Line data Source code
1 : /* Copyright (C) 2014 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 : #include "pari.h"
16 : #include "paripriv.h"
17 :
18 : static GEN
19 218650 : pcp_get_L(GEN G) { return gmael(G,1,1)+1; }
20 : static GEN
21 218650 : pcp_get_n(GEN G) { return gmael(G,1,2)+1; }
22 : static GEN
23 218649 : pcp_get_o(GEN G) { return gmael(G,1,3)+1; }
24 : static long
25 218650 : pcp_get_L0(GEN G) { return mael(G,2,1); }
26 : static long
27 218649 : pcp_get_k(GEN G) { return mael(G,2,2); }
28 : static long
29 218648 : pcp_get_enum_cnt(GEN G) { return mael(G,2,3); }
30 :
31 : /* FIXME: Implement {ascend,descend}_volcano() in terms of the "new"
32 : * volcano traversal functions at the bottom of the file. */
33 :
34 : /* Is j = 0 or 1728 (mod p)? */
35 : INLINE int
36 363191 : is_j_exceptional(ulong j, ulong p) { return j == 0 || j == 1728 % p; }
37 :
38 : INLINE long
39 81427 : node_degree(GEN phi, long L, ulong j, ulong p, ulong pi)
40 : {
41 81427 : pari_sp av = avma;
42 81427 : long n = Flx_nbroots(Flm_Fl_polmodular_evalx(phi, L, j, p, pi), p);
43 81425 : return gc_long(av, n);
44 : }
45 :
46 : /* Given an array path = [j0, j1] of length 2, return the polynomial
47 : *
48 : * \Phi_L(X, j1) / (X - j0)
49 : *
50 : * where \Phi_L(X, Y) is the modular polynomial of level L. An error
51 : * is raised if X - j0 does not divide \Phi_L(X, j1) */
52 : INLINE GEN
53 142840 : nhbr_polynomial(ulong path[], GEN phi, ulong p, ulong pi, long L)
54 : {
55 142840 : GEN modpol = Flm_Fl_polmodular_evalx(phi, L, path[0], p, pi);
56 : ulong rem;
57 142840 : GEN nhbr_pol = Flx_div_by_X_x(modpol, path[-1], p, &rem);
58 : /* If disc End(path[0]) <= L^2, it's possible for path[0] to appear among the
59 : * roots of nhbr_pol. This should have been obviated by earlier choices */
60 142837 : if (rem) pari_err_BUG("nhbr_polynomial: invalid preceding j");
61 142837 : return nhbr_pol;
62 : }
63 :
64 : /* Path is an array with space for at least max_len+1 * elements, whose first
65 : * and second elements are the beginning of the path. I.e., the path starts
66 : * (path[0], path[1])
67 : * If the result is less than max_len, then the last element of path is on the
68 : * floor. If the result equals max_len, then it is unknown whether the last
69 : * element of path is on the floor or not */
70 : static long
71 279420 : extend_path(ulong path[], GEN phi, ulong p, ulong pi, long L, long max_len)
72 : {
73 279420 : pari_sp av = avma;
74 279420 : long d = 1;
75 359914 : for ( ; d < max_len; d++)
76 : {
77 103262 : GEN nhbr_pol = nhbr_polynomial(path + d, phi, p, pi, L);
78 103261 : ulong nhbr = Flx_oneroot_pre(nhbr_pol, p, pi);
79 103259 : set_avma(av);
80 103261 : if (nhbr == p) break; /* no root: we are on the floor. */
81 80494 : path[d+1] = nhbr;
82 : }
83 279419 : return d;
84 : }
85 :
86 : /* This is Sutherland 2009 Algorithm Ascend (p12) */
87 : ulong
88 127367 : ascend_volcano(GEN phi, ulong j, ulong p, ulong pi, long level, long L,
89 : long depth, long steps)
90 : {
91 127367 : pari_sp ltop = avma, av;
92 : /* path will never hold more than max_len < depth elements */
93 127367 : GEN path_g = cgetg(depth + 2, t_VECSMALL);
94 127369 : ulong *path = zv_to_ulongptr(path_g);
95 127369 : long max_len = depth - level;
96 127369 : int first_iter = 1;
97 :
98 127369 : if (steps <= 0 || max_len < 0) pari_err_BUG("ascend_volcano: bad params");
99 127369 : av = avma;
100 294318 : while (steps--)
101 : {
102 127369 : GEN nhbr_pol = first_iter? Flm_Fl_polmodular_evalx(phi, L, j, p, pi)
103 166950 : : nhbr_polynomial(path+1, phi, p, pi, L);
104 166952 : GEN nhbrs = Flx_roots_pre(nhbr_pol, p, pi);
105 166941 : long nhbrs_len = lg(nhbrs)-1, i;
106 166941 : pari_sp btop = avma;
107 166941 : path[0] = j;
108 166941 : first_iter = 0;
109 :
110 166941 : j = nhbrs[nhbrs_len];
111 210445 : for (i = 1; i < nhbrs_len; i++)
112 : {
113 78260 : ulong next_j = nhbrs[i], last_j;
114 : long len;
115 78260 : if (is_j_exceptional(next_j, p))
116 : {
117 : /* Fouquet & Morain, Section 4.3, if j = 0 or 1728, then it is on the
118 : * surface. So we just return it. */
119 33 : if (steps)
120 0 : pari_err_BUG("ascend_volcano: Got to the top with more steps to go!");
121 33 : j = next_j; break;
122 : }
123 78227 : path[1] = next_j;
124 78227 : len = extend_path(path, phi, p, pi, L, max_len);
125 78227 : last_j = path[len];
126 78227 : if (len == max_len
127 : /* Ended up on the surface */
128 78227 : && (is_j_exceptional(last_j, p)
129 78227 : || node_degree(phi, L, last_j, p, pi) > 1)) { j = next_j; break; }
130 43504 : set_avma(btop);
131 : }
132 166939 : path[1] = j; /* For nhbr_polynomial() at the top. */
133 :
134 166939 : max_len++; set_avma(av);
135 : }
136 127368 : return gc_long(ltop, j);
137 : }
138 :
139 : static void
140 206700 : random_distinct_neighbours_of(ulong *nhbr1, ulong *nhbr2,
141 : GEN phi, ulong j, ulong p, ulong pi, long L, long must_have_two_neighbours)
142 : {
143 206700 : pari_sp av = avma;
144 206700 : GEN modpol = Flm_Fl_polmodular_evalx(phi, L, j, p, pi);
145 : ulong rem;
146 206708 : *nhbr1 = Flx_oneroot_pre(modpol, p, pi);
147 206705 : if (*nhbr1 == p) pari_err_BUG("random_distinct_neighbours_of [no neighbour]");
148 206705 : modpol = Flx_div_by_X_x(modpol, *nhbr1, p, &rem);
149 206693 : *nhbr2 = Flx_oneroot_pre(modpol, p, pi);
150 206698 : if (must_have_two_neighbours && *nhbr2 == p)
151 0 : pari_err_BUG("random_distinct_neighbours_of [single neighbour]");
152 206698 : set_avma(av);
153 206698 : }
154 :
155 : /* This is Sutherland 2009 Algorithm Descend (p12) */
156 : ulong
157 2975 : descend_volcano(GEN phi, ulong j, ulong p, ulong pi,
158 : long level, long L, long depth, long steps)
159 : {
160 2975 : pari_sp ltop = avma;
161 : GEN path_g;
162 : ulong *path;
163 : long max_len;
164 :
165 2975 : if (steps <= 0 || level + steps > depth) pari_err_BUG("descend_volcano");
166 2975 : max_len = depth - level;
167 2975 : path_g = cgetg(max_len + 1 + 1, t_VECSMALL);
168 2975 : path = zv_to_ulongptr(path_g);
169 2975 : path[0] = j;
170 : /* level = 0 means we're on the volcano surface... */
171 2975 : if (!level)
172 : {
173 : /* Look for any path to the floor. One of j's first three neighbours leads
174 : * to the floor, since at most two neighbours are on the surface. */
175 2652 : GEN nhbrs = Flx_roots_pre(Flm_Fl_polmodular_evalx(phi, L, j, p, pi), p, pi);
176 : long i;
177 2929 : for (i = 1; i <= 3; i++)
178 : {
179 : long len;
180 2929 : path[1] = nhbrs[i];
181 2929 : len = extend_path(path, phi, p, pi, L, max_len);
182 : /* If nhbrs[i] took us to the floor: */
183 2929 : if (len < max_len || node_degree(phi, L, path[len], p, pi) == 1) break;
184 : }
185 2652 : if (i > 3) pari_err_BUG("descend_volcano [2]");
186 : }
187 : else
188 : {
189 : ulong nhbr1, nhbr2;
190 : long len;
191 323 : random_distinct_neighbours_of(&nhbr1, &nhbr2, phi, j, p, pi, L, 1);
192 323 : path[1] = nhbr1;
193 323 : len = extend_path(path, phi, p, pi, L, max_len);
194 : /* If last j isn't on the floor */
195 323 : if (len == max_len /* Ended up on the surface. */
196 323 : && (is_j_exceptional(path[len], p)
197 271 : || node_degree(phi, L, path[len], p, pi) != 1)) {
198 : /* The other neighbour leads to the floor */
199 140 : path[1] = nhbr2;
200 140 : (void) extend_path(path, phi, p, pi, L, steps);
201 : }
202 : }
203 2975 : return gc_ulong(ltop, path[steps]);
204 : }
205 :
206 : long
207 206391 : j_level_in_volcano(
208 : GEN phi, ulong j, ulong p, ulong pi, long L, long depth)
209 : {
210 206391 : pari_sp av = avma;
211 : GEN chunk;
212 : ulong *path1, *path2;
213 : long lvl;
214 :
215 : /* Fouquet & Morain, Section 4.3, if j = 0 or 1728 then it is on the
216 : * surface. Also, if the volcano depth is zero then j has level 0 */
217 206391 : if (depth == 0 || is_j_exceptional(j, p)) return 0;
218 :
219 206379 : chunk = new_chunk(2 * (depth + 1));
220 206376 : path1 = (ulong *) &chunk[0];
221 206376 : path2 = (ulong *) &chunk[depth + 1];
222 206376 : path1[0] = path2[0] = j;
223 206376 : random_distinct_neighbours_of(&path1[1], &path2[1], phi, j, p, pi, L, 0);
224 206380 : if (path2[1] == p)
225 107475 : lvl = depth; /* Only one neighbour => j is on the floor => level = depth */
226 : else
227 : {
228 98905 : long path1_len = extend_path(path1, phi, p, pi, L, depth);
229 98906 : long path2_len = extend_path(path2, phi, p, pi, L, path1_len);
230 98903 : lvl = depth - path2_len;
231 : }
232 206378 : return gc_long(av, lvl);
233 : }
234 :
235 : INLINE GEN
236 32220499 : Flx_remove_root(GEN f, ulong a, ulong p)
237 : {
238 : ulong r;
239 32220499 : GEN g = Flx_div_by_X_x(f, a, p, &r);
240 32070219 : if (r) pari_err_BUG("Flx_remove_root");
241 32070721 : return g;
242 : }
243 :
244 : INLINE GEN
245 24268250 : get_nbrs(GEN phi, long L, ulong J, const ulong *xJ, ulong p, ulong pi)
246 : {
247 24268250 : pari_sp av = avma;
248 24268250 : GEN f = Flm_Fl_polmodular_evalx(phi, L, J, p, pi);
249 24274811 : if (xJ) f = Flx_remove_root(f, *xJ, p);
250 24177095 : return gc_upto(av, Flx_roots_pre(f, p, pi));
251 : }
252 :
253 : /* Return a path of length n along the surface of an L-volcano of height h
254 : * starting from surface node j0. Assumes (D|L) = 1 where D = disc End(j0).
255 : *
256 : * Actually, if j0's endomorphism ring is a suborder, we return the
257 : * corresponding shorter path. W must hold space for n + h nodes.
258 : *
259 : * TODO: have two versions of this function: one that assumes J has the correct
260 : * endomorphism ring (hence avoiding several branches in the inner loop) and a
261 : * second that does not and accordingly checks for repetitions */
262 : static long
263 214722 : surface_path(
264 : ulong W[],
265 : long n, GEN phi, long L, long h, ulong J, const ulong *nJ, ulong p, ulong pi)
266 : {
267 214722 : pari_sp av = avma, bv;
268 : GEN T, v;
269 : long j, k, w, x;
270 : ulong W0;
271 :
272 214722 : W[0] = W0 = J;
273 214722 : if (n == 1) return 1;
274 :
275 214722 : T = cgetg(h+2, t_VEC); bv = avma;
276 214720 : v = get_nbrs(phi, L, J, nJ, p, pi);
277 : /* Insert known neighbour first */
278 214720 : if (nJ) v = gc_upto(bv, vecsmall_prepend(v, *nJ));
279 214720 : gel(T,1) = v; k = lg(v)-1;
280 :
281 214720 : switch (k) {
282 0 : case 0: pari_err_BUG("surface_path"); /* We must always have neighbours */
283 8992 : case 1:
284 : /* If volcano is not flat, then we must have more than one neighbour */
285 8992 : if (h) pari_err_BUG("surface_path");
286 8992 : W[1] = uel(v, 1);
287 8992 : set_avma(av);
288 : /* Check for bad endo ring */
289 8992 : if (W[1] == W[0]) return 1;
290 8807 : return 2;
291 25266 : case 2:
292 : /* If L=2 the only way we can have 2 neighbours is if we have a double root
293 : * which can only happen for |D| <= 16 (Thm 2.2 of Fouquet-Morain)
294 : * and if it does we must have a 2-cycle. Happens for D=-15. */
295 25266 : if (L == 2)
296 : { /* The double root is the neighbour on the surface, with exactly one
297 : * neighbour other than J; the other neighbour of J has either 0 or 2
298 : * neighbours that are not J */
299 84 : GEN u = get_nbrs(phi, L, uel(v, 1), &J, p, pi);
300 84 : long n = lg(u)-1 - !!vecsmall_isin(u, J);
301 84 : W[1] = n == 1 ? uel(v,1) : uel(v,2);
302 84 : return gc_long(av, 2);
303 : }
304 : /* Volcano is not flat but found only 2 neighbours for the surface node J */
305 25182 : if (h) pari_err_BUG("surface_path");
306 :
307 25182 : W[1] = uel(v,1); /* TODO: Can we use the other root uel(v,2) somehow? */
308 4421355 : for (w = 2; w < n; w++)
309 : {
310 4396804 : v = get_nbrs(phi, L, W[w-1], &W[w-2], p, pi);
311 : /* A flat volcano must have exactly one non-previous neighbour */
312 4396565 : if (lg(v) != 2) pari_err_BUG("surface_path");
313 4396565 : W[w] = uel(v, 1);
314 : /* Detect cycle in case J doesn't have the right endo ring. */
315 4396565 : set_avma(av); if (W[w] == W0) return w;
316 : }
317 24551 : return gc_long(av, n);
318 : }
319 180462 : if (!h) pari_err_BUG("surface_path"); /* Can't have a flat volcano if k > 2 */
320 :
321 : /* At this point, each surface node has L+1 distinct neighbours, 2 of which
322 : * are on the surface */
323 180462 : w = 1;
324 6404400 : for (x = 0;; x++)
325 : {
326 : /* Get next neighbour of last known surface node to attempt to
327 : * extend the path. */
328 6404400 : W[w] = umael(T, ((w-1) % h) + 1, x + 1);
329 :
330 : /* Detect cycle in case the given J didn't have the right endo ring */
331 6404400 : if (W[w] == W0) return gc_long(av,w);
332 :
333 : /* If we have to test the last neighbour, we know it's on the
334 : * surface, and if we're done there's no need to extend. */
335 6404369 : if (x == k-1 && w == n-1) return gc_long(av,n);
336 :
337 : /* Walk forward until we hit the floor or finish. */
338 : /* NB: We don't keep the stack clean here; usage is in the order of Lh,
339 : * i.e. L roots for each level of the volcano of height h. */
340 6285114 : for (j = w;;)
341 13184087 : {
342 : long m;
343 : /* We must get 0 or L neighbours here. */
344 19469201 : v = get_nbrs(phi, L, W[j], &W[j-1], p, pi);
345 19414231 : m = lg(v)-1;
346 19414231 : if (!m) {
347 : /* We hit the floor: save the neighbours of W[w-1] and dump the rest */
348 6223230 : GEN nbrs = gel(T, ((w-1) % h) + 1);
349 6223230 : gel(T, ((w-1) % h) + 1) = gc_upto(bv, nbrs);
350 6223938 : break;
351 : }
352 13191001 : if (m != L) pari_err_BUG("surface_path");
353 :
354 13245268 : gel(T, (j % h) + 1) = v;
355 :
356 13245268 : W[++j] = uel(v, 1);
357 : /* If we have our path by h nodes, we know W[w] is on the surface */
358 13245268 : if (j == w + h) {
359 12246057 : ++w;
360 : /* Detect cycle in case the given J didn't have the right endo ring */
361 12246057 : if (W[w] == W0) return gc_long(av,w);
362 12217623 : x = 0; k = L;
363 : }
364 13216834 : if (w == n) return gc_long(av,w);
365 : }
366 : }
367 : }
368 :
369 : long
370 149973 : next_surface_nbr(
371 : ulong *nJ,
372 : GEN phi, long L, long h, ulong J, const ulong *pJ, ulong p, ulong pi)
373 : {
374 149973 : pari_sp av = avma, bv;
375 : GEN S;
376 : ulong *P;
377 : long i, k;
378 :
379 149973 : S = get_nbrs(phi, L, J, pJ, p, pi); k = lg(S)-1;
380 : /* If there is a double root and pJ is set, then k will be zero. */
381 149963 : if (!k) return gc_long(av,0);
382 149963 : if (k == 1 || ( ! pJ && k == 2)) { *nJ = uel(S, 1); return gc_long(av,1); }
383 25218 : if (!h) pari_err_BUG("next_surface_nbr");
384 :
385 25218 : P = (ulong *) new_chunk(h + 1); bv = avma;
386 25219 : P[0] = J;
387 53923 : for (i = 0; i < k; i++)
388 : {
389 : long j;
390 53921 : P[1] = uel(S, i + 1);
391 84718 : for (j = 1; j <= h; j++)
392 : {
393 59499 : GEN T = get_nbrs(phi, L, P[j], &P[j - 1], p, pi);
394 59501 : if (lg(T) == 1) break;
395 30797 : P[j + 1] = uel(T, 1);
396 : }
397 53923 : if (j < h) pari_err_BUG("next_surface_nbr");
398 53923 : set_avma(bv); if (j > h) break;
399 : }
400 : /* TODO: We could save one get_nbrs call by iterating from i up to k-1 and
401 : * assume that the last (kth) nbr is the one we want. For now we're careful
402 : * and check that this last nbr really is on the surface */
403 25221 : if (i == k) pari_err_BUG("next_surf_nbr");
404 25221 : *nJ = uel(S, i+1); return gc_long(av,1);
405 : }
406 :
407 : /* Return the number of distinct neighbours (1 or 2) */
408 : INLINE long
409 239856 : common_nbr(ulong *nbr,
410 : ulong J1, GEN Phi1, long L1,
411 : ulong J2, GEN Phi2, long L2, ulong p, ulong pi)
412 : {
413 239856 : pari_sp av = avma;
414 : GEN d, f, g, r;
415 : long rlen;
416 :
417 239856 : g = Flm_Fl_polmodular_evalx(Phi1, L1, J1, p, pi);
418 239863 : f = Flm_Fl_polmodular_evalx(Phi2, L2, J2, p, pi);
419 239868 : d = Flx_gcd(f, g, p);
420 239712 : if (degpol(d) == 1) { *nbr = Flx_deg1_root(d, p); return gc_long(av,1); }
421 1174 : if (degpol(d) != 2) pari_err_BUG("common_neighbour");
422 1174 : r = Flx_roots_pre(d, p, pi);
423 1174 : rlen = lg(r)-1;
424 1174 : if (!rlen) pari_err_BUG("common_neighbour");
425 : /* rlen is 1 or 2 depending on whether the root is unique or not. */
426 1174 : nbr[0] = uel(r, 1);
427 1174 : nbr[1] = uel(r, rlen); return gc_long(av,rlen);
428 : }
429 :
430 : /* Return gcd(Phi1(X,J1)/(X - J0), Phi2(X,J2)). Not stack clean. */
431 : INLINE GEN
432 43968 : common_nbr_pred_poly(
433 : ulong J1, GEN Phi1, long L1,
434 : ulong J2, GEN Phi2, long L2, ulong J0, ulong p, ulong pi)
435 : {
436 : GEN f, g;
437 :
438 43968 : g = Flm_Fl_polmodular_evalx(Phi1, L1, J1, p, pi);
439 43969 : g = Flx_remove_root(g, J0, p);
440 43969 : f = Flm_Fl_polmodular_evalx(Phi2, L2, J2, p, pi);
441 43967 : return Flx_gcd(f, g, p);
442 : }
443 :
444 : /* Find common neighbour of J1 and J2, where J0 is an L1 predecessor of J1.
445 : * Return 1 if successful, 0 if not. */
446 : INLINE int
447 43026 : common_nbr_pred(ulong *nbr,
448 : ulong J1, GEN Phi1, long L1,
449 : ulong J2, GEN Phi2, long L2, ulong J0, ulong p, ulong pi)
450 : {
451 43026 : pari_sp av = avma;
452 43026 : GEN d = common_nbr_pred_poly(J1, Phi1, L1, J2, Phi2, L2, J0, p, pi);
453 43022 : int res = (degpol(d) == 1);
454 43022 : if (res) *nbr = Flx_deg1_root(d, p);
455 43027 : return gc_bool(av,res);
456 : }
457 :
458 : INLINE long
459 938 : common_nbr_verify(ulong *nbr,
460 : ulong J1, GEN Phi1, long L1,
461 : ulong J2, GEN Phi2, long L2, ulong J0, ulong p, ulong pi)
462 : {
463 938 : pari_sp av = avma;
464 938 : GEN d = common_nbr_pred_poly(J1, Phi1, L1, J2, Phi2, L2, J0, p, pi);
465 :
466 938 : if (!degpol(d)) return gc_long(av,0);
467 354 : if (degpol(d) > 1) pari_err_BUG("common_neighbour_verify");
468 354 : *nbr = Flx_deg1_root(d, p);
469 354 : return gc_long(av,1);
470 : }
471 :
472 : INLINE ulong
473 541 : Flm_Fl_polmodular_evalxy(GEN Phi, long L, ulong x, ulong y, ulong p, ulong pi)
474 : {
475 541 : pari_sp av = avma;
476 541 : GEN f = Flm_Fl_polmodular_evalx(Phi, L, x, p, pi);
477 541 : return gc_ulong(av, Flx_eval_pre(f, y, p, pi));
478 : }
479 :
480 : /* Find a common L1-neighbor of J1 and L2-neighbor of J2, given J0 an
481 : * L2-neighbor of J1 and an L1-neighbor of J2. Return 1 if successful, 0
482 : * otherwise. Will only fail if initial J-invariant had the wrong endo ring */
483 : INLINE int
484 37367 : common_nbr_corner(ulong *nbr,
485 : ulong J1, GEN Phi1, long L1, long h1,
486 : ulong J2, GEN Phi2, long L2, ulong J0, ulong p, ulong pi)
487 : {
488 : ulong nbrs[2];
489 37367 : if (common_nbr(nbrs, J1,Phi1,L1, J2,Phi2,L2, p, pi) == 2)
490 : {
491 : ulong nJ1, nJ2;
492 698 : if (!next_surface_nbr(&nJ2, Phi1, L1, h1, J2, &J0, p, pi) ||
493 400 : !next_surface_nbr(&nJ1, Phi1, L1, h1, nbrs[0], &J1, p, pi)) return 0;
494 :
495 349 : if (Flm_Fl_polmodular_evalxy(Phi2, L2, nJ1, nJ2, p, pi))
496 157 : nbrs[0] = nbrs[1];
497 384 : else if (!next_surface_nbr(&nJ1, Phi1, L1, h1, nbrs[1], &J1, p, pi) ||
498 243 : !Flm_Fl_polmodular_evalxy(Phi2, L2, nJ1, nJ2, p, pi)) return 0;
499 : }
500 37311 : *nbr = nbrs[0]; return 1;
501 : }
502 :
503 : /* Enumerate a surface L1-cycle using gcds with Phi_L2, where c_L2=c_L1^e and
504 : * |c_L1|=n, where c_a is the class of the pos def reduced primeform <a,b,c>.
505 : * Assumes n > e > 1 and roots[0],...,roots[e-1] are already present in W */
506 : static long
507 94194 : surface_gcd_cycle(
508 : ulong W[], ulong V[], long n,
509 : GEN Phi1, long L1, GEN Phi2, long L2, long e, ulong p, ulong pi)
510 : {
511 94194 : pari_sp av = avma;
512 : long i1, i2, j1, j2;
513 :
514 94194 : i1 = j2 = 0;
515 94194 : i2 = j1 = e - 1;
516 : /* If W != V we assume V actually points to an L2-isogenous parallel L1-path.
517 : * e should be 2 in this case */
518 94194 : if (W != V) { i1 = j1+1; i2 = n-1; }
519 : do {
520 : ulong t0, t1, t2, h10, h11, h20, h21;
521 : long k;
522 : GEN f, g, h1, h2;
523 :
524 4141746 : f = Flm_Fl_polmodular_evalx(Phi2, L2, V[i1], p, pi);
525 4137723 : g = Flm_Fl_polmodular_evalx(Phi1, L1, W[j1], p, pi);
526 4136476 : g = Flx_remove_root(g, W[j1 - 1], p);
527 4123811 : h1 = Flx_gcd(f, g, p);
528 4116884 : if (degpol(h1) != 1) break; /* Error */
529 4116401 : h11 = Flx_coeff(h1, 1);
530 4117393 : h10 = Flx_coeff(h1, 0); set_avma(av);
531 :
532 4118687 : f = Flm_Fl_polmodular_evalx(Phi2, L2, V[i2], p, pi);
533 4137892 : g = Flm_Fl_polmodular_evalx(Phi1, L1, W[j2], p, pi);
534 4137114 : k = j2 + 1;
535 4137114 : if (k == n) k = 0;
536 4137114 : g = Flx_remove_root(g, W[k], p);
537 4125454 : h2 = Flx_gcd(f, g, p);
538 4118722 : if (degpol(h2) != 1) break; /* Error */
539 4118518 : h21 = Flx_coeff(h2, 1);
540 4119424 : h20 = Flx_coeff(h2, 0); set_avma(av);
541 :
542 4122357 : i1++; i2--; if (i2 < 0) i2 = n-1;
543 4122357 : j1++; j2--; if (j2 < 0) j2 = n-1;
544 :
545 4122357 : t0 = Fl_mul_pre(h11, h21, p, pi);
546 4141737 : t1 = Fl_inv(t0, p);
547 4142711 : t0 = Fl_mul_pre(t1, h21, p, pi);
548 4140942 : t2 = Fl_mul_pre(t0, h10, p, pi);
549 4141061 : W[j1] = Fl_neg(t2, p);
550 4140455 : t0 = Fl_mul_pre(t1, h11, p, pi);
551 4142406 : t2 = Fl_mul_pre(t0, h20, p, pi);
552 4142404 : W[j2] = Fl_neg(t2, p);
553 4141648 : } while (j2 > j1 + 1);
554 : /* Usually the loop exits when j2 = j1 + 1, in which case we return n.
555 : * If we break early because of an error, then (j2 - (j1+1)) > 0 is the
556 : * number of elements we haven't calculated yet, and we return n minus that
557 : * quantity */
558 94096 : return gc_long(av, n - j2 + j1 + 1);
559 : }
560 :
561 : static long
562 1072 : surface_gcd_path(
563 : ulong W[], ulong V[], long n,
564 : GEN Phi1, long L1, GEN Phi2, long L2, long e, ulong p, ulong pi)
565 : {
566 1072 : pari_sp av = avma;
567 : long i, j;
568 :
569 1072 : i = 0; j = e;
570 : /* If W != V then assume V actually points to a L2-isogenous
571 : * parallel L1-path. e should be 2 in this case */
572 1072 : if (W != V) i = j;
573 3821 : while (j < n)
574 : {
575 : GEN f, g, d;
576 :
577 2749 : f = Flm_Fl_polmodular_evalx(Phi2, L2, V[i], p, pi);
578 2752 : g = Flm_Fl_polmodular_evalx(Phi1, L1, W[j - 1], p, pi);
579 2750 : g = Flx_remove_root(g, W[j - 2], p);
580 2748 : d = Flx_gcd(f, g, p);
581 2747 : if (degpol(d) != 1) break; /* Error */
582 2747 : W[j] = Flx_deg1_root(d, p);
583 2748 : i++; j++; set_avma(av);
584 : }
585 1072 : return gc_long(av, j);
586 : }
587 :
588 : /* Given a path V of length n on an L1-volcano, and W[0] L2-isogenous to V[0],
589 : * extends the path W to length n on an L1-volcano, with W[i] L2-isogenous
590 : * to V[i]. Uses gcds unless L2 is too large to make it helpful. Always uses
591 : * GCD to get W[1] to ensure consistent orientation.
592 : *
593 : * Returns the new length of W. This will almost always be n, but could be
594 : * lower if V was started with a J-invariant with bad endomorphism ring */
595 : INLINE long
596 202501 : surface_parallel_path(
597 : ulong W[], ulong V[], long n,
598 : GEN Phi1, long L1, GEN Phi2, long L2, ulong p, ulong pi, long cycle)
599 : {
600 : ulong W2, nbrs[2];
601 202501 : if (common_nbr(nbrs, W[0], Phi1, L1, V[1], Phi2, L2, p, pi) == 2)
602 : {
603 629 : if (n <= 2) return 1; /* Error: Two choices with n = 2; ambiguous */
604 629 : if (!common_nbr_verify(&W2,nbrs[0], Phi1,L1,V[2], Phi2,L2,W[0], p,pi))
605 320 : nbrs[0] = nbrs[1]; /* nbrs[1] must be the correct choice */
606 309 : else if (common_nbr_verify(&W2,nbrs[1], Phi1,L1,V[2], Phi2,L2,W[0], p,pi))
607 45 : return 1; /* Error: Both paths extend successfully */
608 : }
609 202460 : W[1] = nbrs[0];
610 202460 : if (n <= 2) return n;
611 94194 : return cycle? surface_gcd_cycle(W, V, n, Phi1, L1, Phi2, L2, 2, p, pi)
612 189466 : : surface_gcd_path (W, V, n, Phi1, L1, Phi2, L2, 2, p, pi);
613 : }
614 :
615 : GEN
616 218650 : enum_roots(ulong J0, norm_eqn_t ne, GEN fdb, GEN G, GEN vshape)
617 : { /* MAX_HEIGHT >= max_{p,n} val_p(n) where p and n are ulongs */
618 : enum { MAX_HEIGHT = BITS_IN_LONG };
619 218650 : pari_sp av, ltop = avma;
620 218650 : long s = !!pcp_get_L0(G);
621 218650 : long *n = pcp_get_n(G)+s, *L = pcp_get_L(G)+s, *o = pcp_get_o(G)+s, k = pcp_get_k(G)-s;
622 218648 : long i, t, vlen, *e, *h, *off, *poff, *M, N = pcp_get_enum_cnt(G);
623 218647 : ulong p = ne->p, pi = ne->pi, *roots;
624 : GEN Phi, vp, ve, roots_;
625 :
626 218647 : if (!k) return mkvecsmall(J0);
627 :
628 214716 : roots_ = cgetg(N + MAX_HEIGHT, t_VECSMALL);
629 214718 : roots = zv_to_ulongptr(roots_);
630 214718 : av = avma;
631 :
632 214718 : if (!vshape) vshape = factoru(ne->v);
633 214719 : vp = gel(vshape, 1); vlen = lg(vp)-1;
634 214719 : ve = gel(vshape, 2);
635 :
636 214719 : Phi = new_chunk(k);
637 214718 : e = new_chunk(k);
638 214716 : off = new_chunk(k);
639 214716 : poff = new_chunk(k);
640 : /* TODO: Surely we can work these out ahead of time? */
641 : /* h[i] is the valuation of p[i] in v */
642 214716 : h = new_chunk(k);
643 496773 : for (i = 0; i < k; ++i) {
644 282051 : h[i] = 0;
645 419749 : for (t = 1; t <= vlen; ++t)
646 327457 : if (vp[t] == L[i]) { h[i] = uel(ve, t); break; }
647 282051 : e[i] = 0;
648 282051 : off[i] = 0;
649 282051 : gel(Phi, i) = polmodular_db_getp(fdb, L[i], p);
650 : }
651 :
652 214722 : t = surface_path(roots, n[0], gel(Phi, 0), L[0], h[0], J0, NULL, p, pi);
653 214726 : if (t < n[0]) return gc_NULL(ltop); /* J0 has bad endo ring */
654 213977 : if (k == 1) { setlg(roots_, t + 1); return gc_const(av,roots_); }
655 :
656 54659 : M = new_chunk(k);
657 120695 : for (M[0] = 1, i = 1; i < k; ++i) M[i] = M[i-1] * n[i-1];
658 54659 : i = 1;
659 257122 : while (i < k) {
660 : long j, t0;
661 229284 : for (j = i + 1; j < k && ! e[j]; ++j);
662 202604 : if (j < k) {
663 80392 : if (e[i]) {
664 43027 : if (! common_nbr_pred(
665 43026 : &roots[t], roots[off[i]], gel(Phi,i), L[i],
666 43026 : roots[t - M[j]], gel(Phi, j), L[j], roots[poff[i]], p, pi)) {
667 0 : break; /* J0 has bad endo ring */
668 : }
669 37362 : } else if ( ! common_nbr_corner(
670 37366 : &roots[t], roots[off[i]], gel(Phi,i), L[i], h[i],
671 37366 : roots[t - M[j]], gel(Phi, j), L[j], roots[poff[j]], p, pi)) {
672 51 : break; /* J0 has bad endo ring */
673 : }
674 188129 : } else if ( ! next_surface_nbr(
675 122212 : &roots[t], gel(Phi,i), L[i], h[i],
676 188126 : roots[off[i]], e[i] ? &roots[poff[i]] : NULL, p, pi))
677 0 : break; /* J0 has bad endo ring */
678 202553 : if (roots[t] == roots[0]) break; /* J0 has bad endo ring */
679 :
680 202506 : poff[i] = off[i];
681 202506 : off[i] = t;
682 202506 : e[i]++;
683 239868 : for (j = i-1; j; --j) { e[j] = 0; off[j] = off[j+1]; }
684 :
685 202506 : t0 = surface_parallel_path(&roots[t], &roots[poff[i]], n[0],
686 202506 : gel(Phi, 0), L[0], gel(Phi, i), L[i], p, pi, n[0] == o[0]);
687 202508 : if (t0 < n[0]) break; /* J0 has bad endo ring */
688 :
689 : /* TODO: Do I need to check if any of the new roots is a repeat in
690 : * the case where J0 has bad endo ring? */
691 202463 : t += n[0];
692 305545 : for (i = 1; i < k && e[i] == n[i]-1; i++);
693 : }
694 54661 : if (t != N) return gc_NULL(ltop); /* J0 has wrong endo ring */
695 54518 : setlg(roots_, t + 1); return gc_const(av,roots_);
696 : }
|
