstark", nf_get_pol(nf), "=", 0);
2365 203 : if (nf_get_r2(nf)) pari_err_DOMAIN("bnrstark", "r2", "!=", gen_0, nf);
2366 :
2367 : /* compute bnr(conductor) */
2368 196 : bnr_subgroup_sanitize(&bnr, &subgrp);
2369 196 : if (gequal1(ZM_det_triangular(subgrp))) { set_avma(av); return pol_x(0); }
2370 :
2371 : /* check the class field */
2372 196 : if (!gequal0(gel(bnr_get_mod(bnr), 2)))
2373 7 : pari_err_DOMAIN("bnrstark", "r2(class field)", "!=", gen_0, bnr);
2374 :
2375 : /* find a suitable extension N */
2376 189 : dtQ = InitQuotient(subgrp);
2377 189 : data = FindModulus(bnr, dtQ, &newprec);
2378 189 : if (!data) return gerepileupto(av, bnrstark_cyclic(bnr, dtQ, prec));
2379 189 : if (DEBUGLEVEL>1 && newprec > prec)
2380 0 : err_printf("new precision: %ld\n", newprec);
2381 189 : return gerepileupto(av, AllStark(data, 0, newprec));
2382 : }
2383 :
2384 : /* For each character of Cl(bnr)/subgp, compute L(1, chi) (or equivalently
2385 : * the first nonzero term c(chi) of the expansion at s = 0).
2386 : * If flag & 1: compute the value at s = 1 (for nontrivial characters),
2387 : * else compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is
2388 : * the order of L(s, chi) at s = 0.
2389 : * If flag & 2: compute the value of the L-function L_S(s, chi) where S is the
2390 : * set of places dividing the modulus of bnr (and the infinite places),
2391 : * else
2392 : * compute the value of the primitive L-function attached to chi,
2393 : * If flag & 4: return also the character */
2394 : GEN
2395 70 : bnrL1(GEN bnr, GEN subgp, long flag, long prec)
2396 : {
2397 : GEN L1, ch, Qt, z;
2398 : long l, h;
2399 70 : pari_sp av = avma;
2400 :
2401 70 : checkbnr(bnr);
2402 70 : if (flag < 0 || flag > 8) pari_err_FLAG("bnrL1");
2403 :
2404 70 : subgp = bnr_subgroup_check(bnr, subgp, NULL);
2405 70 : if (!subgp) subgp = diagonal_shallow(bnr_get_cyc(bnr));
2406 :
2407 70 : Qt = InitQuotient(subgp);
2408 70 : ch = AllChars(bnr, Qt, 0); l = lg(ch);
2409 70 : h = itou(gel(Qt,1));
2410 70 : L1 = cgetg((flag&1)? h: h+1, t_VEC);
2411 70 : if (l > 1)
2412 : {
2413 63 : GEN W, S, T, CR = InitChar(bnr, ch, 1, prec), dataCR = gel(CR,2);
2414 : long i, j;
2415 :
2416 63 : GetST(bnr, &S, &T, CR, prec);
2417 63 : W = AllArtinNumbers(CR, prec);
2418 301 : for (i = j = 1; i < l; i++)
2419 : {
2420 238 : GEN chi = gel(ch,i);
2421 238 : z = GetValue(gel(dataCR,i), gel(W,i), gel(S,i), gel(T,i), flag, prec);
2422 238 : gel(L1,j++) = (flag & 4)? mkvec2(gel(chi,1), z): z;
2423 238 : if (lg(chi) == 4)
2424 : { /* nonreal */
2425 133 : z = conj_i(z);
2426 133 : gel(L1, j++) = (flag & 4)? mkvec2(gel(chi,3), z): z;
2427 : }
2428 : }
2429 : }
2430 70 : if (!(flag & 1))
2431 : { /* trivial character */
2432 35 : z = GetValue1(bnr, flag & 2, prec);
2433 35 : if (flag & 4)
2434 : {
2435 0 : GEN chi = zerovec(lg(bnr_get_cyc(bnr))-1);
2436 0 : z = mkvec2(chi, z);
2437 : }
2438 35 : gel(L1,h) = z;
2439 : }
2440 70 : return gerepilecopy(av, L1);
2441 : }
2442 :
2443 : /*******************************************************************/
2444 : /* */
2445 : /* Hilbert and Ray Class field using Stark */
2446 : /* */
2447 : /*******************************************************************/
2448 : /* P in A[x,y], deg_y P < 2, return P0 and P1 in A[x] such that P = P0 + P1 y */
2449 : static void
2450 133 : split_pol_quad(GEN P, GEN *gP0, GEN *gP1)
2451 : {
2452 133 : long i, l = lg(P);
2453 133 : GEN P0 = cgetg(l, t_POL), P1 = cgetg(l, t_POL);
2454 133 : P0[1] = P1[1] = P[1];
2455 1211 : for (i = 2; i < l; i++)
2456 : {
2457 1078 : GEN c = gel(P,i), c0 = c, c1 = gen_0;
2458 1078 : if (typ(c) == t_POL) /* write c = c1 y + c0 */
2459 945 : switch(degpol(c))
2460 : {
2461 0 : case -1: c0 = gen_0; break;
2462 945 : default: c1 = gel(c,3); /* fall through */
2463 945 : case 0: c0 = gel(c,2); break;
2464 : }
2465 1078 : gel(P0,i) = c0; gel(P1,i) = c1;
2466 : }
2467 133 : *gP0 = normalizepol_lg(P0, l);
2468 133 : *gP1 = normalizepol_lg(P1, l);
2469 133 : }
2470 :
2471 : /* k = nf quadratic field, P relative equation of H_k (Hilbert class field)
2472 : * return T in Z[X], such that H_k / Q is the compositum of Q[X]/(T) and k */
2473 : static GEN
2474 133 : makescind(GEN nf, GEN P)
2475 : {
2476 133 : GEN Pp, p, pol, G, L, a, roo, P0,P1, Ny,Try, nfpol = nf_get_pol(nf);
2477 : long i, is_P;
2478 :
2479 133 : P = lift_shallow(P);
2480 133 : split_pol_quad(P, &P0, &P1);
2481 : /* P = P0 + y P1, Norm_{k/Q}(P) = P0^2 + Tr y P0P1 + Ny P1^2, irreducible/Q */
2482 133 : Ny = gel(nfpol, 2);
2483 133 : Try = negi(gel(nfpol, 3));
2484 133 : pol = RgX_add(RgX_sqr(P0), RgX_Rg_mul(RgX_sqr(P1), Ny));
2485 133 : if (signe(Try)) pol = RgX_add(pol, RgX_Rg_mul(RgX_mul(P0,P1), Try));
2486 : /* pol = rnfequation(nf, P); */
2487 133 : G = galoisinit(pol, NULL);
2488 133 : L = gal_get_group(G);
2489 133 : p = gal_get_p(G);
2490 133 : a = FpX_oneroot(nfpol, p);
2491 : /* P mod a prime \wp above p (which splits) */
2492 133 : Pp = FpXY_evalx(P, a, p);
2493 133 : roo = gal_get_roots(G);
2494 133 : is_P = gequal0( FpX_eval(Pp, remii(gel(roo,1),p), p) );
2495 : /* each roo[i] mod p is a root of P or (exclusive) tau(P) mod \wp */
2496 : /* record whether roo[1] is a root of P or tau(P) */
2497 :
2498 1022 : for (i = 1; i < lg(L); i++)
2499 : {
2500 1022 : GEN perm = gel(L,i);
2501 1022 : long k = perm[1]; if (k == 1) continue;
2502 889 : k = gequal0( FpX_eval(Pp, remii(gel(roo,k),p), p) );
2503 : /* roo[k] is a root of the other polynomial */
2504 889 : if (k != is_P)
2505 : {
2506 133 : ulong o = perm_orderu(perm);
2507 133 : if (o != 2) perm = perm_powu(perm, o >> 1);
2508 : /* perm has order two and doesn't belong to Gal(H_k/k) */
2509 133 : return polredbest(galoisfixedfield(G, perm, 1, varn(P)), 0);
2510 : }
2511 : }
2512 0 : pari_err_BUG("makescind");
2513 : return NULL; /*LCOV_EXCL_LINE*/
2514 : }
2515 :
2516 : /* pbnf = NULL if no bnf is needed, f = NULL may be passed for a trivial
2517 : * conductor */
2518 : static void
2519 847 : quadray_init(GEN *pD, GEN *pbnf, long prec)
2520 : {
2521 847 : GEN D = *pD, nf, bnf = NULL;
2522 847 : if (typ(D) == t_INT)
2523 : {
2524 : int isfund;
2525 812 : if (pbnf) {
2526 252 : bnf = Buchall(quadpoly0(D, 1), nf_FORCE, prec);
2527 252 : nf = bnf_get_nf(bnf);
2528 252 : isfund = equalii(D, nf_get_disc(nf));
2529 : }
2530 : else
2531 560 : isfund = Z_isfundamental(D);
2532 812 : if (!isfund) pari_err_DOMAIN("quadray", "isfundamental(D)", "=",gen_0, D);
2533 : }
2534 : else
2535 : {
2536 35 : bnf = checkbnf(D);
2537 35 : nf = bnf_get_nf(bnf);
2538 35 : if (nf_get_degree(nf) != 2)
2539 7 : pari_err_DOMAIN("quadray", "degree", "!=", gen_2, nf_get_pol(nf));
2540 28 : D = nf_get_disc(nf);
2541 : }
2542 833 : if (pbnf) *pbnf = bnf;
2543 833 : *pD = D;
2544 833 : }
2545 :
2546 : /* compute the polynomial over Q of the Hilbert class field of
2547 : Q(sqrt(D)) where D is a positive fundamental discriminant */
2548 : static GEN
2549 147 : quadhilbertreal(GEN D, long prec)
2550 : {
2551 : GEN bnf, bnr, dtQ, data, M;
2552 : long newprec;
2553 : pari_timer T;
2554 :
2555 147 : quadray_init(&D, &bnf, prec);
2556 147 : switch(itou_or_0(cyc_get_expo(bnf_get_cyc(bnf))))
2557 : {
2558 0 : case 1: return pol_x(0);
2559 14 : case 2: return GenusFieldQuadReal(D);
2560 : }
2561 133 : bnr = Buchray(bnf, gen_1, nf_INIT);
2562 133 : M = diagonal_shallow(bnr_get_cyc(bnr));
2563 133 : dtQ = InitQuotient(M);
2564 :
2565 133 : if (DEBUGLEVEL) timer_start(&T);
2566 133 : data = FindModulus(bnr, dtQ, &newprec);
2567 133 : if (DEBUGLEVEL) timer_printf(&T,"FindModulus");
2568 133 : if (!data) return bnrstark_cyclic(bnr, dtQ, prec);
2569 133 : return makescind(bnf_get_nf(bnf), AllStark(data, 0, newprec));
2570 : }
2571 :
2572 : /*******************************************************************/
2573 : /* */
2574 : /* Hilbert and Ray Class field using CM (Schertz) */
2575 : /* */
2576 : /*******************************************************************/
2577 : /* form^2 = 1 ? */
2578 : static int
2579 813 : hasexp2(GEN form)
2580 : {
2581 813 : GEN a = gel(form,1), b = gel(form,2), c = gel(form,3);
2582 813 : return !signe(b) || absequalii(a,b) || equalii(a,c);
2583 : }
2584 : static int
2585 1323 : uhasexp2(GEN form)
2586 : {
2587 1323 : long a = form[1], b = form[2], c = form[3];
2588 1323 : return !b || a == labs(b) || a == c;
2589 : }
2590 :
2591 : GEN
2592 455 : qfbforms(GEN D)
2593 : {
2594 455 : ulong d = itou(D), dover3 = d/3, t, b2, a, b, c, h;
2595 455 : GEN L = cgetg((long)(sqrt((double)d) * log2(d)), t_VEC);
2596 455 : b2 = b = (d&1); h = 0;
2597 455 : if (!b) /* b = 0 treated separately to avoid special cases */
2598 : {
2599 252 : t = d >> 2; /* (b^2 - D) / 4*/
2600 2954 : for (a=1; a*a<=t; a++)
2601 2702 : if (c = t/a, t == c*a) gel(L,++h) = mkvecsmall3(a,0,c);
2602 252 : b = 2; b2 = 4;
2603 : }
2604 : /* now b > 0, b = D (mod 2) */
2605 8078 : for ( ; b2 <= dover3; b += 2, b2 = b*b)
2606 : {
2607 7623 : t = (b2 + d) >> 2; /* (b^2 - D) / 4*/
2608 : /* b = a */
2609 7623 : if (c = t/b, t == c*b) gel(L,++h) = mkvecsmall3(b,b,c);
2610 : /* b < a < c */
2611 1912029 : for (a = b+1; a*a < t; a++)
2612 1904406 : if (c = t/a, t == c*a)
2613 : {
2614 1057 : gel(L,++h) = mkvecsmall3(a, b,c);
2615 1057 : gel(L,++h) = mkvecsmall3(a,-b,c);
2616 : }
2617 : /* a = c */
2618 7623 : if (a * a == t) gel(L,++h) = mkvecsmall3(a,b,a);
2619 : }
2620 455 : setlg(L,h+1); return L;
2621 : }
2622 :
2623 : /* gcd(n, 24) */
2624 : static long
2625 813 : GCD24(long n)
2626 : {
2627 813 : switch(n % 24)
2628 : {
2629 35 : case 0: return 24;
2630 35 : case 1: return 1;
2631 28 : case 2: return 2;
2632 0 : case 3: return 3;
2633 119 : case 4: return 4;
2634 0 : case 5: return 1;
2635 105 : case 6: return 6;
2636 0 : case 7: return 1;
2637 0 : case 8: return 8;
2638 0 : case 9: return 3;
2639 91 : case 10: return 2;
2640 0 : case 11: return 1;
2641 119 : case 12: return 12;
2642 0 : case 13: return 1;
2643 0 : case 14: return 2;
2644 0 : case 15: return 3;
2645 91 : case 16: return 8;
2646 0 : case 17: return 1;
2647 92 : case 18: return 6;
2648 0 : case 19: return 1;
2649 0 : case 20: return 4;
2650 0 : case 21: return 3;
2651 98 : case 22: return 2;
2652 0 : case 23: return 1;
2653 0 : default: return 0;
2654 : }
2655 : }
2656 :
2657 : struct gpq_data {
2658 : long p, q;
2659 : GEN sqd; /* sqrt(D), t_REAL */
2660 : GEN u, D;
2661 : GEN pq, pq2; /* p*q, 2*p*q */
2662 : GEN qfpq ; /* class of \P * \Q */
2663 : };
2664 :
2665 : /* find P and Q two non principal prime ideals (above p <= q) such that
2666 : * cl(P) = cl(Q) if P,Q have order 2 in Cl(K).
2667 : * Ensure that e = 24 / gcd(24, (p-1)(q-1)) = 1 */
2668 : /* D t_INT, discriminant */
2669 : static void
2670 49 : init_pq(GEN D, struct gpq_data *T)
2671 : {
2672 49 : const long Np = 6547; /* N.B. primepi(50000) = 5133 */
2673 49 : const ulong maxq = 50000;
2674 49 : GEN listp = cgetg(Np + 1, t_VECSMALL); /* primes p */
2675 49 : GEN listP = cgetg(Np + 1, t_VEC); /* primeform(p) if of order 2, else NULL */
2676 49 : GEN gcd24 = cgetg(Np + 1, t_VECSMALL); /* gcd(p-1, 24) */
2677 : forprime_t S;
2678 49 : long l = 1;
2679 49 : double best = 0.;
2680 : ulong q;
2681 :
2682 49 : u_forprime_init(&S, 2, ULONG_MAX);
2683 49 : T->D = D;
2684 49 : T->p = T->q = 0;
2685 : for(;;)
2686 1777 : {
2687 : GEN Q;
2688 : long i, gcdq, mod;
2689 : int order2, store;
2690 : double t;
2691 :
2692 1826 : q = u_forprime_next(&S);
2693 1826 : if (best > 0 && q >= maxq)
2694 : {
2695 0 : if (DEBUGLEVEL)
2696 0 : pari_warn(warner,"possibly suboptimal (p,q) for D = %Ps", D);
2697 0 : break;
2698 : }
2699 1826 : if (kroiu(D, q) < 0) continue; /* inert */
2700 890 : Q = qfbred_i(primeform_u(D, q));
2701 890 : if (is_pm1(gel(Q,1))) continue; /* Q | q is principal */
2702 :
2703 813 : store = 1;
2704 813 : order2 = hasexp2(Q);
2705 813 : gcd24[l] = gcdq = GCD24(q-1);
2706 813 : mod = 24 / gcdq; /* mod must divide p-1 otherwise e > 1 */
2707 813 : listp[l] = q;
2708 813 : gel(listP,l) = order2 ? Q : NULL;
2709 813 : t = (q+1)/(double)(q-1);
2710 2129 : for (i = 1; i < l; i++) /* try all (p, q), p < q in listp */
2711 : {
2712 1660 : long p = listp[i], gcdp = gcd24[i];
2713 : double b;
2714 : /* P,Q order 2 => cl(Q) = cl(P) */
2715 1660 : if (order2 && gel(listP,i) && !gequal(gel(listP,i), Q)) continue;
2716 1653 : if (gcdp % gcdq == 0) store = 0; /* already a better one in the list */
2717 1653 : if ((p-1) % mod) continue;
2718 :
2719 344 : b = (t*(p+1)) / (p-1); /* (p+1)(q+1) / (p-1)(q-1) */
2720 344 : if (b > best) {
2721 98 : store = 0; /* (p,q) always better than (q,r) for r >= q */
2722 98 : best = b; T->q = q; T->p = p;
2723 98 : if (DEBUGLEVEL>2) err_printf("p,q = %ld,%ld\n", p, q);
2724 : }
2725 : /* won't improve with this q as largest member */
2726 344 : if (best > 0) break;
2727 : }
2728 : /* if !store or (q,r) won't improve on current best pair, forget that q */
2729 813 : if (store && t*t > best)
2730 119 : if (++l >= Np) pari_err_BUG("quadhilbert (not enough primes)");
2731 813 : if (!best) /* (p,q) with p < q always better than (q,q) */
2732 : { /* try (q,q) */
2733 140 : if (gcdq >= 12 && umodiu(D, q)) /* e = 1 and unramified */
2734 : {
2735 7 : double b = (t*q) / (q-1); /* q(q+1) / (q-1)^2 */
2736 7 : if (b > best) {
2737 7 : best = b; T->q = T->p = q;
2738 7 : if (DEBUGLEVEL>2) err_printf("p,q = %ld,%ld\n", q, q);
2739 : }
2740 : }
2741 : }
2742 : /* If (p1+1)(q+1) / (p1-1)(q-1) <= best, we can no longer improve
2743 : * even with best p : stop */
2744 813 : if ((listp[1]+1)*t <= (listp[1]-1)*best) break;
2745 : }
2746 49 : if (DEBUGLEVEL>1)
2747 0 : err_printf("(p, q) = %ld, %ld; gain = %f\n", T->p, T->q, 12*best);
2748 49 : }
2749 :
2750 : static GEN
2751 4102 : gpq(GEN form, struct gpq_data *T)
2752 : {
2753 4102 : pari_sp av = avma;
2754 4102 : long a = form[1], b = form[2], c = form[3], p = T->p, q = T->q;
2755 : GEN form2, w, z;
2756 4102 : int fl, real = 0;
2757 :
2758 4102 : form2 = qfbcomp_i(T->qfpq, mkqfb(stoi(a), stoi(-b), stoi(c), T->D));
2759 : /* form2 and form yield complex conjugate roots : only compute for the
2760 : * lexicographically smallest of the 2 */
2761 4102 : fl = cmpis(gel(form2,1), a);
2762 4102 : if (fl <= 0)
2763 : {
2764 2156 : if (fl < 0) return NULL;
2765 210 : fl = cmpis(gel(form2,2), b);
2766 210 : if (fl <= 0)
2767 : {
2768 147 : if (fl < 0) return NULL;
2769 : /* form == form2 : real root */
2770 84 : real = 1;
2771 : }
2772 : }
2773 :
2774 2093 : if (p == 2) { /* (a,b,c) = (1,1,0) mod 2 ? */
2775 203 : if (a % q == 0 && (a & b & 1) && !(c & 1))
2776 : { /* apply S : make sure that (a,b,c) represents odd values */
2777 0 : lswap(a,c); b = -b;
2778 : }
2779 : }
2780 2093 : if (a % p == 0 || a % q == 0)
2781 : { /* apply T^k, look for c' = a k^2 + b k + c coprime to N */
2782 595 : while (c % p == 0 || c % q == 0)
2783 : {
2784 98 : c += a + b;
2785 98 : b += a << 1;
2786 : }
2787 497 : lswap(a, c); b = -b; /* apply S */
2788 : }
2789 : /* now (a,b,c) ~ form and (a,pq) = 1 */
2790 :
2791 : /* gcd(2a, u) = 2, w = u mod 2pq, -b mod 2a */
2792 2093 : w = Z_chinese(T->u, stoi(-b), T->pq2, utoipos(a << 1));
2793 2093 : z = double_eta_quotient(utoipos(a), w, T->D, T->p, T->q, T->pq, T->sqd);
2794 2093 : if (real && typ(z) == t_COMPLEX) z = gcopy(gel(z, 1));
2795 2093 : return gerepileupto(av, z);
2796 : }
2797 :
2798 : /* returns an equation for the Hilbert class field of Q(sqrt(D)), D < 0
2799 : * fundamental discriminant */
2800 : static GEN
2801 462 : quadhilbertimag(GEN D)
2802 : {
2803 : GEN L, P, Pi, Pr, qfp, u;
2804 : long h, i, prec;
2805 : struct gpq_data T;
2806 : pari_timer ti;
2807 :
2808 462 : if (DEBUGLEVEL>1) timer_start(&ti);
2809 462 : if (lgefint(D) == 3)
2810 462 : switch (D[2]) { /* = |D|; special cases where e > 1 */
2811 7 : case 3:
2812 : case 4:
2813 : case 7:
2814 : case 8:
2815 : case 11:
2816 : case 19:
2817 : case 43:
2818 : case 67:
2819 7 : case 163: return pol_x(0);
2820 : }
2821 455 : L = qfbforms(D);
2822 455 : h = lg(L)-1;
2823 455 : if (! (h & (h - 1))) /* power of 2 */
2824 : { /* check whether > |Cl|/2 elements have order <= 2 ==> 2-elementary */
2825 413 : long lim = (h>>1) + 1;
2826 1729 : for (i=1; i <= lim; i++)
2827 1323 : if (!uhasexp2(gel(L,i))) break;
2828 413 : if (i > lim) return GenusFieldQuadImag(D);
2829 : }
2830 49 : if (DEBUGLEVEL>1) timer_printf(&ti,"class number = %ld",h);
2831 49 : init_pq(D, &T);
2832 49 : qfp = primeform_u(D, T.p);
2833 49 : T.pq = muluu(T.p, T.q);
2834 49 : T.pq2 = shifti(T.pq,1);
2835 49 : if (T.p == T.q)
2836 : {
2837 0 : GEN qfbp2 = qfbcompraw(qfp, qfp);
2838 0 : u = gel(qfbp2,2);
2839 0 : T.u = modii(u, T.pq2);
2840 0 : T.qfpq = qfbred_i(qfbp2);
2841 : }
2842 : else
2843 : {
2844 49 : GEN qfq = primeform_u(D, T.q), bp = gel(qfp,2), bq = gel(qfq,2);
2845 49 : T.u = Z_chinese(bp, bq, utoipos(T.p << 1), utoipos(T.q << 1));
2846 : /* T.u = bp (mod 2p), T.u = bq (mod 2q) */
2847 49 : T.qfpq = qfbcomp_i(qfp, qfq);
2848 : }
2849 : /* u modulo 2pq */
2850 49 : prec = LOWDEFAULTPREC;
2851 49 : Pr = cgetg(h+1,t_VEC);
2852 49 : Pi = cgetg(h+1,t_VEC);
2853 : for(;;)
2854 14 : {
2855 63 : long ex, exmax = 0, r1 = 0, r2 = 0;
2856 63 : pari_sp av0 = avma;
2857 63 : T.sqd = sqrtr_abs(itor(D, prec));
2858 4165 : for (i=1; i<=h; i++)
2859 : {
2860 4102 : GEN s = gpq(gel(L,i), &T);
2861 4102 : if (DEBUGLEVEL>3) err_printf("%ld ", i);
2862 4102 : if (!s) continue;
2863 2093 : if (typ(s) != t_COMPLEX) gel(Pr, ++r1) = s; /* real root */
2864 2009 : else gel(Pi, ++r2) = s;
2865 2093 : ex = gexpo(s); if (ex > 0) exmax += ex;
2866 : }
2867 63 : if (DEBUGLEVEL>1) timer_printf(&ti,"roots");
2868 63 : setlg(Pr, r1+1);
2869 63 : setlg(Pi, r2+1);
2870 63 : P = roots_to_pol_r1(shallowconcat(Pr,Pi), 0, r1);
2871 63 : P = grndtoi(P,&exmax);
2872 63 : if (DEBUGLEVEL>1) timer_printf(&ti,"product, error bits = %ld",exmax);
2873 63 : if (exmax <= -10) break;
2874 14 : set_avma(av0); prec += nbits2extraprec(prec2nbits(DEFAULTPREC)+exmax);
2875 14 : if (DEBUGLEVEL) pari_warn(warnprec,"quadhilbertimag",prec);
2876 : }
2877 49 : return P;
2878 : }
2879 :
2880 : GEN
2881 574 : quadhilbert(GEN D, long prec)
2882 : {
2883 574 : pari_sp av = avma;
2884 574 : GEN d = D;
2885 574 : quadray_init(&d, NULL, 0);
2886 973 : return gerepileupto(av, signe(d)>0? quadhilbertreal(D,prec)
2887 413 : : quadhilbertimag(d));
2888 : }
2889 :
2890 : /* return a vector of all roots of 1 in bnf [not necessarily quadratic] */
2891 : static GEN
2892 70 : getallrootsof1(GEN bnf)
2893 : {
2894 70 : GEN T, u, nf = bnf_get_nf(bnf), tu;
2895 70 : long i, n = bnf_get_tuN(bnf);
2896 :
2897 70 : if (n == 2) {
2898 56 : long N = nf_get_degree(nf);
2899 56 : return mkvec2(scalarcol_shallow(gen_m1, N),
2900 : scalarcol_shallow(gen_1, N));
2901 : }
2902 14 : tu = poltobasis(nf, bnf_get_tuU(bnf));
2903 14 : T = zk_multable(nf, tu);
2904 14 : u = cgetg(n+1, t_VEC); gel(u,1) = tu;
2905 56 : for (i=2; i <= n; i++) gel(u,i) = ZM_ZC_mul(T, gel(u,i-1));
2906 14 : return u;
2907 : }
2908 : /* assume bnr has the right conductor */
2909 : static GEN
2910 70 : get_lambda(GEN bnr)
2911 : {
2912 70 : GEN bnf = bnr_get_bnf(bnr), nf = bnf_get_nf(bnf), pol = nf_get_pol(nf);
2913 70 : GEN f = gel(bnr_get_mod(bnr), 1), labas, lamodf, u;
2914 70 : long a, b, f2, i, lu, v = varn(pol);
2915 :
2916 70 : f2 = 2 * itos(gcoeff(f,1,1));
2917 70 : u = getallrootsof1(bnf); lu = lg(u);
2918 238 : for (i=1; i<lu; i++)
2919 168 : gel(u,i) = ZC_hnfrem(gel(u,i), f); /* roots of 1, mod f */
2920 70 : if (DEBUGLEVEL>1)
2921 0 : err_printf("quadray: looking for [a,b] != unit mod 2f\n[a,b] = ");
2922 168 : for (a=0; a<f2; a++)
2923 2576 : for (b=0; b<f2; b++)
2924 : {
2925 2478 : GEN la = deg1pol_shallow(stoi(a), stoi(b), v); /* ax + b */
2926 2478 : if (umodiu(gnorm(mkpolmod(la, pol)), f2) != 1) continue;
2927 224 : if (DEBUGLEVEL>1) err_printf("[%ld,%ld] ",a,b);
2928 :
2929 224 : labas = poltobasis(nf, la);
2930 224 : lamodf = ZC_hnfrem(labas, f);
2931 469 : for (i=1; i<lu; i++)
2932 399 : if (ZV_equal(lamodf, gel(u,i))) break;
2933 224 : if (i < lu) continue; /* la = unit mod f */
2934 70 : if (DEBUGLEVEL)
2935 : {
2936 0 : if (DEBUGLEVEL>1) err_printf("\n");
2937 0 : err_printf("lambda = %Ps\n",la);
2938 : }
2939 70 : return labas;
2940 : }
2941 0 : pari_err_BUG("get_lambda");
2942 : return NULL;/*LCOV_EXCL_LINE*/
2943 : }
2944 :
2945 : static GEN
2946 8778 : to_approx(GEN nf, GEN a)
2947 : {
2948 8778 : GEN M = nf_get_M(nf);
2949 8778 : return gadd(gel(a,1), gmul(gcoeff(M,1,2),gel(a,2)));
2950 : }
2951 : /* Z-basis for a (over C) */
2952 : static GEN
2953 4354 : get_om(GEN nf, GEN a) {
2954 4354 : return mkvec2(to_approx(nf,gel(a,2)),
2955 4354 : to_approx(nf,gel(a,1)));
2956 : }
2957 :
2958 : /* Compute all elts in class group G = [|G|,c,g], c=cyclic factors, g=gens.
2959 : * Set list[j + 1] = g1^e1...gk^ek where j is the integer
2960 : * ek + ck [ e(k-1) + c(k-1) [... + c2 [e1]]...] */
2961 : static GEN
2962 70 : getallelts(GEN bnr)
2963 : {
2964 : GEN nf, C, c, g, list, pows, gk;
2965 : long lc, i, j, no;
2966 :
2967 70 : nf = bnr_get_nf(bnr);
2968 70 : no = itos( bnr_get_no(bnr) );
2969 70 : c = bnr_get_cyc(bnr);
2970 70 : g = bnr_get_gen_nocheck(bnr); lc = lg(c)-1;
2971 70 : list = cgetg(no+1,t_VEC);
2972 70 : gel(list,1) = matid(nf_get_degree(nf)); /* (1) */
2973 70 : if (!no) return list;
2974 :
2975 70 : pows = cgetg(lc+1,t_VEC);
2976 70 : c = leafcopy(c); settyp(c, t_VECSMALL);
2977 140 : for (i=1; i<=lc; i++)
2978 : {
2979 70 : long k = itos(gel(c,i));
2980 70 : c[i] = k;
2981 70 : gk = cgetg(k, t_VEC); gel(gk,1) = gel(g,i);
2982 4284 : for (j=2; j<k; j++)
2983 4214 : gel(gk,j) = idealmoddivisor(bnr, idealmul(nf, gel(gk,j-1), gel(gk,1)));
2984 70 : gel(pows,i) = gk; /* powers of g[i] */
2985 : }
2986 :
2987 70 : C = cgetg(lc+1, t_VECSMALL); C[1] = c[lc];
2988 70 : for (i=2; i<=lc; i++) C[i] = C[i-1] * c[lc-i+1];
2989 : /* C[i] = c(k-i+1) * ... * ck */
2990 : /* j < C[i+1] <==> j only involves g(k-i)...gk */
2991 70 : i = 1;
2992 4354 : for (j=1; j < C[1]; j++)
2993 4284 : gel(list, j+1) = gmael(pows,lc,j);
2994 70 : while(j<no)
2995 : {
2996 : long k;
2997 : GEN a;
2998 0 : if (j == C[i+1]) i++;
2999 0 : a = gmael(pows,lc-i,j/C[i]);
3000 0 : k = j%C[i] + 1;
3001 0 : if (k > 1) a = idealmoddivisor(bnr, idealmul(nf, a, gel(list,k)));
3002 0 : gel(list, ++j) = a;
3003 : }
3004 70 : return list;
3005 : }
3006 :
3007 : /* x quadratic integer (approximate), recognize it. If error return NULL */
3008 : static GEN
3009 4424 : findbezk(GEN nf, GEN x)
3010 : {
3011 4424 : GEN a,b, M = nf_get_M(nf), u = gcoeff(M,1,2);
3012 : long ea, eb;
3013 :
3014 : /* u t_COMPLEX generator of nf.zk, write x ~ a + b u, a,b in Z */
3015 4424 : b = grndtoi(mpdiv(imag_i(x), gel(u,2)), &eb);
3016 4424 : if (eb > -20) return NULL;
3017 4424 : a = grndtoi(mpsub(real_i(x), mpmul(b,gel(u,1))), &ea);
3018 4424 : if (ea > -20) return NULL;
3019 4424 : return signe(b)? coltoalg(nf, mkcol2(a,b)): a;
3020 : }
3021 :
3022 : static GEN
3023 70 : findbezk_pol(GEN nf, GEN x)
3024 : {
3025 70 : long i, lx = lg(x);
3026 70 : GEN y = cgetg(lx,t_POL);
3027 4494 : for (i=2; i<lx; i++)
3028 4424 : if (! (gel(y,i) = findbezk(nf,gel(x,i))) ) return NULL;
3029 70 : y[1] = x[1]; return y;
3030 : }
3031 :
3032 : /* P approximation computed at initial precision prec. Compute needed prec
3033 : * to know P with 1 word worth of trailing decimals */
3034 : static long
3035 0 : get_prec(GEN P, long prec)
3036 : {
3037 0 : long k = gprecision(P);
3038 0 : if (k == 3) return precdbl(prec); /* approximation not trustworthy */
3039 0 : k = prec - k; /* lost precision when computing P */
3040 0 : if (k < 0) k = 0;
3041 0 : k += nbits2prec(gexpo(P) + 128);
3042 0 : if (k <= prec) k = precdbl(prec); /* dubious: old prec should have worked */
3043 0 : return k;
3044 : }
3045 :
3046 : /* Compute data for ellphist */
3047 : static GEN
3048 4354 : ellphistinit(GEN om, long prec)
3049 : {
3050 4354 : GEN res,om1b,om2b, om1 = gel(om,1), om2 = gel(om,2);
3051 :
3052 4354 : if (gsigne(imag_i(gdiv(om1,om2))) < 0) { swap(om1,om2); om = mkvec2(om1,om2); }
3053 4354 : om1b = conj_i(om1);
3054 4354 : om2b = conj_i(om2); res = cgetg(4,t_VEC);
3055 4354 : gel(res,1) = gdivgu(elleisnum(om,2,0,prec),12);
3056 4354 : gel(res,2) = gdiv(PiI2(prec), gmul(om2, imag_i(gmul(om1b,om2))));
3057 4354 : gel(res,3) = om2b; return res;
3058 : }
3059 :
3060 : /* Computes log(phi^*(z,om)), using res computed by ellphistinit */
3061 : static GEN
3062 8708 : ellphist(GEN om, GEN res, GEN z, long prec)
3063 : {
3064 8708 : GEN u = imag_i(gmul(z, gel(res,3)));
3065 8708 : GEN zst = gsub(gmul(u, gel(res,2)), gmul(z,gel(res,1)));
3066 8708 : return gsub(ellsigma(om,z,1,prec),gmul2n(gmul(z,zst),-1));
3067 : }
3068 :
3069 : /* Computes phi^*(la,om)/phi^*(1,om) where (1,om) is an oriented basis of the
3070 : ideal gf*gc^{-1} */
3071 : static GEN
3072 4354 : computeth2(GEN om, GEN la, long prec)
3073 : {
3074 4354 : GEN p1,p2,res = ellphistinit(om,prec);
3075 :
3076 4354 : p1 = gsub(ellphist(om,res,la,prec), ellphist(om,res,gen_1,prec));
3077 4354 : p2 = imag_i(p1);
3078 4354 : if (gexpo(real_i(p1))>20 || gexpo(p2)> prec2nbits(minss(prec,realprec(p2)))-10)
3079 0 : return NULL;
3080 4354 : return gexp(p1,prec);
3081 : }
3082 :
3083 : /* Computes P_2(X)=polynomial in Z_K[X] closest to prod_gc(X-th2(gc)) where
3084 : the product is over the ray class group bnr.*/
3085 : static GEN
3086 70 : computeP2(GEN bnr, long prec)
3087 : {
3088 70 : long clrayno, i, first = 1;
3089 70 : pari_sp av=avma, av2;
3090 70 : GEN listray, P0, P, lanum, la = get_lambda(bnr);
3091 70 : GEN nf = bnr_get_nf(bnr), f = gel(bnr_get_mod(bnr), 1);
3092 70 : listray = getallelts(bnr);
3093 70 : clrayno = lg(listray)-1; av2 = avma;
3094 70 : PRECPB:
3095 70 : if (!first)
3096 : {
3097 0 : if (DEBUGLEVEL) pari_warn(warnprec,"computeP2",prec);
3098 0 : nf = gerepilecopy(av2, nfnewprec_shallow(bnr_get_nf(bnr),prec));
3099 : }
3100 70 : first = 0; lanum = to_approx(nf,la);
3101 70 : P = cgetg(clrayno+1,t_VEC);
3102 4424 : for (i=1; i<=clrayno; i++)
3103 : {
3104 4354 : GEN om = get_om(nf, idealdiv(nf,f,gel(listray,i)));
3105 4354 : GEN s = computeth2(om,lanum,prec);
3106 4354 : if (!s) { prec = precdbl(prec); goto PRECPB; }
3107 4354 : gel(P,i) = s;
3108 : }
3109 70 : P0 = roots_to_pol(P, 0);
3110 70 : P = findbezk_pol(nf, P0);
3111 70 : if (!P) { prec = get_prec(P0, prec); goto PRECPB; }
3112 70 : return gerepilecopy(av, P);
3113 : }
3114 :
3115 : #define nexta(a) (a>0 ? -a : 1-a)
3116 : static GEN
3117 49 : do_compo(GEN A0, GEN B)
3118 : {
3119 49 : long a, i, l = lg(B), v = fetch_var_higher();
3120 : GEN A, z;
3121 : /* now v > x = pol_x(0) > nf variable */
3122 49 : B = leafcopy(B); setvarn(B, v);
3123 210 : for (i = 2; i < l; i++) gel(B,i) = monomial(gel(B,i), l-i-1, 0);
3124 : /* B := x^deg(B) B(v/x) */
3125 49 : A = A0 = leafcopy(A0); setvarn(A0, v);
3126 56 : for (a = 0;; a = nexta(a))
3127 : {
3128 56 : if (a) A = RgX_translate(A0, stoi(a));
3129 56 : z = resultant(A,B); /* in variable 0 */
3130 56 : if (issquarefree(z)) break;
3131 : }
3132 49 : (void)delete_var(); return z;
3133 : }
3134 : #undef nexta
3135 :
3136 : static GEN
3137 14 : galoisapplypol(GEN nf, GEN s, GEN x)
3138 : {
3139 14 : long i, lx = lg(x);
3140 14 : GEN y = cgetg(lx,t_POL);
3141 :
3142 56 : for (i=2; i<lx; i++) gel(y,i) = galoisapply(nf,s,gel(x,i));
3143 14 : y[1] = x[1]; return y;
3144 : }
3145 : /* x quadratic, write it as ua + v, u,v rational */
3146 : static GEN
3147 70 : findquad(GEN a, GEN x, GEN p)
3148 : {
3149 : long tu, tv;
3150 70 : pari_sp av = avma;
3151 : GEN u,v;
3152 70 : if (typ(x) == t_POLMOD) x = gel(x,2);
3153 70 : if (typ(a) == t_POLMOD) a = gel(a,2);
3154 70 : u = poldivrem(x, a, &v);
3155 70 : u = simplify_shallow(u); tu = typ(u);
3156 70 : v = simplify_shallow(v); tv = typ(v);
3157 70 : if (!is_scalar_t(tu)) pari_err_TYPE("findquad", u);
3158 70 : if (!is_scalar_t(tv)) pari_err_TYPE("findquad", v);
3159 70 : x = deg1pol(u, v, varn(a));
3160 70 : if (typ(x) == t_POL) x = gmodulo(x,p);
3161 70 : return gerepileupto(av, x);
3162 : }
3163 : static GEN
3164 14 : findquad_pol(GEN p, GEN a, GEN x)
3165 : {
3166 14 : long i, lx = lg(x);
3167 14 : GEN y = cgetg(lx,t_POL);
3168 84 : for (i=2; i<lx; i++) gel(y,i) = findquad(a, gel(x,i), p);
3169 14 : y[1] = x[1]; return y;
3170 : }
3171 : /* m is 3, 4 or 12 */
3172 : static GEN
3173 35 : compocyclo(GEN D, long m)
3174 35 : { return do_compo(quadhilbertimag(D), polcyclo(m,0)); }
3175 : /* m is prime or 4 * prime */
3176 : static GEN
3177 14 : compocyclop(GEN nf, long m)
3178 : {
3179 14 : GEN sb,a,b,s,p1,p2,p3,res,polL,polLK,nfL, D = nf_get_disc(nf);
3180 : long ell,vx;
3181 :
3182 14 : p1 = quadhilbertimag(D);
3183 14 : p2 = polcyclo(m,0);
3184 14 : ell = odd(m)? m: (m>>2); /* prime */
3185 14 : if (absequalui(ell,D)) /* ell = |D| */
3186 : {
3187 0 : p2 = gcoeff(nffactor(nf,p2),1,1);
3188 0 : return do_compo(p1,p2);
3189 : }
3190 14 : if (ell%4 == 3) ell = -ell;
3191 : /* nf = K = Q(a), L = K(b) quadratic extension = Q(t) */
3192 14 : polLK = quadpoly_i(stoi(ell)); /* relative polynomial */
3193 14 : res = rnfequation2(nf, polLK);
3194 14 : vx = nf_get_varn(nf);
3195 14 : polL = gsubst(gel(res,1),0,pol_x(vx)); /* = charpoly(t) */
3196 14 : a = gsubst(lift_shallow(gel(res,2)), 0,pol_x(vx));
3197 14 : b = gsub(pol_x(vx), gmul(gel(res,3), a));
3198 14 : nfL = nfinit(polL, DEFAULTPREC);
3199 14 : p1 = gcoeff(nffactor(nfL,p1),1,1);
3200 14 : p2 = gcoeff(nffactor(nfL,p2),1,1);
3201 14 : p3 = do_compo(p1,p2); /* relative equation over L */
3202 : /* compute non trivial s in Gal(L / K) */
3203 14 : sb= gneg(gadd(b, RgX_coeff(polLK,1))); /* s(b) = Tr(b) - b */
3204 14 : s = gadd(pol_x(vx), gsub(sb, b)); /* s(t) = t + s(b) - b */
3205 14 : p3 = gmul(p3, galoisapplypol(nfL, s, p3));
3206 14 : return findquad_pol(nf_get_pol(nf), a, p3);
3207 : }
3208 :
3209 : /* I integral ideal in HNF. (x) = I, x small in Z ? */
3210 : static long
3211 119 : isZ(GEN I)
3212 : {
3213 119 : GEN x = gcoeff(I,1,1);
3214 119 : if (signe(gcoeff(I,1,2)) || !equalii(x, gcoeff(I,2,2))) return 0;
3215 105 : return is_bigint(x)? -1: itos(x);
3216 : }
3217 :
3218 : /* Treat special cases directly. return NULL if not special case */
3219 : static GEN
3220 119 : treatspecialsigma(GEN bnr)
3221 : {
3222 119 : GEN bnf = bnr_get_bnf(bnr), nf = bnf_get_nf(bnf);
3223 119 : GEN f = gel(bnr_get_mod(bnr), 1), D = nf_get_disc(nf);
3224 : GEN p1, p2;
3225 119 : long Ds, fl, tryf, i = isZ(f);
3226 :
3227 119 : if (i == 1) return quadhilbertimag(D); /* f = 1 */
3228 :
3229 119 : if (absequaliu(D,3)) /* Q(j) */
3230 : {
3231 0 : if (i == 4 || i == 5 || i == 7) return polcyclo(i,0);
3232 0 : if (!absequaliu(gcoeff(f,1,1),9) || !absequaliu(Z_content(f),3)) return NULL;
3233 : /* f = P_3^3 */
3234 0 : p1 = mkpolmod(bnf_get_tuU(bnf), nf_get_pol(nf));
3235 0 : return gadd(pol_xn(3,0), p1); /* x^3+j */
3236 : }
3237 119 : if (absequaliu(D,4)) /* Q(i) */
3238 : {
3239 14 : if (i == 3 || i == 5) return polcyclo(i,0);
3240 14 : if (i != 4) return NULL;
3241 0 : p1 = mkpolmod(bnf_get_tuU(bnf), nf_get_pol(nf));
3242 0 : return gadd(pol_xn(2,0), p1); /* x^2+i */
3243 : }
3244 105 : Ds = smodis(D,48);
3245 105 : if (i)
3246 : {
3247 91 : if (i==2 && Ds%16== 8) return compocyclo(D, 4);
3248 84 : if (i==3 && Ds% 3== 1) return compocyclo(D, 3);
3249 70 : if (i==4 && Ds% 8== 1) return compocyclo(D, 4);
3250 63 : if (i==6 && Ds ==40) return compocyclo(D,12);
3251 56 : return NULL;
3252 : }
3253 :
3254 14 : p1 = gcoeff(f,1,1); /* integer > 0 */
3255 14 : tryf = itou_or_0(p1); if (!tryf) return NULL;
3256 14 : p2 = gcoeff(f,2,2); /* integer > 0 */
3257 14 : if (is_pm1(p2)) fl = 0;
3258 : else {
3259 0 : if (Ds % 16 != 8 || !absequaliu(Z_content(f),2)) return NULL;
3260 0 : fl = 1; tryf >>= 1;
3261 : }
3262 14 : if (tryf <= 3 || umodiu(D, tryf) || !uisprime(tryf)) return NULL;
3263 14 : if (fl) tryf <<= 2;
3264 14 : return compocyclop(nf, tryf);
3265 : }
3266 :
3267 : GEN
3268 161 : quadray(GEN D, GEN f, long prec)
3269 : {
3270 161 : GEN bnr, y, bnf, H = NULL;
3271 161 : pari_sp av = avma;
3272 :
3273 161 : if (isint1(f)) return quadhilbert(D, prec);
3274 126 : if (typ(D) == t_INT && typ(f) != t_INT)
3275 0 : pari_err_TYPE("quadray [conductor]", f);
3276 126 : quadray_init(&D, &bnf, prec);
3277 126 : bnr = Buchray(bnf, f, nf_INIT|nf_GEN);
3278 126 : if (is_pm1(bnr_get_no(bnr))) { set_avma(av); return pol_x(0); }
3279 126 : if (signe(D) > 0)
3280 7 : y = bnrstark(bnr, H, prec);
3281 : else
3282 : {
3283 119 : bnr_subgroup_sanitize(&bnr, &H);
3284 119 : y = treatspecialsigma(bnr);
3285 119 : if (!y) y = computeP2(bnr, prec);
3286 : }
3287 126 : return gerepileupto(av, y);
3288 : }
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