forked from JohnCremona/eclib
-
Notifications
You must be signed in to change notification settings - Fork 0
/
mrank2.cc
1244 lines (1133 loc) · 34.8 KB
/
mrank2.cc
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
// mrank2.cc: implementation of class rank2 for descent via 2-isogeny
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2012 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
#include <iomanip> // for setbase(), used for hex output of codes
#include <eclib/bitspace.h>
#include <eclib/compproc.h>
#include <eclib/points.h>
#include <eclib/mwprocs.h>
#include <eclib/mquartic.h>
#include <eclib/msoluble.h>
#include <eclib/descent.h>
#include <eclib/mrank2.h>
#include <eclib/sqfdiv.h>
#include <eclib/desc2.h>
#ifndef QSIEVE_OPT
#define QSIEVE_OPT 0 // uses Stoll's sieve
#endif
//#define DEBUG_ROOTS
//#define DEBUG_SQF
//#define DEBUG_ELS
//#define DEBUG_GLS
// bound on first descent search when a second descent is going to be done
#define FIRST_DESCENT_BOUND 8
#define MAX_R 5 // will not attempt to list all coset reps
// for 2E(Q) in E(Q) if rank is more than this
void rank2::makepoint(const bigint& c,const bigint& d1,const bigint& d2,
const bigint& x, const bigint& y, const bigint& z,
int which)
{
Point P(ee);
if (verbose) cout<<" (x:y:z) = ("<<x<<":"<<y<<":"<<z<<")\n";
if(which==0)
{
if(sign(z)!=0) P.init(ee, d1*x*x*z, d1*x*y, pow(z,3));
if(verbose)
{
cout<<"\tCurve E " /* <<(Curve)ee */ <<"\tPoint "<< P;
bigfloat h = height(P);
cout << ", height = " << h;
if(!P.isvalid()) {cout << " --warning: NOT on curve! " ;}
cout << "\n";
}
}
else
{
if(verbose) // display point on isogenous curve too
{
Point Q(eedash);
if(sign(z)!=0) Q.init(eedash, d1*x*x*z, d1*x*y, pow(z,3));
cout<<"\tCurve E' " /* <<(Curve)eedash */ <<"\tPoint "<< Q;
bigfloat h = height(Q);
cout << ", height = " << h;
if(!Q.isvalid()) {cout << " --warning: NOT on curve! " ;}
cout << "\n";
}
const bigint& xz=x*z, x2=x*x, z2=z*z;
if(sign(xz)!=0) P.init(ee,2*y*y*xz,y*(d1*x2*x2-d2*z2*z2),pow(2*xz,3));
if(verbose)
{
cout<<"\tCurve E " /* <<(Curve)ee */ <<"\tPoint "<< P;
bigfloat h = height(P);
cout << ", height = " << h;
if(!P.isvalid()) {cout << " --warning: NOT on curve! " ;}
cout << "\n";
}
}
if(order(P)<0) {pointlist.push_back(P); npoints++;} // else torsion so ignore
}
int rank2::testquartic(const bigint& c,const bigint& d1,const bigint& d2,int which)
{
// creates quartic (d1,0,c,0,d2), assumed els, and tries to find a rational point
// returns +1 if rational point found (handled my makepoint())
// 0 if undecided
static const bigint zero = BIGINT(0);
static const bigint one = BIGINT(1);
quartic q(d1, zero, c, zero, d2);
if (verbose) cout<<q<<": ";
bigint x,y,z;
// First a quick search for a small point:
if (ratpoint(q,one,BIGINT(lim1),x,y,z))
{
makepoint(c,d1,d2,x,y,z,which);
return 1;
}
// Next a more serious search using a sieve,
// but not too far if we are going to use second descent
quartic_sieve qs(&q,QSIEVE_OPT,0);
long lim3=lim2;
if(do_second_descent) // save the long search till later...
if(lim3>FIRST_DESCENT_BOUND) lim3=FIRST_DESCENT_BOUND;
if(qs.search(lim3, 1, 1)) // maxnpoints, pos-x-only
{
qs.getpoint(x,y,z);
makepoint(c,d1,d2,x,y,z,which);
return 1;
}
if (verbose)
cout<<" no rational point found (hlim="<<lim3<<")\n";
return 0;
} /* end of testquartic*/
int rank2::second_descent(const bigint& c, const bigint& d1, const bigint& d2, int which)
// creates quartic (d1,0,c,0,d2), assumed els, and tries to find a rational point
// by second descent (after testquartic() has failed using direct search)
// returns +1 if rational point found (handled by makepoint())
// 0 if undecided
// -1 if no rational point exists (proved by second descent)
{
int res, verb=verbose; if(verb) verb--; // reduced verbosity level within desc2()
bigint x,y,z;
if(verbose) cout<<"d1="<<d1<<": "<<flush;
if(which)
res = desc2(c,d1,d2,badprimes, supp0, elsgens0, mask0, lim2,x,y,z,verb);
else
res = desc2(c,d1,d2,badprimes, supp1, elsgens1, mask1, lim2,x,y,z,verb);
if(verbose==1) cout<<endl;
switch(res)
{
case 1: // Positive success, point found
makepoint(c,d1,d2,x,y,z,which);
if(verbose)
cout<<"Second descent successfully found rational point for d1="<<d1<<"\n\n";
break;
case -1: // Negative success, no point exists
if(verbose)
cout<<"Second descent shows that no rational point exists for d1="<<d1<<"\n\n";
break;
case 0: // Undecided
if (verbose)
cout<<"Second descent inconclusive for d1="<<d1<<": ELS descendents exist but no rational point found\n\n";
}
return res;
}
// First local descent: determine
// S^(phi)(E') if which==0
// S^(phi')(E) if which==1
void rank2::find_elsgens(int which, const bigint& c, const bigint& d)
{
static const bigint zero = BIGINT(0);
if (verbose>1)
{
if(which) cout<<"\n";
cout<<"Finding els gens for E"; if(which) cout<<"'";
cout<<" (c"; if(which) cout<<"'";
cout<<"= "<<c<<", d"; if(which) cout<<"'";
cout<<"= "<<d<<")"<<endl;
}
bigint ddash = c*c-4*d;
int posd1, istep=1;
if (sign(ddash)< 0)
posd1=1;
else
posd1= (c+sqrt(ddash)< 0);
if(posd1) istep=2; // this will skip negative d1
int extra2torsion=0;
bigint ee2, ee3;
if(which)
{
if(d_is_sq)
{
extra2torsion=1;
ee2 = e2dash;
ee3 = e3dash;
}
}
else
{
if(ddash_is_sq)
{
extra2torsion=1;
ee2 = e2;
ee3 = e3;
}
}
bigint d1, d2, badp;
const vector<bigint>& supp = (which? supp1 : supp0);
long ns = supp.size();
if(verbose>1)
{
cout<<"Support (length "<<ns<<"): "<<supp<<endl;
}
if(ns>=NTL_BITS_PER_LONG) {cout<<"Too many primes dividing d!\n"; abort();}
long mask=0, maxn=1<<ns, index, j, nelsgens=0;
vector<bigint> elsgens;
// use all torsion: added 24/6/02
// Find and process torsion
vector<Point> torsion;
if(which)
torsion=torsion_points(eedash);
else
torsion=torsion_points(ee);
long it,ntorsion=torsion.size();
#ifdef DEBUG_ELS
cout<<"Number of torsion points = "<<ntorsion<<endl;
cout<<"Torsion subgroup is ";
if(extra2torsion) cout<<"not ";
cout<<"cyclic"<<endl;
#endif
// find generators of torsion mod 2*torsion
Point T1,T2; long orderT1=1;
for(it=0; it<ntorsion; it++)
{
Point T = torsion[it];
long orderT = order(T);
if(orderT>orderT1) {T1=T; orderT1=orderT;}
}
#ifdef DEBUG_ELS
cout<<"Generator of maximal order = "<<T1<<" of order "<<orderT1<<endl;
#endif
if(extra2torsion)
// then we need a second generator, for which we can take either of
// the 2-torsion points which is not the multiple of T1
{
T2=(ntorsion/4)*T1;
bigint xT2=getX(T2);
if(xT2==zero)
T2.init(ee,ee2,zero);
else
T2.init(ee,zero,zero);
#ifdef DEBUG_ELS
cout<<"Second generator = "<<T2<<" of order 2"<<endl;
#endif
ntorsion=2;
torsion[0]=T1;
torsion[1]=T2;
}
else
{
ntorsion=1;
torsion[0]=T1;
}
#ifdef DEBUG_ELS
cout<<"Processing generating torsion points"<<endl;
#endif
bigint d1x;
for(it=0; it<ntorsion; it++)
{
Point T = torsion[it];
if(T.iszero()) continue;
d1=getX(T);
if(d1==zero) d1=d;
#ifdef DEBUG_ELS
if(verbose>1) cout<<"Processing torsion d1 = " << d1 << endl;
#endif
index=makeindex(supp, d1, d1x);
d1=d1x; // the square-free part, = d1/square
#ifdef DEBUG_ELS
if(verbose>1) cout<<"Squarefree part of d1 = " << d1 << endl;
#endif
if(mask&index) continue;
if(index)
{
elsgens.push_back(d1); nelsgens++;
if(verbose>1)
cout<<"Adding (torsion) els generator #"<<nelsgens<<": d1 = " << d1;
for(j=ns-1; j>=0; j--)
// if(testbit(index,j)) {setbit(mask,j); break;}
if(index&(1<<j)) {mask|=(1<<j); break;}
#ifdef DEBUG_ELS
if(verbose>1) cout<<" (pivot = "<<j<<": "<<supp[j]<<")";
#endif
if(verbose>1) cout<<endl;
}
}
// Record that the first 0, 1 or 2 elsgens come from known torsion points,
// so can be ingored in the gls step later:
if(which)
{
nt2gens1=nelsgens;
}
else
{
nt2gens0=nelsgens;
}
// Now systematically go through square-free divisors d1 of d, always modulo
// the subgroup of those which we know are els
int res;
for(index=istep; index<maxn; index+=istep)
{
if(mask&index) continue;
d1 = makenum(supp,index);
d2 = d/d1;
#ifdef DEBUG_ELS
if(verbose>1) cout<<"Testing d1 = "<<d1<<":\t";
#endif
res = locallysoluble(d1,c,d2,badprimes,badp);
if(res)
{
elsgens.push_back(d1); nelsgens++;
if(verbose>1)
cout<<"Adding els generator #"<<nelsgens<<": d1 = " << d1;
for(j=ns-1; j>=0; j--)
// if(testbit(index,j)) {setbit(mask,j); break;}
if(index&(1<<j)) {mask|=(1<<j); break;}
#ifdef DEBUG_ELS
if(verbose>1) cout<<" (pivot = "<<j<<": "<<supp[j]<<")";
#endif
if(verbose>1) cout<<endl;
}
else
{
#ifdef DEBUG_ELS
if(verbose>1) cout<<"not locally soluble at p = "<<badp<<endl;
#endif
}
}
if(verbose>1)
{
cout<<"After els sieving, nelsgens = " << nelsgens;
cout << endl;
cout<<"2-rank of S^{phi"; if(which) cout<<"'";
cout<<"}(E"; if(!which) cout<<"'";
cout<<") = "<<nelsgens<<endl;
if(nelsgens>0) cout<<"(els)gens: "<<elsgens<<endl;
// cout<<"mask = "<<mask<<endl;
}
if(which)
{
elsgens1=elsgens; mask1=mask; els1=nelsgens;
}
else
{
elsgens0=elsgens; mask0=mask; els0=nelsgens;
}
}
// Second local descent: determine
// phi'(S^2(E)) in S^(phi)(E') if which==0
// phi(S^2(E')) in S^(phi')(E) if which==1
void rank2::find_els2gens(int which, const bigint& c, const bigint& d)
{
if (verbose>1)
{
if(which) cout<<"\n";
cout<<"Finding els2 gens for E"; if(which) cout<<"'";
cout<<" (c"; if(which) cout<<"'";
cout<<"= "<<c<<", d"; if(which) cout<<"'";
cout<<"= "<<d<<") which lift to S^2(E";
if(which) cout<<"'"; cout<<")"<<endl;
}
vector<bigint>& elsgens = (which? elsgens1: elsgens0);
long nelsgens = (which? els1: els0);
long nt2gens = (which? nt2gens1: nt2gens0);
bigint d1, d2, badp, x,y,z;
unsigned long els2mask; long index;
long maxn=1<<nelsgens, nels2gens=0, els2piv;
vector<bigint> els2gens;
bitspace els2_space(nelsgens);
// first record the torsion contribution:
for(index=0; index<nt2gens; index++)
{
#ifdef DEBUG_ELS
d1=elsgens[index];
if(verbose>1) cout<<"Processing torsion d1 = " << d1 << ":"<<endl;
#endif
els2mask=(1<<index);
if(els2_space.mask(els2mask)) continue; // we work mod the els2 subgp
els2piv=els2_space.reduce(els2mask);
if(els2piv<0) continue; // means we're in the subgp; won't happen
#ifndef DEBUG_ELS // else done earlier
d1=elsgens[index];
#endif
els2gens.push_back(d1); nels2gens++;
els2_space.augment(els2mask,els2piv);
if(verbose>1) cout<<"Adding (torsion) els2 generator #"<<(nels2gens)
<<": d1 = " << d1 <<endl;
#ifdef DEBUG_ELS
else
cout<<"Just incremented nels2gens to "<<nels2gens<<endl;
cout<<"now bitmask = "<<els2_space.getbitmask()<<endl;
#endif
} // end of torsion loop
int res, verb=0; if(verbose>2)verb=verbose-2;
// reduced verbosity level within desc2()
for(index=1; (index<maxn)&&(nels2gens<nelsgens); index++)
{
// First mask against known els2 subgroup:
if(els2_space.mask(index)) continue; // we work mod the els2 subgp
els2mask=index;
els2piv=els2_space.reduce(els2mask);
#ifdef DEBUG_ELS
cout<<"index = "<<index<<endl;
cout<<"els2mask = "<<els2mask<<endl;
cout<<"els2piv = "<<els2piv <<endl;
#endif
if(els2piv<0) continue; // means w're in the els2 subgp; won't happen
d1 = makenum(elsgens,index);
d2 = d/d1;
#ifdef DEBUG_ELS
if(verbose>1) cout<<"Processing d1 = " << d1 << ":\t";
#endif
if(which)
res = 1+desc2(c,d1,d2,badprimes,supp0,elsgens0,mask0,lim2,x,y,z,verb,1);
else
res = 1+desc2(c,d1,d2,badprimes,supp1,elsgens1,mask1,lim2,x,y,z,verb,1);
#ifdef DEBUG_ELS
if(verbose>1) cout<<"res = " << res << endl;
#endif
if(res)
{
els2gens.push_back(d1); nels2gens++;
els2_space.augment(els2mask,els2piv);
#ifdef DEBUG_ELS
cout<<"now bitmask = "<<els2_space.getbitmask()<<endl;
#endif
if(verbose>1)
cout<<"Adding els2 generator #"<<nels2gens<<": d1 = " << d1<<endl;
}
}
if(verbose>1)
{
cout<<"After els2 sieving, nels2gens = " << nels2gens;
cout << endl;
cout<<"2-rank of phi";if(!which) cout<<"'";
cout<<"(S^2(E"; if(which) cout<<"'";
cout<<")) = "<<nels2gens<<endl;
if(nels2gens>0) cout<<"(els2)gens: "<<els2gens<<endl;
}
if(which)
{
els2gens1=els2gens; els21=nels2gens;
}
else
{
els2gens0=els2gens; els20=nels2gens;
}
}
void rank2::find_glsgens(int which, const bigint& c, const bigint& d)
{
vector<bigint>& elsgens = (which? els2gens1: els2gens0);
long nelsgens = (which? els21: els20);
long nt2gens = (which? nt2gens1: nt2gens0);
vector<bigint> gls_gens;
bitspace gls_space(nelsgens);
long glspiv, maxn = 1<<nelsgens;
long nglsgens=0;
unsigned long glsmask;
long index, stage, nstages=1; if(do_second_descent) nstages=2;
long shortfall1, shortfall2;
bigint d1, d2;
int res;
// first record the torsion contribution:
for(index=0; index<nt2gens; index++)
{
glsmask=(1<<index);
d1=elsgens[index];
glspiv=gls_space.reduce(glsmask);
if(glspiv<0)
{
#ifdef DEBUG_GLS
cout<<"d1="<<d1<<": known gls (torsion)"<<endl;
#endif
continue; // as we are certainly in the gls subgroup
}
if(verbose>1)
{
cout<<"Adding (torsion) gls generator #"<<(nglsgens+1)
<<": d1 = " << d1 <<endl;
}
gls_gens.push_back(d1); nglsgens++;
gls_space.augment(glsmask,glspiv);
#ifdef DEBUG_GLS
cout<<"Just incremented nglsgens to "<<nglsgens<<endl;
#endif
} // end of torsion loop
// Next test all first descent curves for global solubility (stage 1)
// and (optionally) do a second descent (stage 2) on unsuccessful ones.
// The next lines are in case torsion accounts for everything,
// as then the following loop is not executed at all:
shortfall1 = 0; shortfall2 = 0;
if(which)
{gls1=gls21=nglsgens;}
else
{gls0=gls20=nglsgens;}
if((nglsgens==nelsgens)&&verbose)
cout<<"This component of the rank is 0\n";
for(stage=1; (stage<=nstages)&&(nglsgens<nelsgens); stage++)
{
if(verbose&&do_second_descent)
{
if(stage==1)
cout << "First stage (no second descent yet)...\n";
else
cout << "Second stage (using second descent)...\n";
}
for(index=1; (index<maxn)&&(nglsgens<nelsgens); index++)
{
#ifdef DEBUG_GLS
d1 = makenum(elsgens,index); // do it now for output purposes
cout<<"index="<<setbase(16)<<index<<setbase(10);
cout<<"\td1 = "<<d1<<":\t";
#endif
// First mask against gls subgroup:
glsmask=index;
glspiv=gls_space.reduce(glsmask);
if(glspiv<0)
{
#ifdef DEBUG_GLS
cout<<"known to be gls"<<endl;
#endif
continue; // as we are certainly in the gls subgroup
}
#ifdef DEBUG_GLS
cout<<"not known gls"
<<", mask="<<setbase(16)<<glsmask<<setbase(10)
<<endl;
cout<<"Keeping j="<<glspiv
<<" with mask "<<setbase(16)<<glsmask<<setbase(10)
<<" for possible use as pivot"<<endl;
#else
d1 = makenum(elsgens,index); // else we did it earlier
#endif
d2 = d/d1;
if(stage==1) res = testquartic(c,d1,d2,which);
else res = second_descent(c,d1,d2,which);
#ifdef DEBUG_GLS
cout<<"result = "<<res<<endl;
#endif
switch(res){
case -1: // should not happen
{
cout<<"Problem in 2nd descent!" <<endl;
}
case 0: // no point found, nothing to do
{
break;
}
case 1:
{
gls_space.augment(glsmask, glspiv);
gls_gens.push_back(d1);
nglsgens++;
#ifdef DEBUG_GLS
cout<<"Just incremented nglsgens to "<<nglsgens<<endl;
#endif
if(verbose>1)
{
cout<<"Adding gls generator #"<<(nglsgens)
<<": d1 = " << d1;
#ifdef DEBUG_GLS
cout<<" (g_pos = "<<glspiv<<", g_gen = "
<<setbase(16)<<glsmask<<setbase(10)<<")";
#endif
cout<<endl;
}
break;
}
} // end of switch(res)
} // end of index loop
#ifdef DEBUG_GLS
cout<<"At bottom of index loop\n";
cout<<"nglsgens = "<<nglsgens<<"\tgls_gens = ";
cout << gls_gens << endl;
#endif
if(stage==1) // finished stage 1, i.e. first descent
{
shortfall1 = nelsgens-nglsgens;
if(verbose)
{
cout << "After first global descent, this component of the rank";
if(shortfall1)
{
cout << "\n\thas lower bound " << nglsgens-nt2gens
<< "\n\tand upper bound " << nelsgens-nt2gens
<< "\n\t(difference = " << shortfall1 << ")\n";
if(nstages==2)
cout<<"Second descent will attempt to reduce this\n";
}
else
cout << " = " << nelsgens <<endl;
if(verbose>1) cout<<"\n";
}
if(which)
gls1=gls21=nglsgens;
else
gls0=gls20=nglsgens;
}
else // finished stage==2, i.e. 2nd descent
{
shortfall2 = nelsgens-nglsgens;
if(verbose)
{
cout << "After second global descent, this component of the rank";
if(shortfall2)
{
cout << "\n\thas lower bound " << nglsgens-nt2gens
<< "\n\tand upper bound " << nelsgens-nt2gens
<< "\n\t(difference = " << shortfall2 << ")\n";
}
else
cout << " = " << nelsgens <<endl;
if(verbose>1) cout<<"\n";
}
if(which)
gls21=nglsgens;
else
gls20=nglsgens;
}
} // end of stage loop
if(which)
glsgens1=gls_gens;
else
glsgens0=gls_gens;
if(verbose>1)
{
cout<<"After gls sieving, nglsgens = " << nglsgens << endl;
cout<< "shortfall in rank from first descent = "<<shortfall1<<endl;
if(do_second_descent)
cout<< "shortfall in rank from second descent = "<<shortfall2<<endl;
if(nglsgens>0) cout<<"(gls)gens: "<<gls_gens<<endl;
}
}
void rank2::local_descent(const bigint& x0)
{
bigint c,d,cdash,ddash,disc,rootd;
const bigint zero = BIGINT(0), two=BIGINT(2), minusone=BIGINT(-1);
c = 3 * x0 + s2;
d = x0*(c+s2) + s4;
cdash = - 2 * c;
ddash = c*c - 4*d;
disc = 2*d*ddash;
if (is_zero(disc)) // this should have been caught by the calling program!
{
cout << "Curve is singular\n";
success = 0;
return;
}
if(verbose>1)
{
cout<<"(c,d) =("<<c<<","<<d<<")"<<endl;
cout<<"(c',d')=("<<cdash<<","<<ddash<<")"<<endl;
}
ee = Curvedata(zero, c, zero, d, zero);
eedash = Curvedata(zero,cdash,zero,ddash,zero);
Eprime = Curvedata(eedash,1); // minimal model
if(verbose)
{
cout<<"Using 2-isogenous curve "<<(Curve)(eedash);
if (eedash!=Eprime)
cout<<" (minimal model "<<(Curve)(Eprime)<<")";
cout<<endl;
}
supp0 = pdivs(d);
supp1 = pdivs(ddash);
badprimes.clear();
set_union(supp0.begin(),supp0.end(),supp1.begin(),supp1.end(),back_inserter(badprimes));
if(find(badprimes.begin(),badprimes.end(),two)==badprimes.end())
badprimes.insert(badprimes.begin(),two);
// cout<<"supp0="<<supp0<<", supp1="<<supp1<<endl;
// cout<<"badprimes="<<badprimes<<endl;
// NB -1 MUST be the first entry in the supports!
supp0.insert(supp0.begin(),minusone);
supp1.insert(supp1.begin(),minusone);
// cout<<"supp0="<<supp0<<", supp1="<<supp1<<endl;
d_is_sq = ddash_is_sq = 0;
if(isqrt(ddash,rootd))
{
ddash_is_sq=1;
e2 = (-c-rootd)/2;
e3 = (-c+rootd)/2;
}
if(isqrt(d,rootd))
{
d_is_sq=1;
e2dash = c-2*rootd;
e3dash = c+2*rootd;
}
if(verbose)
{
cout<<"-------------------------------------------------------\n";
cout<<"First step, determining 1st descent Selmer groups\n";
cout<<"-------------------------------------------------------\n";
}
// cout<<"Calling find_elsgens(0,...)"<<endl;
find_elsgens(0,c,d);
// cout<<"Calling find_elsgens(1,...)"<<endl;
find_elsgens(1,cdash,ddash);
selmer_rank_phi_Eprime = els0;
selmer_rank_phiprime_E = els1;
rank_bound = els0+els1-2;
if(verbose)
{
cout<<"After first local descent, rank bound = "<<rank_bound<<"\n";
cout<<"rk(S^{phi}(E'))=\t"<<selmer_rank_phi_Eprime<<endl;
cout<<"rk(S^{phi'}(E))=\t"<<selmer_rank_phiprime_E<<endl;
cout<<endl;
cout<<"-------------------------------------------------------\n";
cout<<"Second step, determining 2nd descent Selmer groups\n";
cout<<"-------------------------------------------------------\n";
}
if((rank_bound==0)||(!do_second_descent))
{
els20=els0; els21=els1;
els2gens0=elsgens0; els2gens1=elsgens1;
if(verbose)
{
if(do_second_descent)
cout<<"...skipping since we already know rank=0"<<endl;
else
cout<<"...skipping -- no second descent requested"<<endl;
}
}
else
{
find_els2gens(0,c,d);
find_els2gens(1,cdash,ddash);
rank_bound = els20+els21-2;
}
// NB these are only upper bounds if !do_second_descent
selmer_rank = els1+els20-1+ddash_is_sq;
selmer_rank_Eprime = els0+els21-1+d_is_sq;
if(verbose)
{
if(do_second_descent)
{
cout<<"After second local descent, rank bound = "<<rank_bound<<"\n";
cout<<"rk(phi'(S^{2}(E)))=\t"<<els20<<endl;
cout<<"rk(phi(S^{2}(E')))=\t"<<els21<<endl;
cout<<"rk(S^{2}(E))=\t"<<selmer_rank<<endl;
cout<<"rk(S^{2}(E'))=\t"<<selmer_rank_Eprime<<endl;
cout<<endl;
}
else
{
cout<<"No second local descent done, rank bound = "<<rank_bound<<"\n";
cout<<"rk(phi'(S^{2}(E)))<=\t"<<els20<<endl;
cout<<"rk(phi(S^{2}(E')))<=\t"<<els21<<endl;
cout<<"rk(S^{2}(E))<=\t"<<selmer_rank<<endl;
cout<<"rk(S^{2}(E'))<=\t"<<selmer_rank_Eprime<<endl;
cout<<endl;
}
}
}
rank2::rank2(Curvedata* ec, int verb, int sel, long l1, long l2, int second)
: rank12(ec,verb,sel,l1,l2,0,second)
{
bigint a1, a2, a3, a4, a6;
ec->getai(a1,a2,a3,a4,a6);
fullnpoints = npoints = 0;
rank = 0; // default value if failure occurs
int best_isogeny=0, best_rank_bound=999999;
success = 1;
int n, scaled=0;
if (odd(a1) || odd(a3))
{ s2= a1*a1+ 4*a2;
s4= 8*(a1*a3+ 2*a4);
s6= 16*(a3*a3+ 4*a6);
scaled=1;
}
else
{ s2=a2; s4=a4; s6=a6;
}
vector<bigint> xlist = Introotscubic(s2,s4,s6);
ntwo_torsion = xlist.size();
if (ntwo_torsion==0)
{
success=0;
if (verbose) cout << "No points of order 2\n";
return;
}
if(verbose) cout << "\n" << ntwo_torsion << " points of order 2:\n";
// If there are 3 points of order 2, we order them (for consistency:
// otherwise the order can be machine dependent)
if(ntwo_torsion==3) sort(xlist.begin(),xlist.end());
two_torsion.resize(ntwo_torsion);
for(n=0; n<ntwo_torsion; n++)
{
bigint ei = xlist[n];
if(scaled) two_torsion[n].init(the_curve,2*ei,-a1*ei-4*a3,BIGINT(8));
else two_torsion[n].init(the_curve,ei,BIGINT(0),BIGINT(1));
if(verbose)
{ if(n>0) cout<<", "; cout<<two_torsion[n];}
}
if(verbose)cout<<endl<<endl;
long mw0,mw1,sel0,sel1,sha0,sha1,sel20,sel21,sha20,sha21,lindex;
// we loop over the different 2-isogenies (if there are 3), doing the
// local descent 3 times to get the best rank bound (or stopping if
// that bound is 0).
// Fudge: we have to repeat the best one if it was not the last in
// order to get the class-global variables right for the global
// descent. This should obviously be fixed.
// The relevant variables are:
// ee,eedash,supp0,supp1,badprimes,d_is_sq,ddash_is_sq,
// e2,e3,e2dash,e3dash,els20,els21,elsgens0,elsgens1
for(n=0; n<ntwo_torsion; n++)
{
if(verbose&&(ntwo_torsion>1))
{
cout<<"****************************"<<endl;
cout<<"* Using 2-isogeny number "<<(n+1)<<" *"<<endl;
cout<<"****************************\n"<<endl;
}
local_descent(xlist[n]);
if((n==0)||(rank_bound<=best_rank_bound))
{
best_rank_bound=rank_bound;
best_isogeny=n;
}
if(best_rank_bound==0) break;
}
int rerun_needed=(rank_bound>best_rank_bound);
rank_bound=best_rank_bound;
if(verbose&&best_isogeny>0)
{
cout<<"After second local descent, combined upper bound on rank = "
<<rank_bound<<"\n";
}
if(selmer_only) // Nothing more to do in this case
{
rank=0;
certain=(rank_bound==0);
}
else // do the global stages...
{
// Redo the best local descent if necessary
bigint x0=xlist[best_isogeny];
if(rerun_needed)
{
if(verbose) cout<<"Rerunning the local descent for isogeny number "
<<(best_isogeny+1)
<< ", which gave the best upper bound on the rank"
<<endl;
local_descent(x0);
}
bigint c = 3 * x0 + s2;
bigint d = x0*(c+s2) + s4;
bigint cdash = - 2 * c;
bigint ddash = c*c - 4*d;
if(verbose)
{
cout<<"Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))\n";
cout<<"-------------------------------------------------------\n";
cout<<"1. E(Q)/phi(E'(Q))\n";
cout<<"-------------------------------------------------------\n";
}
if(verbose)
{
cout<<"(c,d) =("<<c<<","<<d<<")"<<endl;
cout<<"(c',d')=("<<cdash<<","<<ddash<<")"<<endl;
}
find_glsgens(0,c,d);
npoints1=npoints;
if(verbose)
{
cout<<"-------------------------------------------------------\n";
cout<<"2. E'(Q)/phi'(E(Q))\n";
cout<<"-------------------------------------------------------\n";
}
find_glsgens(1,cdash,ddash);
//Debug only:
/*
cout<<"gls20 = "<<gls20<<endl;
cout<<"gls21 = "<<gls21<<endl;
cout<<"els0 = "<<els0<<endl;
cout<<"els1 = "<<els1<<endl;
cout<<"els20 = "<<els20<<endl;
cout<<"els21 = "<<els21<<endl;
*/
rank = gls20+gls21-2; // lower bound for rank
//rank_bound = els20+els21-2; // upper bound for rank, from above
mw0 = 1<<gls20;
sel0 = 1<<els0; sha0=sel0/mw0;
sel20 = 1<<els20; sha20=sel20/mw0;
// gls20 is the 2-rank of E/phi(E') (lower bound)
// els0 is the 2-rank of S^(phi)(E') (exact)
// els20 is the 2-rank of its subgroup phi'(S^(2)(E)) (exact if 2nd descent)
// els0-gls20 is the 2-rank of III(E')[phi) (upper bound)
// els20-gls20 is the 2-rank of its subgroup phi'(III(E)[2]) (upper bound)
mw1 = 1<<gls21;
sel1 = 1<<els1; sha1=sel1/mw1;
sel21 = 1<<els21; sha21=sel21/mw1;
// gls21 is the 2-rank of E'/phi'(E) (lower bound)
// els1 is the 2-rank of S^(phi')(E) (exact)
// els21 is the 2-rank of its subgroup phi(S^(2)(E')) (exact if 2nd descent)
// els1-gls21 is the 2-rank of III(E)[phi'] (upper bound)
// els21-gls21 is the 2-rank of its subgroup phi(III(E')[2]) (upper bound)
index2=mw0*mw1;
lindex=gls20+gls21;
if(!ddash_is_sq) {index2/=2; lindex-=1;}
int certain0 = (els20==gls20); // i.e. descent was conclusive
int certain1 = (els21==gls21); //
certain = certain0&&certain1;
if (verbose)
{
cout<<"\n";
cout<<"-------------------------------------------------------\n";
cout<<"Summary of results:\n";
cout<<"-------------------------------------------------------\n";
if(certain) cout << "\trank(E) " << "= " << rank;
else cout << "\t"<<rank<< " <= rank(E) <= " << rank_bound;
cout<<"\n";
cout << "\t#E(Q)/2E(Q) ";
if(!certain) cout<<">";
cout<<"= " << index2 << "\n\n";