Line data Source code
1 : /* Copyright (C) 2010 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 : /** L functions of elliptic curves **/
18 : /** **/
19 : /********************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_ellanal
24 :
25 : struct baby_giant
26 : {
27 : GEN baby, giant, sum;
28 : GEN bnd, rbnd;
29 : };
30 :
31 : /* Generic Buhler-Gross algorithm */
32 :
33 : struct bg_data
34 : {
35 : GEN E, N; /* ell, conductor */
36 : GEN bnd; /* t_INT; will need all an for n <= bnd */
37 : ulong rootbnd; /* sqrt(bnd) */
38 : GEN an; /* t_VECSMALL: cache of an, n <= rootbnd */
39 : GEN p; /* t_VECSMALL: primes <= rootbnd */
40 : };
41 :
42 : typedef void bg_fun(void*el, GEN n, GEN a);
43 :
44 : /* a = a_n, where p = bg->pp[i] divides n, and lasta = a_{n/p}.
45 : * Call fun(E, N, a_N), for all N, n | N, P^+(N) <= p, a_N != 0,
46 : * i.e. assumes that fun accumulates a_N * w(N) */
47 :
48 : static void
49 1424256 : gen_BG_add(void *E, bg_fun *fun, struct bg_data *bg, GEN n, long i, GEN a, GEN lasta)
50 : {
51 1424256 : pari_sp av = avma;
52 : long j;
53 1424256 : ulong nn = itou_or_0(n);
54 1424256 : if (nn && nn <= bg->rootbnd) bg->an[nn] = itos(a);
55 :
56 1424256 : if (signe(a))
57 : {
58 358800 : fun(E, n, a);
59 358800 : j = 1;
60 : }
61 : else
62 1065456 : j = i;
63 2845458 : for(; j <= i; j++)
64 : {
65 2261340 : ulong p = bg->p[j];
66 2261340 : GEN nexta, pn = mului(p, n);
67 2261340 : if (cmpii(pn, bg->bnd) > 0) return;
68 1421202 : nexta = mulis(a, bg->an[p]);
69 1421202 : if (i == j && umodiu(bg->N, p)) nexta = subii(nexta, mului(p, lasta));
70 1421202 : gen_BG_add(E, fun, bg, pn, j, nexta, a);
71 1421202 : set_avma(av);
72 : }
73 : }
74 :
75 : static void
76 36 : gen_BG_init(struct bg_data *bg, GEN E, GEN N, GEN bnd)
77 : {
78 36 : bg->E = E;
79 36 : bg->N = N;
80 36 : bg->bnd = bnd;
81 36 : bg->rootbnd = itou(sqrtint(bnd));
82 36 : bg->p = primes_upto_zv(bg->rootbnd);
83 36 : bg->an = ellanQ_zv(E, bg->rootbnd);
84 36 : }
85 :
86 : static void
87 36 : gen_BG_rec(void *E, bg_fun *fun, struct bg_data *bg)
88 : {
89 36 : long i, j, lp = lg(bg->p)-1;
90 36 : GEN bndov2 = shifti(bg->bnd, -1);
91 36 : pari_sp av = avma, av2;
92 : GEN p;
93 : forprime_t S;
94 36 : (void)forprime_init(&S, utoipos(bg->p[lp]+1), bg->bnd);
95 36 : av2 = avma;
96 36 : if (DEBUGLEVEL)
97 0 : err_printf("1st stage, using recursion for p <= %ld\n", bg->p[lp]);
98 3090 : for (i = 1; i <= lp; i++)
99 : {
100 3054 : ulong pp = bg->p[i];
101 3054 : long ap = bg->an[pp];
102 3054 : gen_BG_add(E, fun, bg, utoipos(pp), i, stoi(ap), gen_1);
103 3054 : set_avma(av2);
104 : }
105 36 : if (DEBUGLEVEL) err_printf("2nd stage, looping for p <= %Ps\n", bndov2);
106 992286 : while ( (p = forprime_next(&S)) )
107 : {
108 : long jmax;
109 992286 : GEN ap = ellap(bg->E, p);
110 992286 : pari_sp av3 = avma;
111 992286 : if (!signe(ap)) continue;
112 :
113 495870 : jmax = itou( divii(bg->bnd, p) ); /* 2 <= jmax <= el->rootbound */
114 495870 : fun(E, p, ap);
115 7901220 : for (j = 2; j <= jmax; j++)
116 : {
117 7405350 : long aj = bg->an[j];
118 : GEN a, n;
119 7405350 : if (!aj) continue;
120 1158936 : a = mulis(ap, aj);
121 1158936 : n = muliu(p, j);
122 1158936 : fun(E, n, a);
123 1158936 : set_avma(av3);
124 : }
125 495870 : set_avma(av2);
126 495870 : if (abscmpii(p, bndov2) >= 0) break;
127 : }
128 36 : if (DEBUGLEVEL) err_printf("3nd stage, looping for p <= %Ps\n", bg->bnd);
129 892686 : while ( (p = forprime_next(&S)) )
130 : {
131 892650 : GEN ap = ellap(bg->E, p);
132 892650 : if (!signe(ap)) continue;
133 446034 : fun(E, p, ap);
134 446034 : set_avma(av2);
135 : }
136 36 : set_avma(av);
137 36 : }
138 :
139 : /******************************************************************
140 : *
141 : * L functions of elliptic curves
142 : * Pascal Molin (molin.maths@gmail.com) 2014
143 : *
144 : ******************************************************************/
145 :
146 : struct lcritical
147 : {
148 : GEN h; /* real */
149 : long cprec; /* computation prec */
150 : long L; /* number of points */
151 : GEN K; /* length of series */
152 : long real;
153 : };
154 :
155 : static double
156 210 : logboundG0(long e, double aY)
157 : {
158 : double cla, loggam;
159 210 : cla = 1 + 1/sqrt(aY);
160 210 : if (e) cla = ( cla + 1/(2*aY) ) / (2*sqrt(aY));
161 210 : loggam = (e) ? M_LN2-aY : -aY + log( log( 1+1/aY) );
162 210 : return log(cla) + loggam;
163 : }
164 :
165 : static void
166 210 : param_points(GEN N, double Y, double tmax, long bprec, long *cprec, long *L,
167 : GEN *K, double *h)
168 : {
169 : double D, a, aY, X, logM;
170 210 : long d = 2, w = 1;
171 210 : tmax *= d;
172 210 : D = bprec * M_LN2 + M_PI/4*tmax + 2;
173 210 : *cprec = nbits2prec(ceil(D / M_LN2) + 5);
174 210 : a = 2 * M_PI / sqrt(gtodouble(N));
175 210 : aY = a * cos(M_PI/2*Y);
176 210 : logM = 2*M_LN2 + logboundG0(w+1, aY) + tmax * Y * M_PI/2;
177 210 : *h = ( 2 * M_PI * M_PI / 2 * Y ) / ( D + logM );
178 210 : X = log( D / a);
179 210 : *L = ceil( X / *h);
180 210 : *K = ceil_safe(dbltor( D / a ));
181 210 : }
182 :
183 : static GEN
184 210 : vecF2_lk(GEN E, GEN K, GEN rbnd, GEN Q, GEN sleh, long prec)
185 : {
186 : pari_sp av;
187 210 : long l, L = lg(K)-1;
188 210 : GEN a = ellanQ_zv(E, itos(gel(K,1)));
189 210 : GEN S = cgetg(L+1, t_VEC);
190 :
191 11808 : for (l = 1; l <= L; l++) gel(S,l) = cgetr(prec);
192 210 : av = avma;
193 11808 : for (l = 1; l <= L; l++)
194 : {
195 : GEN e1, Sl, z, zB;
196 11598 : long aB, b, A, B, Kl = itou(gel(K,l));
197 : pari_sp av2;
198 : /* FIXME: could reduce prec here (useful for large prec) */
199 11598 : e1 = gel(Q, l);
200 11598 : Sl = real_0(prec);;
201 : /* baby-step giant step */
202 11598 : B = A = rbnd[l];
203 11598 : z = powersr(e1, B); zB = gel(z, B+1);
204 11598 : av2 = avma;
205 269292 : for (aB = A*B; aB >= 0; aB -= B)
206 : {
207 257694 : GEN s = real_0(prec); /* could change also prec here */
208 32599302 : for (b = B; b > 0; --b)
209 : {
210 32341608 : long k = aB+b;
211 32341608 : if (k <= Kl && a[k]) s = addrr(s, mulsr(a[k], gel(z, b+1)));
212 32341608 : if (gc_needed(av2, 1)) (void)gc_all(av2, 2, &s, &Sl);
213 : }
214 257694 : Sl = addrr(mulrr(Sl, zB), s);
215 : }
216 11598 : affrr(mulrr(Sl, gel(sleh,l)), gel(S, l)); /* to avoid copying all S */
217 11598 : set_avma(av);
218 : }
219 210 : return S;
220 : }
221 :
222 : /* Return C, C[i][j] = Q[j]^i, i = 1..nb */
223 : static void
224 0 : baby_init(struct baby_giant *bb, GEN Q, GEN bnd, GEN rbnd, long prec)
225 : {
226 0 : long i, j, l = lg(Q);
227 : GEN R, C;
228 0 : C = cgetg(l,t_VEC);
229 0 : for (i = 1; i < l; ++i) gel(C, i) = powersr(gel(Q, i), rbnd[i]);
230 0 : R = cgetg(l,t_VEC);
231 0 : for (i = 1; i < l; ++i)
232 : {
233 0 : gel(R, i) = cgetg(rbnd[i]+1, t_VEC);
234 0 : gmael(R, i, 1) = rtor(gmael(C, i, 2), prec);
235 0 : for (j = 2; j <= rbnd[i]; j++) gmael(R, i, j) = stor(0, prec);
236 : }
237 0 : bb->baby = C; bb->giant = R;
238 0 : bb->bnd = bnd; bb->rbnd = rbnd;
239 0 : }
240 :
241 : static long
242 210 : baby_size(GEN rbnd, long Ks, long prec)
243 : {
244 210 : long i, s, m, l = lg(rbnd);
245 11808 : for (s = 0, i = 1; i < l; ++i)
246 11598 : s += rbnd[i];
247 210 : m = 2*s*prec + 3*l + s;
248 210 : if (DEBUGLEVEL)
249 0 : err_printf("ellL1: BG_add: %ld words, ellan: %ld words\n", m, Ks);
250 210 : return m;
251 : }
252 :
253 : static void
254 0 : ellL1_add(void *E, GEN n, GEN a)
255 : {
256 0 : pari_sp av = avma;
257 0 : struct baby_giant *bb = (struct baby_giant*) E;
258 0 : long j, l = lg(bb->giant);
259 0 : for (j = 1; j < l; j++)
260 0 : if (cmpii(n, gel(bb->bnd,j)) <= 0)
261 : {
262 0 : ulong r, q = uabsdiviu_rem(n, bb->rbnd[j], &r);
263 0 : GEN giant = gel(bb->giant, j), baby = gel(bb->baby, j);
264 0 : affrr(addrr(gel(giant, q+1), mulri(gel(baby, r+1), a)), gel(giant, q+1));
265 0 : set_avma(av);
266 0 : } else break;
267 0 : }
268 :
269 : static GEN
270 0 : vecF2_lk_bsgs(GEN E, GEN bnd, GEN rbnd, GEN Q, GEN sleh, GEN N, long prec)
271 : {
272 : struct bg_data bg;
273 : struct baby_giant bb;
274 0 : long k, L = lg(bnd)-1;
275 : GEN S;
276 0 : baby_init(&bb, Q, bnd, rbnd, prec);
277 0 : gen_BG_init(&bg, E, N, gel(bnd,1));
278 0 : gen_BG_rec((void*) &bb, ellL1_add, &bg);
279 0 : S = cgetg(L+1, t_VEC);
280 0 : for (k = 1; k <= L; ++k)
281 : {
282 0 : pari_sp av = avma;
283 0 : long j, g = rbnd[k];
284 0 : GEN giant = gmael(bb.baby, k, g+1), Sl = gmael(bb.giant, k, g);
285 0 : for (j = g-1; j >=1; j--) Sl = addrr(mulrr(Sl, giant), gmael(bb.giant,k,j));
286 0 : gel(S, k) = gc_leaf(av, mulrr(gel(sleh,k), Sl));
287 : }
288 0 : return S;
289 : }
290 :
291 : static long
292 11598 : _sqrt(GEN x) { pari_sp av = avma; return gc_long(av, itou(sqrtint(x))); }
293 :
294 : static GEN
295 210 : vecF(struct lcritical *C, GEN E)
296 : {
297 210 : pari_sp av = avma;
298 210 : long prec = C->cprec, Ks = itos_or_0(C->K), L = C->L, l;
299 210 : GEN N = ellQ_get_N(E), PiN;
300 210 : GEN e = mpexp(C->h), elh = powersr(e, L-1), Q, bnd, rbnd, vec;
301 :
302 210 : PiN = divrr(Pi2n(1,prec), sqrtr_abs(itor(N, prec)));
303 210 : setsigne(PiN, -1); /* - 2Pi/sqrt(N) */
304 210 : bnd = gpowers0(invr(e), L-1, C->K); /* bnd[i] = K exp(-(i-1)h) */
305 210 : rbnd = cgetg(L+1, t_VECSMALL);
306 210 : Q = cgetg(L+1, t_VEC);
307 11808 : for (l = 1; l <= L; l++)
308 : {
309 11598 : gel(bnd,l) = ceil_safe(gel(bnd,l));
310 11598 : rbnd[l] = _sqrt(gel(bnd,l)) + 1;
311 11598 : gel(Q, l) = mpexp(mulrr(PiN, gel(elh, l)));
312 : }
313 210 : if (Ks && baby_size(rbnd, Ks, prec) > (Ks>>1))
314 210 : vec = vecF2_lk(E, bnd, rbnd, Q, elh, prec);
315 : else
316 0 : vec = vecF2_lk_bsgs(E, bnd, rbnd, Q, elh, N, prec);
317 210 : return gc_upto(av, vec);
318 : }
319 :
320 : /* Lambda function by Fourier inversion. vec is a grid, t a scalar or t_SER */
321 : static GEN
322 234 : glambda(GEN t, GEN vec, GEN h, long real, long prec)
323 : {
324 234 : GEN z, r, e = gexp(gmul(mkcomplex(gen_0,h), t), prec);
325 234 : long n = lg(vec)-1, i;
326 :
327 234 : r = real == 1? gmul2n(real_i(gel(vec, 1)), -1): gen_0;
328 234 : z = real == 1? e: gmul(powIs(3), e);
329 : /* FIXME: summing backward may be more stable */
330 13944 : for (i = 2; i <= n; i++)
331 : {
332 13710 : r = gadd(r, real_i(gmul(gel(vec,i), z)));
333 13710 : if (i < n) z = gmul(z, e);
334 : }
335 234 : return gmul(mulsr(4, h), r);
336 : }
337 :
338 : static GEN
339 210 : Lpoints(struct lcritical *C, GEN e, double tmax, long bprec)
340 : {
341 210 : double h = 0, Y = .97;
342 210 : GEN N = ellQ_get_N(e);
343 210 : param_points(N, Y, tmax, bprec, &C->cprec, &C->L, &C->K, &h);
344 210 : C->real = ellrootno_global(e);
345 210 : C->h = rtor(dbltor(h), C->cprec);
346 210 : return vecF(C, e);
347 : }
348 :
349 : static GEN
350 234 : Llambda(GEN vec, struct lcritical *C, GEN t, long prec)
351 : {
352 234 : GEN lambda = glambda(gprec_w(t, C->cprec), vec, C->h, C->real, C->cprec);
353 234 : return gprec_w(lambda, prec);
354 : }
355 :
356 : /* 2*(2*Pi)^(-s)*gamma(s)*N^(s/2); */
357 : static GEN
358 234 : ellgammafactor(GEN N, GEN s, long prec)
359 : {
360 234 : GEN c = gpow(divrr(gsqrt(N,prec), Pi2n(1,prec)), s, prec);
361 234 : return gmul(gmul2n(c,1), ggamma(s, prec));
362 : }
363 :
364 : static GEN
365 234 : ellL1_eval(GEN e, GEN vec, struct lcritical *C, GEN t, long prec)
366 : {
367 234 : GEN g = ellgammafactor(ellQ_get_N(e), gaddgs(gmul(gen_I(),t), 1), prec);
368 234 : return gdiv(Llambda(vec, C, t, prec), g);
369 : }
370 :
371 : static GEN
372 234 : ellL1_der(GEN e, GEN vec, struct lcritical *C, GEN t, long der, long prec)
373 : {
374 234 : GEN r = polcoef_i(ellL1_eval(e, vec, C, t, prec), der, 0);
375 234 : r = gmul(r,powIs(C->real == 1 ? -der: 1-der));
376 234 : return gmul(real_i(r), mpfact(der));
377 : }
378 :
379 : GEN
380 198 : ellL1(GEN E, long r, long bitprec)
381 : {
382 198 : pari_sp av = avma;
383 : struct lcritical C;
384 198 : long prec = nbits2prec(bitprec);
385 : GEN e, vec, t;
386 198 : if (r < 0)
387 6 : pari_err_DOMAIN("ellL1", "derivative order", "<", gen_0, stoi(r));
388 192 : e = ellanal_globalred(E, NULL);
389 192 : if (r == 0 && ellrootno_global(e) < 0) { set_avma(av); return gen_0; }
390 180 : vec = Lpoints(&C, e, 0., bitprec);
391 180 : t = r ? scalarser(gen_1, 0, r): zeroser(0, 0);
392 180 : setvalser(t, 1);
393 180 : return gc_upto(av, ellL1_der(e, vec, &C, t, r, prec));
394 : }
395 :
396 : GEN
397 30 : ellanalyticrank(GEN E, GEN eps, long bitprec)
398 : {
399 30 : pari_sp av = avma, av2;
400 30 : long prec = nbits2prec(bitprec);
401 : struct lcritical C;
402 : pari_timer ti;
403 : GEN e, vec;
404 : long rk;
405 30 : if (DEBUGLEVEL) timer_start(&ti);
406 30 : if (!eps)
407 30 : eps = real2n(-bitprec/2+1, DEFAULTPREC);
408 : else
409 0 : if (typ(eps) != t_REAL) {
410 0 : eps = gtofp(eps, DEFAULTPREC);
411 0 : if (typ(eps) != t_REAL) pari_err_TYPE("ellanalyticrank", eps);
412 : }
413 30 : e = ellanal_globalred(E, NULL);
414 30 : vec = Lpoints(&C, e, 0., bitprec);
415 30 : if (DEBUGLEVEL) timer_printf(&ti, "init L");
416 30 : av2 = avma;
417 30 : for (rk = C.real>0 ? 0: 1; ; rk += 2)
418 24 : {
419 : GEN Lrk;
420 54 : GEN t = rk ? scalarser(gen_1, 0, rk): zeroser(0, 0);
421 54 : setvalser(t, 1);
422 54 : Lrk = ellL1_der(e, vec, &C, t, rk, prec);
423 54 : if (DEBUGLEVEL) timer_printf(&ti, "L^(%ld)=%Ps", rk, Lrk);
424 54 : if (abscmprr(Lrk, eps) > 0)
425 30 : return gc_GEN(av, mkvec2(stoi(rk), Lrk));
426 24 : set_avma(av2);
427 : }
428 : }
429 :
430 : /* Heegner point computation
431 :
432 : This section is a C version by Bill Allombert of a GP script by
433 : Christophe Delaunay which was based on a GP script by John Cremona.
434 : Reference: Henri Cohen's book GTM 239.
435 : */
436 :
437 : static void
438 0 : heegner_L1_bg(void*E, GEN n, GEN a)
439 : {
440 0 : struct baby_giant *bb = (struct baby_giant*) E;
441 0 : long j, l = lg(bb->giant);
442 0 : for (j = 1; j < l; j++)
443 0 : if (cmpii(n, gel(bb->bnd,j)) <= 0)
444 : {
445 0 : ulong r, q = uabsdiviu_rem(n, bb->rbnd[j], &r);
446 0 : GEN giant = gel(bb->giant, j), baby = gel(bb->baby, j);
447 0 : affgc(gadd(gel(giant, q+1), gdiv(gmul(gel(baby, r+1), a), n)),
448 0 : gel(giant, q+1));
449 : }
450 0 : }
451 :
452 : static void
453 2459640 : heegner_L1(void*E, GEN n, GEN a)
454 : {
455 2459640 : struct baby_giant *bb = (struct baby_giant*) E;
456 2459640 : long j, l = lg(bb->giant);
457 14407476 : for (j = 1; j < l; j++)
458 11947836 : if (cmpii(n, gel(bb->bnd,j)) <= 0)
459 : {
460 10193778 : ulong r, q = uabsdiviu_rem(n, bb->rbnd[j], &r);
461 10193778 : GEN giant = gel(bb->giant, j), baby = gel(bb->baby, j);
462 10193778 : GEN ex = mulreal(gel(baby, r+1), gel(giant, q+1));
463 10193778 : affrr(addrr(gel(bb->sum, j), divri(mulri(ex, a), n)),
464 10193778 : gel(bb->sum, j));
465 : }
466 2459640 : }
467 : /* export ? */
468 : static GEN
469 0 : ctoc(GEN x, long prec)
470 0 : { GEN y = cgetc(prec); affgc(x, y); return y; }
471 :
472 : /* [powers(x[i], n[i]), i=1..#x] */
473 : static GEN
474 36 : RgV_zv_powers(GEN x, GEN n)
475 162 : { pari_APPLY_same(gpowers(gel(x,i), n[i])); }
476 :
477 : /* Return C, C[i][j] = Q[j]^i, i = 1..nb */
478 : static void
479 0 : baby_init2(struct baby_giant *bb, GEN Q, GEN bnd, GEN rbnd, long prec)
480 : {
481 0 : long i, j, l = lg(Q);
482 0 : GEN C = RgV_zv_powers(Q, rbnd), R = cgetg(l,t_VEC);
483 0 : for (i = 1; i < l; ++i)
484 : {
485 0 : gel(R, i) = cgetg(rbnd[i]+1, t_VEC);
486 0 : gmael(R, i, 1) = ctoc(gmael(C, i, 2), prec);
487 0 : for (j = 2; j <= rbnd[i]; j++) gmael(R, i, j) = ctoc(gen_0, prec);
488 : }
489 0 : bb->baby = C; bb->giant = R;
490 0 : bb->bnd = bnd; bb->rbnd = rbnd;
491 0 : }
492 :
493 : /* Return C, C[i][j] = Q[j]^i, i = 1..nb */
494 : static void
495 36 : baby_init3(struct baby_giant *bb, GEN Q, GEN bnd, GEN rbnd, long prec)
496 : {
497 36 : long i, l = lg(Q);
498 36 : GEN S, C = RgV_zv_powers(Q, rbnd), R = cgetg(l,t_VEC);
499 162 : for (i = 1; i < l; ++i)
500 126 : gel(R, i) = gpowers(gmael(C, i, 1+rbnd[i]), rbnd[i]);
501 36 : S = cgetg(l,t_VEC);
502 162 : for (i = 1; i < l; ++i) gel(S, i) = rtor(real_i(gmael(C, i, 2)), prec);
503 36 : bb->baby = C; bb->giant = R; bb->sum = S;
504 36 : bb->bnd = bnd; bb->rbnd = rbnd;
505 36 : }
506 :
507 : /* ymin a t_REAL */
508 : static GEN
509 36 : heegner_psi(GEN E, GEN N, GEN points, long bitprec)
510 : {
511 36 : pari_sp av = avma, av2;
512 : struct baby_giant bb;
513 : struct bg_data bg;
514 36 : long l, k, L = lg(points)-1, prec = nbits2prec(bitprec)+EXTRAPREC64;
515 36 : GEN Q, pi2 = Pi2n(1, prec), bnd, rbnd, bndmax;
516 36 : GEN B = divrr(mulur(bitprec,mplog2(DEFAULTPREC)), pi2);
517 :
518 36 : rbnd = cgetg(L+1, t_VECSMALL); av2 = avma;
519 36 : bnd = cgetg(L+1, t_VEC);
520 36 : Q = cgetg(L+1, t_VEC);
521 162 : for (l = 1; l <= L; ++l)
522 : {
523 126 : gel(bnd,l) = ceil_safe(divrr(B,imag_i(gel(points, l))));
524 126 : rbnd[l] = itou(sqrtint(gel(bnd,l)))+1;
525 126 : gel(Q, l) = expIxy(pi2, gel(points, l), prec);
526 : }
527 36 : (void)gc_all(av2, 2, &bnd, &Q);
528 36 : bndmax = gel(bnd,vecindexmax(bnd));
529 36 : gen_BG_init(&bg, E, N, bndmax);
530 36 : if (bitprec >= 1900)
531 : {
532 0 : GEN S = cgetg(L+1, t_VEC);
533 0 : baby_init2(&bb, Q, bnd, rbnd, prec);
534 0 : gen_BG_rec((void*)&bb, heegner_L1_bg, &bg);
535 0 : for (k = 1; k <= L; ++k)
536 : {
537 0 : pari_sp av2 = avma;
538 0 : long j, g = rbnd[k];
539 0 : GEN giant = gmael(bb.baby, k, g+1), Sl = real_0(prec);
540 0 : for (j = g; j >=1; j--) Sl = gadd(gmul(Sl, giant), gmael(bb.giant,k,j));
541 0 : gel(S, k) = gc_upto(av2, real_i(Sl));
542 : }
543 0 : return gc_upto(av, S);
544 : }
545 : else
546 : {
547 36 : baby_init3(&bb, Q, bnd, rbnd, prec);
548 36 : gen_BG_rec((void*)&bb, heegner_L1, &bg);
549 36 : return gc_GEN(av, bb.sum);
550 : }
551 : }
552 :
553 : /*Returns lambda_bad list for one prime p, nv = localred(E, p) */
554 : static GEN
555 78 : lambda1(GEN E, GEN nv, GEN p, long prec)
556 : {
557 : pari_sp av;
558 : GEN res, lp;
559 78 : long kod = itos(gel(nv, 2));
560 78 : if (kod==2 || kod ==-2) return cgetg(1,t_VEC);
561 78 : av = avma; lp = glog(p, prec);
562 78 : if (kod > 4)
563 : {
564 12 : long n = Z_pval(ell_get_disc(E), p);
565 12 : long j, m = kod - 4, nl = 1 + (m >> 1L);
566 12 : res = cgetg(nl, t_VEC);
567 30 : for (j = 1; j < nl; j++)
568 18 : gel(res, j) = gmul(lp, gsubgs(gdivgu(sqru(j), n), j)); /* j^2/n - j */
569 : }
570 66 : else if (kod < -4)
571 12 : res = mkvec2(negr(lp), shiftr(mulrs(lp, kod), -2));
572 : else
573 : {
574 54 : const long lam[] = {8,9,0,6,0,0,0,3,4};
575 54 : long m = -lam[kod+4];
576 54 : res = mkvec(divru(mulrs(lp, m), 6));
577 : }
578 78 : return gc_GEN(av, res);
579 : }
580 :
581 : static GEN
582 36 : lambdalist(GEN E, long prec)
583 : {
584 36 : pari_sp ltop = avma;
585 36 : GEN glob = ellglobalred(E), plist = gmael(glob,4,1), L = gel(glob,5);
586 36 : GEN res, v, D = ell_get_disc(E);
587 36 : long i, j, k, l, m, n, np = lg(plist), lr = 1;
588 36 : v = cgetg(np, t_VEC);
589 126 : for (j = 1, i = 1 ; j < np; ++j)
590 : {
591 90 : GEN p = gel(plist, j);
592 90 : if (dvdii(D, sqri(p)))
593 : {
594 78 : GEN la = lambda1(E, gel(L,j), p, prec);
595 78 : gel(v, i++) = la;
596 78 : lr *= lg(la);
597 : }
598 : }
599 36 : np = i;
600 36 : res = cgetg(lr+1, t_VEC);
601 36 : gel(res, 1) = gen_0; n = 1; m = 1;
602 114 : for (j = 1; j < np; ++j)
603 : {
604 78 : GEN w = gel(v, j);
605 78 : long lw = lg(w);
606 264 : for (k = 1; k <= n; k++)
607 : {
608 186 : GEN t = gel(res, k);
609 390 : for (l = 1, m = n; l < lw; l++, m+=n)
610 204 : gel(res, k + m) = mpadd(t, gel(w, l));
611 : }
612 78 : n = m;
613 : }
614 36 : return gc_GEN(ltop, res);
615 : }
616 :
617 : /* P a t_INT or t_FRAC, return its logarithmic height */
618 : static GEN
619 84 : heightQ(GEN P, long prec)
620 : {
621 : long s;
622 84 : if (typ(P) == t_FRAC)
623 : {
624 48 : GEN a = gel(P,1), b = gel(P,2);
625 48 : P = abscmpii(a,b) > 0 ? a: b;
626 : }
627 84 : s = signe(P);
628 84 : if (!s) return real_0(prec);
629 72 : if (s < 0) P = negi(P);
630 72 : return glog(P, prec);
631 : }
632 :
633 : /* t a t_INT or t_FRAC, returns max(1, log |t|), returns a t_REAL */
634 : static GEN
635 126 : logplusQ(GEN t, long prec)
636 : {
637 126 : if (typ(t) == t_INT)
638 : {
639 36 : if (!signe(t)) return real_1(prec);
640 24 : t = absi_shallow(t);
641 : }
642 : else
643 : {
644 90 : GEN a = gel(t,1), b = gel(t,2);
645 90 : if (abscmpii(a, b) < 0) return real_1(prec);
646 48 : if (signe(a) < 0) t = gneg(t);
647 : }
648 72 : return glog(t, prec);
649 : }
650 :
651 : /* See GTM239, p532, Th 8.1.18
652 : * Return M such that h_naive <= M */
653 : GEN
654 84 : hnaive_max(GEN ell, GEN ht)
655 : {
656 84 : const long prec = LOWDEFAULTPREC; /* minimal accuracy */
657 84 : GEN b2 = ell_get_b2(ell), j = ell_get_j(ell);
658 84 : GEN logd = glog(absi_shallow(ell_get_disc(ell)), prec);
659 84 : GEN logj = logplusQ(j, prec);
660 84 : GEN hj = heightQ(j, prec);
661 42 : GEN logb2p = signe(b2)? addrr(logplusQ(gdivgu(b2, 12),prec), mplog2(prec))
662 84 : : real_1(prec);
663 84 : GEN mu = addrr(divru(addrr(logd, logj),6), logb2p);
664 84 : return addrs(addrr(addrr(ht, divru(hj,12)), mu), 2);
665 : }
666 :
667 : static GEN
668 126 : qfb_root(GEN Q, GEN vDi)
669 : {
670 126 : GEN a2 = shifti(gel(Q, 1),1), b = gel(Q, 2);
671 126 : return mkcomplex(gdiv(negi(b),a2),divri(vDi,a2));
672 : }
673 :
674 : static GEN
675 21144 : qimag2(GEN Q)
676 : {
677 21144 : pari_sp av = avma;
678 21144 : GEN z = gdiv(negi(qfb_disc(Q)), shifti(sqri(gel(Q, 1)),2));
679 21144 : return gc_upto(av, z);
680 : }
681 :
682 : /***************************************************/
683 : /*Routines for increasing the imaginary parts using*/
684 : /*Atkin-Lehner operators */
685 : /***************************************************/
686 :
687 : static GEN
688 21144 : qfb_mult(GEN Q, GEN a, GEN b, GEN c, GEN d)
689 : {
690 21144 : GEN A = gel(Q, 1) , B = gel(Q, 2), C = gel(Q, 3), D = qfb_disc(Q);
691 21144 : GEN a2 = sqri(a), b2 = sqri(b), c2 = sqri(c), d2 = sqri(d);
692 21144 : GEN ad = mulii(d, a), bc = mulii(b, c), e = subii(ad, bc);
693 21144 : GEN W1 = addii(addii(mulii(a2, A), mulii(mulii(c, a), B)), mulii(c2, C));
694 21144 : GEN W3 = addii(addii(mulii(b2, A), mulii(mulii(d, b), B)), mulii(d2, C));
695 21144 : GEN W2 = addii(addii(mulii(mulii(shifti(b,1), a), A),
696 : mulii(addii(ad, bc), B)),
697 : mulii(mulii(shifti(d,1), c), C));
698 21144 : if (!equali1(e)) {
699 19020 : W1 = diviiexact(W1,e);
700 19020 : W2 = diviiexact(W2,e);
701 19020 : W3 = diviiexact(W3,e);
702 : }
703 21144 : return mkqfb(W1, W2, W3, D);
704 : }
705 :
706 : #ifdef DEBUG
707 : static void
708 : best_point_old(GEN Q, GEN NQ, GEN f, GEN *u, GEN *v)
709 : {
710 : long n, k;
711 : GEN U, c, d, A = gel(f,1), B = gel(f,2), C = gel(f,3), D = qfb_disc(f);
712 : GEN q = mkqfb(mulii(NQ, C), negi(B), diviiexact(A, NQ), D);
713 : redimagsl2(q, &U);
714 : *u = c = gcoeff(U, 1, 1);
715 : *v = d = gcoeff(U, 2, 1);
716 : if (equali1(gcdii(mulii(*u, NQ), mulii(*v, Q)))) return;
717 : for (n = 1;; n++)
718 : {
719 : for (k = -n; k <= n; k++)
720 : {
721 : *u = addis(c, k); *v = addiu(d, n);
722 : if (equali1(gcdii(mulii(*u, NQ), mulii(*v, Q)))) return;
723 : *v = subiu(d, n);
724 : if (equali1(gcdii(mulii(*u, NQ), mulii(*v, Q)))) return;
725 : *u = addiu(c, n); *v = addis(d, k);
726 : if (equali1(gcdii(mulii(*u, NQ), mulii(*v, Q)))) return;
727 : *u = subiu(c, n);
728 : if (equali1(gcdii(mulii(*u, NQ), mulii(*v, Q)))) return;
729 : }
730 : }
731 : }
732 : /* q(x,y) = ax^2 + bxy + cy^2 */
733 : static GEN
734 : qfb_eval(GEN q, GEN x, GEN y)
735 : {
736 : GEN a = gel(q,1), b = gel(q,2), c = gel(q,3);
737 : GEN x2 = sqri(x), y2 = sqri(y), xy = mulii(x,y);
738 : return addii(addii(mulii(a, x2), mulii(b,xy)), mulii(c, y2));
739 : }
740 : #endif
741 :
742 : static long
743 5652 : nexti(long i) { return i>0 ? -i : 1-i; }
744 :
745 : /* q0 + i q1 + i^2 q2 */
746 : static GEN
747 10572 : qfmin_eval(GEN q0, GEN q1, GEN q2, long i)
748 10572 : { return addii(mulis(addii(mulis(q2, i), q1), i), q0); }
749 :
750 : /* assume a > 0, return gcd(a,b,c) */
751 : static ulong
752 14094 : gcduii(ulong a, GEN b, GEN c)
753 : {
754 14094 : a = ugcdiu(b, a);
755 14094 : return a == 1? 1: ugcdiu(c, a);
756 : }
757 :
758 : static void
759 21144 : best_point(GEN Q, GEN NQ, GEN f, GEN *pu, GEN *pv)
760 : {
761 21144 : GEN a = mulii(NQ, gel(f,3)), b = negi(gel(f,2)), c = diviiexact(gel(f,1), NQ);
762 21144 : GEN D = qfb_disc(f);
763 21144 : GEN U, qr = redimagsl2(mkqfb(a, b, c, D), &U);
764 21144 : GEN A = gel(qr,1), B = gel(qr,2), A2 = shifti(A,1), AA4 = sqri(A2);
765 : GEN V, best;
766 : long y;
767 :
768 21144 : D = absi_shallow(D);
769 : /* 4A qr(x,y) = (2A x + By)^2 + D y^2
770 : * Write x = x0(y) + i, where x0 is an integer minimum
771 : * (the smallest in case of tie) of x-> qr(x,y), for given y.
772 : * 4A qr(x,y) = ((2A x0 + By)^2 + Dy^2) + 4A i (2A x0 + By) + 4A^2 i^2
773 : * = q0(y) + q1(y) i + q2 i^2
774 : * Loop through (x,y), y>0 by (roughly) increasing values of qr(x,y) */
775 :
776 : /* We must test whether [X,Y]~ := U * [x,y]~ satisfy (X NQ, Y Q) = 1
777 : * This is equivalent to (X,Y) = 1 (note that (X,Y) = (x,y)), and
778 : * (X, Q) = (Y, NQ) = 1.
779 : * We have U * [x0+i, y]~ = U * [x0,y]~ + i U[,1] =: V0 + i U[,1] */
780 :
781 : /* try [1,0]~ = first minimum */
782 21144 : V = gel(U,1); /* U *[1,0]~ */
783 21144 : *pu = gel(V,1);
784 21144 : *pv = gel(V,2);
785 26526 : if (is_pm1(gcdii(*pu, Q)) && is_pm1(gcdii(*pv, NQ))) return;
786 :
787 : /* try [0,1]~ = second minimum */
788 10230 : V = gel(U,2); /* U *[0,1]~ */
789 10230 : *pu = gel(V,1);
790 10230 : *pv = gel(V,2);
791 10230 : if (is_pm1(gcdii(*pu, Q)) && is_pm1(gcdii(*pv, NQ))) return;
792 :
793 : /* (X,Y) = (1, \pm1) always works. Try to do better now */
794 4848 : best = subii(addii(a, c), absi_shallow(b));
795 4848 : *pu = gen_1;
796 4848 : *pv = signe(b) < 0? gen_1: gen_m1;
797 :
798 4848 : for (y = 1;; y++)
799 7830 : {
800 : GEN Dy2, r, By, x0, q0, q1, V0;
801 : long i;
802 12678 : if (y > 1)
803 : {
804 9174 : if (gcduii(y, gcoeff(U,1,1), Q) != 1) continue;
805 6264 : if (gcduii(y, gcoeff(U,2,1), NQ) != 1) continue;
806 : }
807 9774 : Dy2 = mulii(D, sqru(y));
808 9774 : if (cmpii(Dy2, best) >= 0) break; /* we won't improve. STOP */
809 4926 : By = muliu(B,y), x0 = truedvmdii(negi(By), A2, &r);
810 4926 : if (cmpii(r, A) >= 0) { x0 = subiu(x0,1); r = subii(r, A2); }
811 : /* (2A x + By)^2 + Dy^2, minimal at x = x0. Assume A2 > 0 */
812 : /* r = 2A x0 + By */
813 4926 : q0 = addii(sqri(r), Dy2); /* minimal value for this y, at x0 */
814 4926 : if (cmpii(q0, best) >= 0) continue; /* we won't improve for this y */
815 4920 : q1 = shifti(mulii(A2, r), 1);
816 :
817 4920 : V0 = ZM_ZC_mul(U, mkcol2(x0, utoipos(y)));
818 10572 : for (i = 0;; i = nexti(i))
819 5652 : {
820 10572 : pari_sp av2 = avma;
821 10572 : GEN x, N = qfmin_eval(q0, q1, AA4, i);
822 10572 : if (cmpii(N , best) >= 0) break;
823 10542 : x = addis(x0, i);
824 10542 : if (ugcdiu(x, y) == 1)
825 : {
826 : GEN u, v;
827 10506 : V = ZC_add(V0, ZC_z_mul(gel(U,1), i)); /* [X, Y] */
828 10506 : u = gel(V,1);
829 10506 : v = gel(V,2);
830 10506 : if (is_pm1(gcdii(u, Q)) && is_pm1(gcdii(v, NQ)))
831 : {
832 4890 : *pu = u;
833 4890 : *pv = v;
834 4890 : best = N; break;
835 : }
836 : }
837 5652 : set_avma(av2);
838 : }
839 : }
840 : #ifdef DEBUG
841 : {
842 : GEN oldu, oldv, F = mkqfb(a, b, c, qfb_disc(f));
843 : best_point_old(Q, NQ, f, &oldu, &oldv);
844 : if (!equalii(oldu, *pu) || !equalii(oldv, *pv))
845 : {
846 : if (!equali1(gcdii(mulii(*pu, NQ), mulii(*pv, Q))))
847 : pari_err_BUG("best_point (gcd)");
848 : if (cmpii(qfb_eval(F, *pu,*pv), qfb_eval(F, oldu, oldv)) > 0)
849 : {
850 : pari_warn(warner, "%Ps,%Ps,%Ps, %Ps > %Ps",
851 : Q,NQ,f, mkvec2(*pu,*pv), mkvec2(oldu,oldv));
852 : pari_err_BUG("best_point (too large)");
853 : }
854 : }
855 : }
856 : #endif
857 : }
858 :
859 : static GEN
860 21144 : best_lift(GEN Q, GEN NQ, GEN f)
861 : {
862 : GEN a, b, c, d, dQ, cNQ;
863 21144 : best_point(Q, NQ, f, &c, &d);
864 21144 : dQ = mulii(d, Q); cNQ = mulii(NQ, c);
865 21144 : (void)bezout(dQ, cNQ, &a, &b);
866 21144 : return qfb_mult(f, dQ, b, mulii(negi(Q),cNQ), mulii(a,Q));
867 : }
868 :
869 : static GEN
870 2124 : lift_points(GEN listQ, GEN f, GEN *pt, GEN *pQ)
871 : {
872 2124 : pari_sp av = avma;
873 2124 : GEN yf = gen_0, tf = NULL, Qf = NULL;
874 2124 : long k, l = lg(listQ);
875 23268 : for (k = 1; k < l; ++k)
876 : {
877 21144 : GEN c = gel(listQ, k), Q = gel(c,1), NQ = gel(c,2);
878 21144 : GEN t = best_lift(Q, NQ, f), y = qimag2(t);
879 21144 : if (gcmp(y, yf) > 0) { yf = y; Qf = Q; tf = t; }
880 : }
881 2124 : *pt = tf; *pQ = Qf; return gc_all(av, 3, &yf, pt, pQ);
882 : }
883 :
884 : /***************************/
885 : /* Twists */
886 : /***************************/
887 :
888 : static GEN
889 48 : ltwist1(GEN E, GEN d, long bitprec)
890 : {
891 48 : pari_sp av = avma;
892 48 : GEN Ed = elltwist(E, d), z = ellL1(Ed, 0, bitprec);
893 48 : obj_free(Ed); return gc_leaf(av, z);
894 : }
895 :
896 : /* Return O_re*c(E)/(4*O_vol*|E_t|^2) */
897 :
898 : static GEN
899 36 : heegner_indexmult(GEN om, long t, GEN tam, long prec)
900 : {
901 36 : pari_sp av = avma;
902 36 : GEN Ovr = gabs(imag_i(gel(om, 2)), prec); /* O_vol/O_re, t_REAL */
903 36 : return gc_upto(av, divru(divir(tam, Ovr), 4*t*t));
904 : }
905 :
906 : /* omega(gcd(D, N)), given faN = factor(N) */
907 : static long
908 48 : omega_N_D(GEN faN, ulong D)
909 : {
910 48 : GEN P = gel(faN, 1);
911 48 : long i, l = lg(P), w = 0;
912 168 : for (i = 1; i < l; i++)
913 120 : if (dvdui(D, gel(P,i))) w++;
914 48 : return w;
915 : }
916 :
917 : static GEN
918 48 : heegner_indexmultD(GEN faN, GEN a, long D, GEN sqrtD)
919 : {
920 48 : pari_sp av = avma;
921 : GEN c;
922 : long w;
923 48 : switch(D)
924 : {
925 0 : case -3: w = 9; break;
926 0 : case -4: w = 4; break;
927 48 : default: w = 1;
928 : }
929 48 : c = shifti(stoi(w), omega_N_D(faN,-D)); /* (w(D)/2)^2 * 2^omega(gcd(D,N)) */
930 48 : return gc_upto(av, mulri(mulrr(a, sqrtD), c));
931 : }
932 :
933 : static GEN
934 342 : nf_to_basis(GEN nf, GEN x)
935 : {
936 342 : x = nf_to_scalar_or_basis(nf, x);
937 342 : if (typ(x)!=t_COL)
938 258 : x = scalarcol(x, nf_get_degree(nf));
939 342 : return x;
940 : }
941 :
942 : static GEN
943 168 : etnf_to_basis(GEN et, GEN x)
944 : {
945 168 : long i, l = lg(et);
946 168 : GEN V = cgetg(l, t_VEC);
947 510 : for (i = 1; i < l; i++)
948 342 : gel(V,i) = nf_to_basis(gel(et,i), x);
949 168 : return shallowconcat1(V);
950 : }
951 :
952 : static GEN
953 120 : etnf_get_M(GEN et)
954 : {
955 120 : long i, l = lg(et);
956 120 : GEN V = cgetg(l, t_VEC);
957 384 : for (i = 1; i < l; i++)
958 264 : gel(V,i)=nf_get_M(gel(et,i));
959 120 : return shallowmatconcat(diagonal(V));
960 : }
961 :
962 : static long
963 42 : etnf_get_varn(GEN et)
964 : {
965 42 : return nf_get_varn(gel(et,1));
966 : }
967 :
968 : static GEN
969 84 : heegner_descent_try_point(GEN nfA, GEN z, GEN den, long prec)
970 : {
971 84 : pari_sp av = avma;
972 84 : GEN etal = gel(nfA,1), A = gel(nfA,2), cb = gel(nfA,3);
973 84 : GEN al = gel(nfA,4), th = gel(nfA, 5);
974 84 : GEN et = gel(etal,1), zk = gel(etal, 2), T = gel(etal,3);
975 84 : GEN M = etnf_get_M(et);
976 84 : long i, j, n = lg(th)-1, l = lg(al);
977 84 : GEN u2 = gsqr(gel(cb,1)), r = gel(cb,2);
978 84 : GEN zz = gdiv(gsub(z,r), u2);
979 84 : GEN be = cgetg(n+1, t_COL);
980 132 : for (j = 1; j < l; j++)
981 : {
982 84 : GEN aj = gel(al, j), Aj = gel(A,j);
983 318 : for (i = 1; i <= n; i++)
984 234 : gel(be,i) = gsqrt(gmul(gsub(zz, gel(th,i)), gel(aj,i)), prec);
985 288 : for (i = 0; i <= (1<<(n-1))-1; i++)
986 : {
987 : long eps;
988 240 : GEN s = gmul(den, RgM_solve_realimag(M, be));
989 240 : GEN S = grndtoi(s, &eps), V, S2;
990 240 : gel(be,1+odd(i)) = gneg(gel(be,1+odd(i)));
991 240 : if (eps > -7) continue;
992 36 : S2 = QXQ_sqr(RgV_RgC_mul(zk, S), T);
993 36 : V = gdiv(QXQ_mul(S2, Aj, T), sqri(den));
994 36 : if (typ(V) != t_POL || degpol(V) != 1) continue;
995 36 : if (gequalm1(gel(V,3)))
996 36 : return gc_upto(av,gadd(gmul(gel(V,2),u2),r));
997 : }
998 : }
999 48 : return gc_NULL(av);
1000 : }
1001 :
1002 : static GEN
1003 1530 : heegner_try_point(GEN E, GEN nfA, GEN lambdas, GEN ht, GEN z, long prec)
1004 : {
1005 1530 : long l = lg(lambdas);
1006 : long i, eps;
1007 1530 : GEN P = real_i(pointell(E, z, prec)), x = gel(P,1);
1008 1530 : GEN rh = subrr(ht, shiftr(ellheightoo(E, P, prec),1));
1009 22776 : for (i = 1; i < l; ++i)
1010 : {
1011 21282 : GEN logd = shiftr(gsub(rh, gel(lambdas, i)), -1);
1012 21282 : GEN d, approxd = gexp(logd, prec);
1013 21282 : d = grndtoi(approxd, &eps);
1014 21282 : if (signe(d) > 0 && eps<-10)
1015 : {
1016 : GEN X, ylist;
1017 84 : if (DEBUGLEVEL > 2)
1018 0 : err_printf("\nTrying lambda number %ld, logd=%Ps, approxd=%Ps\n", i, logd, approxd);
1019 84 : X = heegner_descent_try_point(nfA, x, d, prec);
1020 84 : if (X)
1021 : {
1022 36 : ylist = ellordinate(E, X, prec);
1023 36 : if (lg(ylist) > 1)
1024 : {
1025 36 : GEN P = mkvec2(X, gel(ylist, 1));
1026 36 : GEN hp = ellheight(E,P,prec);
1027 36 : if (signe(hp) && cmprr(hp, shiftr(ht,1)) < 0 && cmprr(hp, shiftr(ht,-1)) > 0)
1028 36 : return P;
1029 0 : if (DEBUGLEVEL)
1030 0 : err_printf("found non-Heegner point %Ps\n", P);
1031 : }
1032 : }
1033 : }
1034 : }
1035 1494 : return NULL;
1036 : }
1037 :
1038 : static GEN
1039 36 : heegner_find_point(GEN e, GEN nfA, GEN om, GEN ht, GEN z1, long k, long prec)
1040 : {
1041 36 : GEN lambdas = lambdalist(e, prec);
1042 36 : pari_sp av = avma;
1043 : long m;
1044 36 : GEN Ore = gel(om, 1), Oim = gel(om, 2);
1045 36 : if (DEBUGLEVEL)
1046 0 : err_printf("%ld*%ld multipliers to test: ",k,lg(lambdas)-1);
1047 828 : for (m = 0; m < k; m++)
1048 : {
1049 828 : GEN P, z2 = divru(addrr(z1, mulsr(m, Ore)), k);
1050 828 : if (DEBUGLEVEL > 2)
1051 0 : err_printf("%ld ",m);
1052 828 : P = heegner_try_point(e, nfA, lambdas, ht, z2, prec);
1053 828 : if (P) return P;
1054 798 : if (signe(ell_get_disc(e)) > 0)
1055 : {
1056 702 : z2 = gadd(z2, gmul2n(Oim, -1));
1057 702 : P = heegner_try_point(e, nfA, lambdas, ht, z2, prec);
1058 702 : if (P) return P;
1059 : }
1060 792 : set_avma(av);
1061 : }
1062 0 : pari_err_BUG("ellheegner, point not found");
1063 : return NULL; /* LCOV_EXCL_LINE */
1064 : }
1065 :
1066 : /* N > 1, fa = factor(N), return factor(4*N) */
1067 : static GEN
1068 36 : fa_shift2(GEN fa)
1069 : {
1070 36 : GEN P = gel(fa,1), E = gel(fa,2);
1071 36 : if (absequaliu(gcoeff(fa,1,1), 2))
1072 : {
1073 18 : E = shallowcopy(E);
1074 18 : gel(E,1) = addiu(gel(E,1), 2);
1075 : }
1076 : else
1077 : {
1078 18 : P = shallowconcat(gen_2, P);
1079 18 : E = shallowconcat(gen_2, E);
1080 : }
1081 36 : return mkmat2(P, E);
1082 : }
1083 :
1084 : /* P = prime divisors of N(E). Return the product of primes p in P, a_p != -1
1085 : * HACK: restrict to small primes since large ones won't divide our C-long
1086 : * discriminants */
1087 : static GEN
1088 36 : get_bad(GEN E, GEN P)
1089 : {
1090 36 : long k, l = lg(P), ibad = 1;
1091 36 : GEN B = cgetg(l, t_VECSMALL);
1092 126 : for (k = 1; k < l; k++)
1093 : {
1094 90 : GEN p = gel(P,k);
1095 90 : long pp = itos_or_0(p);
1096 90 : if (!pp) break;
1097 90 : if (! equalim1(ellap(E,p))) B[ibad++] = pp;
1098 : }
1099 36 : setlg(B, ibad); return ibad == 1? NULL: zv_prod_Z(B);
1100 : }
1101 :
1102 : /* factorization into primary factors */
1103 : static GEN
1104 36 : to_primary(GEN fa)
1105 : {
1106 36 : GEN Q, P = gel(fa,1), E = gel(fa,2);
1107 36 : long i, l = lg(P);
1108 36 : Q = cgetg(l, t_COL);
1109 126 : for (i = 1; i < l; i++) gel(Q,i) = powii(gel(P,i), gel(E,i));
1110 36 : return mkmat2(Q, const_col(l-1, gen_1));
1111 : }
1112 :
1113 : /* list of pairs [Q,N/Q], where Q || N, sorted by increasing Q. */
1114 : static GEN
1115 36 : find_div(GEN faN)
1116 : {
1117 36 : GEN L, Q = divisors(to_primary(faN));
1118 36 : long k, l = lg(Q);
1119 36 : L = cgetg(l, t_VEC);
1120 276 : for (k = 1; k < l; k++) gel(L, k) = mkvec2(gel(Q,k), gel(Q,l-k));
1121 36 : return L;
1122 : }
1123 :
1124 : static long
1125 7416 : testDisc(GEN bad, long d) { return !bad || ugcdiu(bad, -d) == 1; }
1126 : /* bad = product of bad primes. Return the NDISC largest fundamental
1127 : * discriminants D < d such that (D,bad) = 1 and d is a square mod 4N */
1128 : static GEN
1129 36 : listDisc(GEN fa4N, GEN bad, long d, long ndisc)
1130 : {
1131 36 : GEN v = cgetg(ndisc+1, t_VECSMALL);
1132 36 : pari_sp av = avma;
1133 36 : long j = 1;
1134 : for(;;)
1135 : {
1136 7416 : d -= odd(d)? 1: 3;
1137 7416 : if (testDisc(bad,d) && unegisfundamental(-d) && Zn_issquare(stoi(d), fa4N))
1138 : {
1139 360 : v[j++] = d;
1140 360 : if (j > ndisc) break;
1141 : }
1142 7380 : set_avma(av);
1143 : }
1144 36 : set_avma(av); return v;
1145 : }
1146 :
1147 : /* as normforms, but only return solution so that b = beta [mod Q] */
1148 : static GEN
1149 4560 : normformsbeta(GEN D, GEN fa, GEN beta, GEN Q)
1150 : {
1151 4560 : pari_sp av = avma;
1152 : long i, j, k, lB, aN, sa;
1153 : GEN a, L, V, B, N, N2;
1154 4560 : int D_odd = mpodd(D);
1155 4560 : a = typ(fa) == t_INT ? fa: typ(fa) == t_VEC? gel(fa,1): factorback(fa);
1156 4560 : sa = signe(a);
1157 4560 : if (sa==0 || (signe(D)<0 && sa<0)) return NULL;
1158 4392 : V = D_odd? Zn_quad_roots(fa, gen_1, shifti(subsi(1, D), -2))
1159 4560 : : Zn_quad_roots(fa, gen_0, negi(shifti(D, -2)));
1160 4560 : if (!V) return NULL;
1161 3420 : N = gel(V,1); B = gel(V,2); lB = lg(B);
1162 3420 : N2 = shifti(N,1);
1163 3420 : aN = itou(diviiexact(a, N)); /* |a|/N */
1164 3420 : L = cgetg((lB-1)*aN+1, t_VEC);
1165 51828 : for (k = 1, i = 1; i < lB; i++)
1166 : {
1167 48408 : GEN b = shifti(gel(B,i), 1), c, C;
1168 48408 : if (D_odd) b = addiu(b , 1);
1169 48408 : c = diviiexact(shifti(subii(sqri(b), D), -2), a);
1170 48408 : for (j = 0;; b = addii(b, N2))
1171 : {
1172 48648 : if (dvdii(subii(b,beta),Q))
1173 4338 : gel(L, k++) = mkqfb(a, b, c, D);
1174 48648 : if (++j == aN) break;
1175 240 : C = addii(b, N); if (aN > 1) C = diviuexact(C, aN);
1176 240 : c = sa > 0? addii(c, C): subii(c, C);
1177 : }
1178 : }
1179 3420 : setlg(L, k); return gc_GEN(av, L);
1180 : }
1181 :
1182 : /* faN4 = factor(4*N) */
1183 : static GEN
1184 360 : listheegner(GEN N, GEN faN4, GEN listQ, GEN D)
1185 : {
1186 360 : pari_sp av = avma;
1187 : hashtable H;
1188 360 : ulong h = itou(quadclassno(D));
1189 360 : GEN ymin, beta = Zn_sqrt(D, faN4), N2 = shifti(N, 1), L, V;
1190 : long l, k, n;
1191 360 : hash_init_GEN(&H, h, gequal, 1);
1192 4920 : for (k = 1; H.nb < h ; k++)
1193 : {
1194 4560 : GEN LF = normformsbeta(D, mulis(N, k), beta, N2);
1195 4560 : if (LF)
1196 : {
1197 3420 : long i, l = lg(LF);
1198 7674 : for (i = 1; i < l && H.nb < h; i++)
1199 : {
1200 4254 : GEN F = gel(LF, i), f = qfi_red(F), k = hash_haskey_GEN(&H, f);
1201 4254 : if (!k)
1202 : {
1203 2124 : GEN a = gel(F,1), b = gel(F,2), c = gel(F,3);
1204 2124 : GEN F2 = mkqfb(diviiexact(a,N), negi(b), mulii(c,N), D);
1205 2124 : GEN f2 = qfi_red(F2);
1206 2124 : if (gequal(f, f2))
1207 276 : hash_insert(&H, f, mkvec(F));
1208 : else
1209 : {
1210 1848 : hash_insert(&H, f, mkvec2(F, F2));
1211 1848 : hash_insert(&H, f2, cgetg(1,t_VEC));
1212 : }
1213 : }
1214 : }
1215 : }
1216 : }
1217 360 : ymin = NULL;
1218 360 : V = hash_values_GEN(&H); l = lg(V);
1219 360 : L = cgetg(H.nb+1, t_VEC);
1220 4332 : for (k = n = 1; k < l; k++)
1221 : {
1222 3972 : GEN t, Q, Vk = gel(V,k), y;
1223 3972 : long nk = lg(Vk) - 1;
1224 3972 : if (nk)
1225 : {
1226 2124 : y = lift_points(listQ, gel(Vk,1), &t, &Q);
1227 2124 : gel(L, n++) = mkvec3(t, stoi(nk), Q);
1228 2124 : if (!ymin || gcmp(y, ymin) < 0) ymin = y;
1229 : }
1230 : }
1231 360 : setlg(L, n); /* H.nb/2 <= n-1 <= H.nb */
1232 360 : if (DEBUGLEVEL > 1)
1233 0 : err_printf("Disc %Ps : N*ymin = %Pg\n", D,
1234 : gmul(gsqrt(ymin, DEFAULTPREC),N));
1235 360 : return gc_GEN(av, mkvec3(ymin, L, D));
1236 : }
1237 :
1238 : /* Q | N, P = prime divisors of N, R[i] = local epsilon-factor at P[i].
1239 : * Return \prod_{p | Q} R[i] */
1240 : static long
1241 126 : rootno(GEN Q, GEN P, GEN R)
1242 : {
1243 126 : long s = 1, i, l = lg(P);
1244 498 : for (i = 1; i < l; i++)
1245 372 : if (dvdii(Q, gel(P,i))) s *= R[i];
1246 126 : return s;
1247 : }
1248 :
1249 : static void
1250 36 : heegner_find_disc(GEN *points, GEN *coefs, long *pind, GEN E,
1251 : GEN indmult, long ndisc, long prec)
1252 : {
1253 36 : long d = 0;
1254 : GEN faN4, bad, N, faN, listQ, listR;
1255 :
1256 36 : ellQ_get_Nfa(E, &N, &faN);
1257 36 : faN4 = fa_shift2(faN);
1258 36 : listQ = find_div(faN);
1259 36 : bad = get_bad(E, gel(faN, 1));
1260 36 : listR = gel(obj_check(E, Q_ROOTNO), 2);
1261 : for(;;)
1262 0 : {
1263 36 : pari_sp av = avma;
1264 36 : GEN list, listD = listDisc(faN4, bad, d, ndisc);
1265 36 : long k, l = lg(listD);
1266 36 : list = cgetg(l, t_VEC);
1267 396 : for (k = 1; k < l; ++k)
1268 360 : gel(list, k) = listheegner(N, faN4, listQ, stoi(listD[k]));
1269 36 : list = vecsort0(list, gen_1, 0);
1270 48 : for (k = l-1; k > 0; --k)
1271 : {
1272 48 : long bprec = 8;
1273 48 : GEN Lk = gel(list,k), D = gel(Lk,3);
1274 48 : GEN sqrtD = sqrtr_abs(itor(D, prec)); /* sqrt(|D|) */
1275 48 : GEN indmultD = heegner_indexmultD(faN, indmult, itos(D), sqrtD);
1276 : do
1277 : {
1278 : GEN mulf, indr;
1279 : pari_timer ti;
1280 48 : if (DEBUGLEVEL) timer_start(&ti);
1281 48 : mulf = ltwist1(E, D, bprec+expo(indmultD));
1282 48 : if (DEBUGLEVEL) timer_printf(&ti,"ellL1twist");
1283 48 : indr = mulrr(indmultD, mulf);
1284 48 : if (DEBUGLEVEL) err_printf("Disc = %Ps, Index^2 = %Ps\n", D, indr);
1285 48 : if (signe(indr)>0 && expo(indr) >= -1) /* indr >=.5 */
1286 : {
1287 : long e, i, l;
1288 36 : GEN pts, cfs, L, indi = grndtoi(sqrtr_abs(indr), &e);
1289 36 : if (e > expi(indi)-7)
1290 : {
1291 0 : bprec++;
1292 0 : pari_warn(warnprec, "ellL1",bprec);
1293 0 : continue;
1294 : }
1295 36 : *pind = itos(indi);
1296 36 : L = gel(Lk, 2); l = lg(L);
1297 36 : pts = cgetg(l, t_VEC);
1298 36 : cfs = cgetg(l, t_VECSMALL);
1299 162 : for (i = 1; i < l; ++i)
1300 : {
1301 126 : GEN P = gel(L,i), z = gel(P,2), Q = gel(P,3); /* [1 or 2, Q] */
1302 : long c;
1303 126 : gel(pts, i) = qfb_root(gel(P,1), sqrtD);
1304 126 : c = rootno(Q, gel(faN,1), listR);
1305 126 : if (!equali1(z)) c *= 2;
1306 126 : cfs[i] = c;
1307 : }
1308 36 : if (DEBUGLEVEL)
1309 0 : err_printf("N = %Ps, ymin*N = %Ps\n",N,
1310 0 : gmul(gsqrt(gel(Lk, 1), prec),N));
1311 36 : *coefs = cfs; *points = pts; return;
1312 : }
1313 : } while(0);
1314 : }
1315 0 : d = listD[l-1]; set_avma(av);
1316 : }
1317 : }
1318 :
1319 : GEN
1320 132 : ellanal_globalred_all(GEN e, GEN *cb, GEN *N, GEN *tam)
1321 : {
1322 132 : GEN E = ellanal_globalred(e, cb), red = obj_check(E, Q_GLOBALRED);
1323 132 : *N = gel(red, 1);
1324 132 : *tam = gel(red,2);
1325 132 : if (signe(ell_get_disc(E))>0) *tam = shifti(*tam,1);
1326 132 : return E;
1327 : }
1328 :
1329 : static GEN
1330 36 : vecelnfembed(GEN x, GEN M, GEN et)
1331 78 : { pari_APPLY_same(gmul(M, etnf_to_basis(et, gel(x,i)))) }
1332 :
1333 : static GEN
1334 36 : QXQV_inv(GEN x, GEN T)
1335 78 : { pari_APPLY_same(QXQ_inv(gel(x,i), T)) }
1336 :
1337 : static GEN
1338 36 : etnfnewprec(GEN x, long prec)
1339 108 : { pari_APPLY_same(nfnewprec(gel(x,i),prec)) }
1340 :
1341 : static GEN
1342 42 : vec_etnf_to_basis(GEN et, GEN x)
1343 132 : { pari_APPLY_same(etnf_to_basis(et,gel(x,i))) }
1344 :
1345 : static GEN
1346 36 : makenfA(GEN sel, GEN A, GEN cb)
1347 : {
1348 36 : GEN etal = gel(sel,1), T = gel(etal,3);
1349 36 : GEN et = gel(etal,1), M = etnf_get_M(et);
1350 36 : long v = etnf_get_varn(et);
1351 36 : GEN al = vecelnfembed(A, M, et);
1352 36 : GEN th = gmul(M, etnf_to_basis(et, pol_x(v)));
1353 36 : return mkvec5(etal,QXQV_inv(A, T),cb,al,th);
1354 : }
1355 :
1356 : GEN
1357 48 : ellheegner(GEN E)
1358 : {
1359 48 : pari_sp av = avma;
1360 : GEN z, P, ht, points, coefs, s, om, indmult;
1361 : GEN sel, etal, et, cbb, A, dAi, T, Ag, At;
1362 : long ind, indx, lint, k, l, wtor, etor, ndisc, ltors2, selrank;
1363 48 : long bitprec = 16, prec = nbits2prec(bitprec) + EXTRAPRECWORD;
1364 : pari_timer ti;
1365 : GEN N, cb, tam, torsion, nfA;
1366 48 : E = ellanal_globalred_all(E, &cb, &N, &tam);
1367 48 : if (ellrootno_global(E) == 1)
1368 6 : pari_err_DOMAIN("ellheegner", "(analytic rank)%2","=",gen_0,E);
1369 42 : torsion = elltors(E);
1370 42 : wtor = itos( gel(torsion,1) ); /* #E(Q)_tor */
1371 42 : etor = wtor > 1? itou(gmael(torsion, 2, 1)): 1; /* exponent of E(Q)_tor */
1372 42 : sel = ell2selmer_basis(E, &cbb, prec);
1373 42 : etal = gel(sel,1); A = gel(sel,2); et = gel(etal,1); T = gel(etal,3);
1374 42 : ltors2 = lg(et)-2; selrank = lg(A)-1;
1375 42 : Ag = selrank > ltors2+1 ? pol_1(etnf_get_varn(et)): gel(A,selrank);
1376 42 : At = vecslice(A,1,ltors2);
1377 42 : dAi = gsupnorm(vec_etnf_to_basis(et,A),prec);
1378 : while (1)
1379 36 : {
1380 : GEN hnaive, l1;
1381 : long bitneeded;
1382 78 : if (DEBUGLEVEL) timer_start(&ti);
1383 78 : l1 = ellL1(E, 1, bitprec);
1384 78 : if (DEBUGLEVEL) timer_printf(&ti,"ellL1");
1385 78 : if (expo(l1) < 1 - bitprec/2)
1386 6 : pari_err_DOMAIN("ellheegner", "analytic rank",">",gen_1,E);
1387 72 : om = ellR_omega(E,prec);
1388 72 : ht = divrr(mulru(l1, wtor * wtor), mulri(gel(om,1), tam));
1389 72 : if (DEBUGLEVEL) err_printf("Expected height=%Ps\n", ht);
1390 72 : hnaive = hnaive_max(E, ht);
1391 72 : if (DEBUGLEVEL) err_printf("Naive height <= %Ps\n", hnaive);
1392 72 : hnaive = gadd(shiftr(hnaive,-1),glog(dAi,prec));
1393 72 : bitneeded = itos(gceil(divrr(hnaive, mplog2(prec)))) + 32;
1394 72 : if (DEBUGLEVEL) err_printf("precision = %ld\n", bitneeded);
1395 72 : if (bitprec>=bitneeded) break;
1396 36 : bitprec = bitneeded;
1397 36 : prec = nbits2prec(bitprec) + EXTRAPRECWORD;
1398 : }
1399 36 : indmult = heegner_indexmult(om, wtor, tam, prec);
1400 36 : ndisc = maxss(10, (long) rtodbl(ht)/10);
1401 36 : heegner_find_disc(&points, &coefs, &ind, E, indmult, ndisc, prec);
1402 36 : if (DEBUGLEVEL) timer_start(&ti);
1403 36 : s = heegner_psi(E, N, points, bitprec);
1404 36 : if (DEBUGLEVEL) timer_printf(&ti,"heegner_psi");
1405 36 : l = lg(points);
1406 36 : z = mulsr(coefs[1], gel(s, 1));
1407 126 : for (k = 2; k < l; ++k) z = addrr(z, mulsr(coefs[k], gel(s, k)));
1408 36 : z = gsub(z, gmul(gel(om,1), ground(gdiv(z, gel(om,1)))));
1409 36 : if (DEBUGLEVEL) err_printf("z=%.*Pg\n",nbits2ndec(bitprec), z);
1410 36 : lint = wtor > 1 ? ugcd(ind, etor): 1;
1411 36 : indx = lint*2*ind;
1412 36 : if (vals(indx) >= vals(etor))
1413 30 : A = mkvec(Ag);
1414 : else
1415 6 : A = mkvec2(Ag, QXQ_mul(Ag, gel(At,1), T));
1416 36 : gmael(sel,1,1) = etnfnewprec(et, prec);
1417 36 : nfA = makenfA(sel, A, cbb);
1418 36 : P = heegner_find_point(E, nfA, om, ht, gmulsg(2*lint, z), indx, prec);
1419 36 : if (DEBUGLEVEL) timer_printf(&ti,"heegner_find_point");
1420 36 : if (cb) P = ellchangepointinv(P, cb);
1421 36 : return gc_GEN(av, P);
1422 : }
1423 :
1424 : /* Modular degree */
1425 :
1426 : static GEN
1427 60 : ellisobound(GEN e)
1428 : {
1429 60 : GEN M = gel(ellisomat(e,0,1),2);
1430 60 : return vecmax(gel(M,1));
1431 : }
1432 : /* 4Pi^2 / E.area */
1433 : static GEN
1434 120 : getA(GEN E, long prec) { return mpdiv(sqrr(Pi2n(1,prec)), ellR_area(E, prec)); }
1435 :
1436 : /* Modular degree of elliptic curve e over Q, assuming Manin constant = 1
1437 : * (otherwise multiply by square of Manin constant). */
1438 : GEN
1439 60 : ellmoddegree(GEN E)
1440 : {
1441 60 : pari_sp av = avma;
1442 : GEN N, tam, mc2, d;
1443 : long b;
1444 60 : E = ellanal_globalred_all(E, NULL, &N, &tam);
1445 60 : mc2 = sqri(ellisobound(E));
1446 60 : b = expi(mulii(N, mc2)) + maxss(0, expo(getA(E, LOWDEFAULTPREC))) + 16;
1447 : for(;;)
1448 0 : {
1449 60 : long prec = nbits2prec(b), e, s;
1450 60 : GEN deg = mulri(mulrr(lfunellmfpeters(E, b), getA(E, prec)), mc2);
1451 60 : d = grndtoi(deg, &e);
1452 60 : if (DEBUGLEVEL) err_printf("ellmoddegree: %Ps, bit=%ld, err=%ld\n",deg,b,e);
1453 60 : s = expo(deg);
1454 60 : if (e <= -8 && s <= b-8) return gc_upto(av, gdiv(d,mc2));
1455 0 : b = maxss(s, b+e) + 16;
1456 : }
1457 : }
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