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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Path: gap4r8 / pkg / congruence-1.2.1 / doc / cong.xml
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1
<Chapter Label="CongruenceSubgroups">
2
<Heading>Construction of congruence subgroups</Heading>
3

4
<Index Key="IsCongruenceSubgroup"><C>IsCongruenceSubgroup</C></Index>
5
The package &Congruence; provides functions to construct several
6
types of canonical congruence subgroups in <M>SL_2(&ZZ;)</M>, and also 
7
intersections of a finite number of such subgroups. They will return 
8
a matrix group in the category <C>IsCongruenceSubgroup</C>,
9
which is defined as a subcategory of <C>IsMatrixGroup</C>, and
10
which will have a distinguishing property determining whether it is
11
a congruence subgroup of one of the canonical types, or an intersection
12
of such congruence subgroups (if it can not be reduced to one of the
13
canonical congruence subgroups).
14

15
To start to work with the package, you need first to load it as follows:
16

17
<Log>
18
<![CDATA[
19
gap> LoadPackage("congruence");
20
-----------------------------------------------------------------------------
21
Loading  Congruence 1.1.0 (Congruence subgroups of SL(2,Integers))
22
by Ann Dooms (http://homepages.vub.ac.be/~andooms),
23
   Eric Jespers (http://homepages.vub.ac.be/~efjesper),
24
   Alexander Konovalov (http://www.cs.st-andrews.ac.uk/~alexk/), and
25
   Helena Verrill (http://www.math.lsu.edu/~verrill).
26
-----------------------------------------------------------------------------
27
true
28
]]>
29
</Log>
30
 
31
<Section Label="CongConstr">
32
<Heading>Construction of congruence subgroups</Heading>
33
            
34
<ManSection>
35
  <Oper Name="PrincipalCongruenceSubgroup" 
36
         Arg="N"  
37
        Comm="" />
38
  <Description>
39
    Returns the principal congruence subgroup <M>\Gamma(N)</M> of level <A>N</A> 
40
    in <M>SL_2(&ZZ;)</M>.<P/>
41
    This subgroup consists of 
42
    all matrices of the form
43
<Alt Only="LaTeX">
44
  <Display>
45
  <![CDATA[
46
        \left( 
47
           \begin{array}{rr}
48
              1+N a & N b \\
49
              N c   & 1+N d
50
           \end{array} 
51
        \right)
52
  ]]>
53
  </Display>
54
</Alt>
55
<Alt Only="Text,HTML"><Verb><![CDATA[
56
                         [1+N*a    N*b]
57
                         [  N*c  1+N*d]
58
]]></Verb></Alt>   
59
where <M>a</M>,<M>b</M>,<M>c</M>,<M>d</M> are integers. 
60
       The returned group will have the property
61
    <Ref Prop="IsPrincipalCongruenceSubgroup" />.
62
  </Description>
63
</ManSection>       
64

65
<Example>
66
<![CDATA[
67
gap> G_8:=PrincipalCongruenceSubgroup(8);
68
<principal congruence subgroup of level 8 in SL_2(Z)>
69
gap> IsGroup(G_8);
70
true
71
gap> IsMatrixGroup(G_8);
72
true
73
gap> DimensionOfMatrixGroup(G_8);
74
2
75
gap> MultiplicativeNeutralElement(G_8);
76
[ [ 1, 0 ], [ 0, 1 ] ]
77
gap> One(G);
78
[ [ 1, 0 ], [ 0, 1 ] ]
79
gap> [[1,2],[3,4]] in G_8;
80
false
81
gap> [[1,8],[8,65]] in G_8;
82
true
83
gap> SL_2:=SL(2,Integers);
84
SL(2,Integers)
85
gap> IsSubgroup(SL_2,G_8);
86
true
87
]]>
88
</Example>
89

90

91
<ManSection>
92
  <Oper Name="CongruenceSubgroupGamma0" 
93
         Arg="N"  
94
        Comm="" />
95
  <Description>
96
    Returns the congruence subgroup <M>\Gamma_0(N)</M> 
97
    of level <A>N</A> in <M>SL_2(&ZZ;)</M>.<P/>
98
    This subgroup consists of 
99
    all matrices of the form
100
<Alt Only="LaTeX">
101
  <Display>
102
  <![CDATA[
103
        \left( 
104
           \begin{array}{rr}
105
              a & b \\
106
              N c   & d
107
           \end{array} 
108
        \right)
109
  ]]>
110
  </Display>
111
</Alt>
112
<Alt Only="Text,HTML"><Verb><![CDATA[
113
                         [a    b]
114
                         [N*c  d]
115
]]></Verb></Alt>   
116
where <M>a</M>,<M>b</M>,<M>c</M>,<M>d</M> are integers. 
117
         The returned group will have the property
118
    <Ref Prop="IsCongruenceSubgroupGamma0" />.   
119
  </Description>
120
</ManSection> 
121

122
<Example>
123
<![CDATA[
124
gap> G0_4:=CongruenceSubgroupGamma0(4);
125
<congruence subgroup CongruenceSubgroupGamma_0(4) in SL_2(Z)>
126
]]>
127
</Example>
128

129

130
<ManSection>
131
  <Oper Name="CongruenceSubgroupGammaUpper0" 
132
         Arg="N"  
133
        Comm="" />
134
  <Description>
135
    Returns the congruence subgroup <M>\Gamma^0(N)</M> 
136
    of level <A>N</A> in <M>SL_2(&ZZ;)</M>.<P/>
137
    This subgroup consists of 
138
    all matrices of the form
139
<Alt Only="LaTeX">
140
  <Display>
141
  <![CDATA[
142
        \left( 
143
           \begin{array}{rr}
144
              a & N b \\
145
              c   & d
146
           \end{array} 
147
        \right)
148
  ]]>
149
  </Display>
150
</Alt>
151
<Alt Only="Text,HTML"><Verb><![CDATA[
152
                         [a  N*b]
153
                         [c    d]
154
]]></Verb></Alt>   
155
where <M>a</M>,<M>b</M>,<M>c</M>,<M>d</M> are integers. 
156
             The returned group will have the property
157
    <Ref Prop="IsCongruenceSubgroupGammaUpper0" />. 
158
  </Description>
159
</ManSection>
160

161
<Example>
162
<![CDATA[
163
gap> GU0_2:=CongruenceSubgroupGammaUpper0(2);
164
<congruence subgroup CongruenceSubgroupGamma^0(2) in SL_2(Z)>
165
]]>
166
</Example>
167

168

169
<ManSection>
170
  <Oper Name="CongruenceSubgroupGamma1" 
171
         Arg="N"  
172
        Comm="" />
173
  <Description>
174
    Returns the congruence subgroup <M>\Gamma_1(N)</M> 
175
    of level <A>N</A> in <M>SL_2(&ZZ;)</M>.<P/>
176
         This subgroup consists of 
177
    all matrices of the form
178
<Alt Only="LaTeX">
179
  <Display>
180
  <![CDATA[
181
        \left( 
182
           \begin{array}{rr}
183
              1+N a & b \\
184
              N c   & 1+N d
185
           \end{array} 
186
        \right)
187
  ]]>
188
  </Display>
189
</Alt>
190
<Alt Only="Text,HTML"><Verb><![CDATA[
191
                         [1+N*a      b]
192
                         [  N*c  1+N*d]
193
]]></Verb></Alt>   
194
where <M>a</M>,<M>b</M>,<M>c</M>,<M>d</M> are integers. 
195
    The returned group will have the property
196
    <Ref Prop="IsCongruenceSubgroupGamma1" />.            
197
  </Description>
198
</ManSection> 
199

200
<Example>
201
<![CDATA[
202
gap> G1_6:=CongruenceSubgroupGamma1(6);
203
<congruence subgroup CongruenceSubgroupGamma_1(6) in SL_2(Z)>
204
]]>
205
</Example>
206

207

208
<ManSection>
209
  <Oper Name="CongruenceSubgroupGammaUpper1" 
210
         Arg="N"  
211
        Comm="" />
212
  <Description>
213
    Returns the congruence subgroup <M>\Gamma^1(N)</M> 
214
    of level <A>N</A> in <M>SL_2(&ZZ;)</M>.<P/>
215
         This subgroup consists of 
216
    all matrices of the form
217
<Alt Only="LaTeX">
218
  <Display>
219
  <![CDATA[
220
        \left( 
221
           \begin{array}{rr}
222
              1+N a & N b \\
223
               c   & 1+N d
224
           \end{array} 
225
        \right)
226
  ]]>
227
  </Display>
228
</Alt>
229
<Alt Only="Text,HTML"><Verb><![CDATA[
230
                         [1+N*a    N*b]
231
                         [    c  1+N*d]
232
]]></Verb></Alt>   
233
where <M>a</M>,<M>b</M>,<M>c</M>,<M>d</M> are integers. 
234
    The returned group will have the property
235
    <Ref Prop="IsCongruenceSubgroupGammaUpper1" />.               
236
  </Description>
237
</ManSection>
238

239
<Example>
240
<![CDATA[
241
gap> GU1_4:=CongruenceSubgroupGammaUpper1(4);
242
<congruence subgroup CongruenceSubgroupGamma^1(4) in SL_2(Z)>
243
]]>
244
</Example>
245

246

247
<ManSection>
248
  <Func Name="IntersectionOfCongruenceSubgroups" 
249
         Arg="G1, G2, ..., GN" />
250
  <Func Name="Intersection" 
251
         Arg="G1, G2, ..., GN" />         
252
  <Description>
253
    Returns the intersection of its arguments, which can be
254
    congruence subgroups or their intersections, constructed
255
    with the same function. It is not necessary for the user 
256
    to use <C>IntersectionOfCongruenceSubgroups</C>, since
257
    it will be called automatically from <C>Intersection</C>.<P/>
258
    
259
    The returned group will have the property
260
    <Ref Prop="IsIntersectionOfCongruenceSubgroups" />.<P/>
261
    
262
    The list of congruence subgroups that form the intersection
263
    can be obtained using <Ref Attr="DefiningCongruenceSubgroups"/>.
264
    
265
    Note, that when the intersection appears to be one of the 
266
    canonical congruence subgroups, the package will recognize 
267
    this and will return a canonical subgroup of the 
268
    appropriate type.
269
         
270
  </Description>
271
</ManSection>
272

273
<Example>
274
<![CDATA[
275
gap> I:=IntersectionOfCongruenceSubgroups(G0_4,GU1_4);
276
<principal congruence subgroup of level 4 in SL_2(Z)>
277
gap> J:=IntersectionOfCongruenceSubgroups(G0_4,G1_6);
278
<intersection of congruence subgroups of resulting level 12 in SL_2(Z)>
279
]]>
280
</Example>
281

282
</Section>
283

284
<!-- ********************************************************* -->
285

286
<Section Label="CongProperties">
287
<Heading>Properties of congruence subgroups</Heading>
288

289
A congruence subgroup constructed by one of the five above listed functions
290
will have certain properties determining its type. These properties will be
291
used for method selection by &Congruence; algorithms. Note that they do not
292
provide an actual test whether a certain matrix group is a congruence subgroup 
293
or not.
294

295
<ManSection>
296
  <Prop Name="IsPrincipalCongruenceSubgroup" 
297
         Arg="G"/>                            
298
  <Description>
299
    For a congruence subgroup <A>G</A> in the category 
300
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if <A>G</A> was
301
    constructed by <Ref Func="PrincipalCongruenceSubgroup" /> 
302
    (or reduced to one as a result of an intersection) and 
303
    returns <K>false</K> otherwise. 
304
  </Description>
305
</ManSection>
306

307
<Example>
308
<![CDATA[
309
gap> IsPrincipalCongruenceSubgroup(G_8);
310
true
311
gap> IsPrincipalCongruenceSubgroup(G0_4);
312
false
313
gap> IsPrincipalCongruenceSubgroup(I);
314
true
315
]]>
316
</Example>
317

318

319
<ManSection>
320
  <Prop Name="IsCongruenceSubgroupGamma0" 
321
         Arg="G"/>
322
  <Description>
323
    For a congruence subgroup <A>G</A> in the category 
324
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if 
325
    <A>G</A> was constructed by <Ref Func="CongruenceSubgroupGamma0" /> (or reduced to one as a result of an intersection)
326
    and returns <K>false</K> otherwise.  
327
  </Description>
328
</ManSection>
329

330

331
<ManSection>
332
  <Prop Name="IsCongruenceSubgroupGammaUpper0" 
333
         Arg="G"/>
334
  <Description>
335
    For a congruence subgroup <A>G</A> in the category 
336
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if 
337
    <A>G</A> was constructed by <Ref Func="CongruenceSubgroupGammaUpper0" /> (or reduced to one as a result of an intersection)
338
    and returns <K>false</K> otherwise.
339
  </Description>
340
</ManSection>
341

342

343
<ManSection>
344
  <Prop Name="IsCongruenceSubgroupGamma1" 
345
         Arg="G"/>
346
  <Description>
347
    For a congruence subgroup <A>G</A> in the category 
348
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if
349
    <A>G</A> was constructed by <Ref Func="CongruenceSubgroupGamma1" /> (or reduced to one as a result of an intersection)
350
    and returns <K>false</K> otherwise.
351
  </Description>
352
</ManSection>
353

354

355
<ManSection>
356
  <Prop Name="IsCongruenceSubgroupGammaUpper1" 
357
         Arg="G"/>                                    
358
  <Description>
359
    For a congruence subgroup <A>G</A> in the category 
360
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if 
361
    <A>G</A> was constructed by <Ref Func="CongruenceSubgroupGammaUpper1" /> (or reduced to one as a result of an intersection)
362
    and returns <K>false</K> otherwise.
363
  </Description>
364
</ManSection>
365

366

367
<ManSection>
368
  <Prop Name="IsIntersectionOfCongruenceSubgroups" 
369
         Arg="G"/>                                    
370
  <Description>
371
    For a congruence subgroup <A>G</A> in the category 
372
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if
373
    <A>G</A> was constructed by 
374
    <Ref Func="IntersectionOfCongruenceSubgroups"/> and
375
    without being one of the canonical congruence
376
    subgroups, otherwise it returns <K>false</K>.
377
  </Description>
378
</ManSection>
379

380
<Example>
381
<![CDATA[
382
gap> IsIntersectionOfCongruenceSubgroups(I);
383
false
384
gap> IsIntersectionOfCongruenceSubgroups(J);
385
true
386
]]>
387
</Example>
388

389
</Section>
390

391
<!-- ********************************************************* -->
392

393
<Section Label="CongAttributes">
394
<Heading>Attributes of congruence subgroups</Heading>
395

396
The next three attributes store key properties of congruence subgroups.
397

398
<ManSection>
399
  <Attr Name="LevelOfCongruenceSubgroup" 
400
         Arg="G"/>                                    
401
  <Description>
402
    Stores the level of the congruence subgroup <A>G</A>. The (arithmetic) level of a congruence subgroup G is the smallest positive
403
number N such that G contains the principal congruence subgroup of level N.
404
  </Description>
405
</ManSection>
406

407
<Example>
408
<![CDATA[
409
gap> LevelOfCongruenceSubgroup(G_8);
410
8
411
gap> LevelOfCongruenceSubgroup(G1_6);
412
6
413
gap> LevelOfCongruenceSubgroup(I);
414
4
415
gap> LevelOfCongruenceSubgroup(J);
416
12
417
]]>
418
</Example>
419

420

421
<ManSection>
422
  <Attr Name="IndexInSL2Z" 
423
         Arg="G"/>                                    
424
  <Description>
425
    Stores the index of the congruence subgroup <A>G</A> in <M>SL_2(&ZZ;)</M>.
426
  </Description>
427
</ManSection>
428

429
<Example>
430
<![CDATA[
431
gap> IndexInSL2Z(G_8);
432
384
433
gap> G_2:=PrincipalCongruenceSubgroup(2);
434
<principal congruence subgroup of level 2 in SL_2(Z)>
435
gap> IndexInSL2Z(G_2);
436
12
437
gap> IndexInSL2Z(GU1_4);
438
12
439
]]>
440
</Example>
441

442
<ManSection>
443
  <Attr Name="DefiningCongruenceSubgroups" 
444
         Arg="G" />
445
  <Returns>
446
    list of congruence subgroups
447
  </Returns>          
448
  <Description>
449
    For an intersection of congruence subgroups, returns 
450
    the list of congruence subgroups forming this intersection.
451
    For a canonical congruence subgroup returns a list of length
452
    one containing that subgroup.
453
  </Description>
454
</ManSection>
455

456
<Example>
457
<![CDATA[
458
gap> DefiningCongruenceSubgroups(J);
459
[ <congruence subgroup CongruenceSubgroupGamma_0(4) in SL_2(Z)>,
460
  <congruence subgroup CongruenceSubgroupGamma_1(6) in SL_2(Z)> ]
461
gap> P:=PrincipalCongruenceSubgroup(6);
462
<principal congruence subgroup of level 6 in SL_2(Z)>
463
gap> Q:=PrincipalCongruenceSubgroup(10); 
464
<principal congruence subgroup of level 10 in SL_2(Z)>
465
gap> G:=IntersectionOfCongruenceSubgroups(Q,P);  
466
<principal congruence subgroup of level 30 in SL_2(Z)>
467
gap> DefiningCongruenceSubgroups(G);
468
[ <principal congruence subgroup of level 30 in SL_2(Z)> ] 
469
]]>
470
</Example>
471

472
</Section>
473

474
<!-- ********************************************************* -->
475

476
<Section Label="CongMethods">
477
<Heading>Operations for congruence subgroups</Heading>
478

479
&Congruence; installs several special methods for operations already 
480
available in &GAP;.
481

482
<ManSection>
483
  <Oper Name="Random" 
484
       Label="one and two argument versions"
485
         Arg="G"/> 
486
  <Oper Name="Random" 
487
         Arg="G m"/>                                             
488
  <Description>
489
    For a congruence subgroup <A>G</A> in the category 
490
    <C>IsCongruenceSubgroup</C>, returns random element.
491
    In the two-argument form, the second parameter will
492
    control the absolute value of randomly selected 
493
    entries of the matrix.
494
  </Description>
495
</ManSection>
496

497
<Example>
498
<![CDATA[
499
gap> Random(G_2) in G_2;
500
true
501
gap> Random(G_8,2) in G_8;
502
true
503
]]>
504
</Example>
505

506

507
<ManSection>
508
  <Oper Name="\in" 
509
         Arg="m G"/> 
510
  <Description>
511
    It is easy to implement the membership test for
512
    congruence subgroups and their intersections.
513
  </Description>
514
</ManSection>
515

516
<Example>
517
<![CDATA[
518
gap> \in([ [ 21, 10 ], [ 2, 1 ] ],G_2);
519
true
520
gap> \in([ [ 21, 10 ], [ 2, 1 ] ],G_8);
521
false
522
]]>
523
</Example>
524

525

526
<ManSection>
527
  <Oper Name="CanEasilyCompareCongruenceSubgroups" 
528
         Arg="G H"/>                                    
529
  <Description>
530
  For congruence subgroups <A>G,H</A> in the category 
531
    <C>IsCongruenceSubgroup</C>, returns <K>true</K> if
532
    <A>G</A> and <A>H</A> are of the same type listed in
533
     <Ref Func="PrincipalCongruenceSubgroup" /> --> 
534
    <Ref Func="CongruenceSubgroupGammaUpper1" />
535
     and have the same
536
    <Ref Func="LevelOfCongruenceSubgroup"/> or if <A>G</A> and <A>H</A> 
537
    are of the type <Ref Func="IntersectionOfCongruenceSubgroups"/> and
538
    the groups from  
539
    <Ref Func="DefiningCongruenceSubgroups"/> are in one 
540
    to one correspondence, otherwise it returns <K>false</K>. 
541
  </Description>
542
</ManSection>
543

544

545
<Example>
546
<![CDATA[
547
gap> CanEasilyCompareCongruenceSubgroups(G_8,I);
548
false
549
]]>
550
</Example>
551

552
<ManSection>
553
  <Oper Name="IsSubset" 
554
         Arg="G H"/> 
555
  <Description>
556
    &Congruence; provides methods for <C>IsSubset</C>
557
    for congruence subgroups. <C>IsSubset</C> returns <K>true</K> 
558
    if <A>H</A> is a subset of <A>G</A>. These methods make it possible
559
    to use <C>IsSubgroup</C> operation for congruence subgroups.
560
  </Description>
561
</ManSection>
562

563

564
<Example>
565
<![CDATA[
566
gap> IsSubset(G_2,G_8);
567
true
568
gap> IsSubset(G_8,G_2);
569
false
570
gap> f:=[PrincipalCongruenceSubgroup,CongruenceSubgroupGamma1,CongruenceSubgroupGammaUpper1,CongruenceSubgroupGamma0,CongruenceSubgroupGammaUpper0];;
571
gap> g1:=List(f, t -> t(2));;
572
gap> g2:=List(f, t -> t(4));;
573
gap> for g in g2 do
574
> Print( List( g1, x -> IsSubgroup(x,g) ), "\n");
575
> od;
576
[ true, true, true, true, true ]
577
[ false, true, false, true, false ]
578
[ false, false, true, false, true ]
579
[ false, false, false, true, false ]
580
[ false, false, false, false, true ]
581
]]>
582
</Example>
583

584

585
<ManSection>
586
  <Oper Name="Index" 
587
         Arg="G H"/> 
588
  <Description>
589
    If a congruence subgroup <A>H</A> is a subgroup of a congruence 
590
    subgroup <A>G</A>, we can easily compute the index of <A>H</A> in
591
    <A>G</A>, since we know the index of both subgroups in <M>SL_2(&ZZ;)</M>.
592
  </Description>
593
</ManSection>
594

595
<Example>
596
<![CDATA[
597
gap> Index(G_2,G_8);
598
32
599
]]>
600
</Example>
601

602
</Section>
603
 
604
</Chapter>
605

606
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