Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

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

1001

1002

1003

1004

1005

1006

1007

1008

1009

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

1038

1039

1040

1041

1042

1043

1044

1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

1081

1082

1083

1084

1085

1086

1087

1088

1089

1090

1091

1092

1093

1094

1095

1096

1097

1098

1099

1100

1101

1102

1103

1104

1105

1106

1107

1108

1109

1110

1111

""" 

Local Density Congruence 

""" 

 

 

########################################################################## 

## Methods which compute the local densities for representing a number 

## by a quadratic form at a prime (possibly subject to additional 

## congruence conditions). 

########################################################################## 

from __future__ import print_function 

 

from copy import deepcopy 

 

from sage.sets.set import Set 

from sage.rings.rational_field import QQ 

from sage.arith.all import valuation 

from sage.misc.misc import verbose 

 

from sage.quadratic_forms.count_local_2 import count_modp__by_gauss_sum 

 

 

 

 

def count_modp_solutions__by_Gauss_sum(self, p, m): 

""" 

Returns the number of solutions of `Q(x) = m (mod p)` of a 

non-degenerate quadratic form over the finite field `Z/pZ`, 

where `p` is a prime number > 2. 

 

Note: We adopt the useful convention that a zero-dimensional 

quadratic form has exactly one solution always (i.e. the empty 

vector). 

 

These are defined in Table 1 on p363 of Hanke's "Local 

Densities..." paper. 

 

INPUT: 

 

`p` -- a prime number > 2 

 

`m` -- an integer 

 

OUTPUT: 

 

an integer >= 0 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1]) 

sage: [Q.count_modp_solutions__by_Gauss_sum(3, m) for m in range(3)] 

[9, 6, 12] 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,2]) 

sage: [Q.count_modp_solutions__by_Gauss_sum(3, m) for m in range(3)] 

[9, 12, 6] 

 

""" 

if self.dim() == 0: 

return 1 

else: 

return count_modp__by_gauss_sum(self.dim(), p, m, self.Gram_det()) 

 

 

 

 

 

def local_good_density_congruence_odd(self, p, m, Zvec, NZvec): 

""" 

Finds the Good-type local density of Q representing `m` at `p`. 

(Assuming that `p` > 2 and Q is given in local diagonal form.) 

 

The additional congruence condition arguments Zvec and NZvec can 

be either a list of indices or None. Zvec = [] is equivalent to 

Zvec = None which both impose no additional conditions, but NZvec 

= [] returns no solutions always while NZvec = None imposes no 

additional condition. 

 

TO DO: Add type checking for Zvec, NZvec, and that Q is in local 

normal form. 

 

INPUT: 

 

Q -- quadratic form assumed to be diagonal and p-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_good_density_congruence_odd(3, 1, None, None) 

2/3 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_good_density_congruence_odd(3, 1, None, None) 

8/9 

 

""" 

n = self.dim() 

 

## Put the Zvec congruence condition in a standard form 

if Zvec is None: 

Zvec = [] 

 

 

## Sanity Check on Zvec and NZvec: 

## ------------------------------- 

Sn = Set(range(n)) 

if (Zvec is not None) and (len(Set(Zvec) + Sn) > n): 

raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.") 

if (NZvec is not None) and (len(Set(NZvec) + Sn) > n): 

raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.") 

 

 

 

## Assuming Q is diagonal, find the indices of the p-unit (diagonal) entries 

UnitVec = [i for i in range(n) if (self[i,i] % p) != 0] 

NonUnitVec = list(Set(range(n)) - Set(UnitVec)) 

 

 

## Take cases on the existence of additional non-zero congruence conditions (mod p) 

UnitVec_minus_Zvec = list(Set(UnitVec) - Set(Zvec)) 

NonUnitVec_minus_Zvec = list(Set(NonUnitVec) - Set(Zvec)) 

Q_Unit_minus_Zvec = self.extract_variables(UnitVec_minus_Zvec) 

 

if (NZvec is None): 

if m % p != 0: 

total = Q_Unit_minus_Zvec.count_modp_solutions__by_Gauss_sum(p, m) * p**len(NonUnitVec_minus_Zvec) ## m != 0 (mod p) 

else: 

total = (Q_Unit_minus_Zvec.count_modp_solutions__by_Gauss_sum(p, m) - 1) * p**len(NonUnitVec_minus_Zvec) ## m == 0 (mod p) 

 

else: 

UnitVec_minus_ZNZvec = list(Set(UnitVec) - (Set(Zvec) + Set(NZvec))) 

NonUnitVec_minus_ZNZvec = list(Set(NonUnitVec) - (Set(Zvec) + Set(NZvec))) 

Q_Unit_minus_ZNZvec = self.extract_variables(UnitVec_minus_ZNZvec) 

 

if m % p != 0: ## m != 0 (mod p) 

total = Q_Unit_minus_Zvec.count_modp_solutions__by_Gauss_sum(p, m) * p**len(NonUnitVec_minus_Zvec) \ 

- Q_Unit_minus_ZNZvec.count_modp_solutions__by_Gauss_sum(p, m) * p**len(NonUnitVec_minus_ZNZvec) 

else: ## m == 0 (mod p) 

total = (Q_Unit_minus_Zvec.count_modp_solutions__by_Gauss_sum(p, m) - 1) * p**len(NonUnitVec_minus_Zvec) \ 

- (Q_Unit_minus_ZNZvec.count_modp_solutions__by_Gauss_sum(p, m) - 1) * p**len(NonUnitVec_minus_ZNZvec) 

 

## Return the Good-type representation density 

good_density = QQ(total) / p**(n-1) 

return good_density 

 

 

 

 

def local_good_density_congruence_even(self, m, Zvec, NZvec): 

""" 

Finds the Good-type local density of Q representing `m` at `p=2`. 

(Assuming Q is given in local diagonal form.) 

 

 

The additional congruence condition arguments Zvec and NZvec can 

be either a list of indices or None. Zvec = [] is equivalent to 

Zvec = None which both impose no additional conditions, but NZvec 

= [] returns no solutions always while NZvec = None imposes no 

additional condition. 

 

WARNING: Here the indices passed in Zvec and NZvec represent 

indices of the solution vector `x` of Q(`x`) = `m (mod p^k)`, and *not* 

the Jordan components of Q. They therefore are required (and 

assumed) to include either all or none of the indices of a given 

Jordan component of Q. This is only important when `p=2` since 

otherwise all Jordan blocks are 1x1, and so there the indices and 

Jordan blocks coincide. 

 

TO DO: Add type checking for Zvec, NZvec, and that Q is in local 

normal form. 

 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and 2-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_good_density_congruence_even(1, None, None) 

1 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_good_density_congruence_even(1, None, None) 

1 

sage: Q.local_good_density_congruence_even(2, None, None) 

3/2 

sage: Q.local_good_density_congruence_even(3, None, None) 

1 

sage: Q.local_good_density_congruence_even(4, None, None) 

1/2 

 

:: 

 

sage: Q = QuadraticForm(ZZ, 4, range(10)) 

sage: Q[0,0] = 5 

sage: Q[1,1] = 10 

sage: Q[2,2] = 15 

sage: Q[3,3] = 20 

sage: Q 

Quadratic form in 4 variables over Integer Ring with coefficients: 

[ 5 1 2 3 ] 

[ * 10 5 6 ] 

[ * * 15 8 ] 

[ * * * 20 ] 

sage: Q.theta_series(20) 

1 + 2*q^5 + 2*q^10 + 2*q^14 + 2*q^15 + 2*q^16 + 2*q^18 + O(q^20) 

sage: Q.local_normal_form(2) 

Quadratic form in 4 variables over Integer Ring with coefficients: 

[ 0 1 0 0 ] 

[ * 0 0 0 ] 

[ * * 0 1 ] 

[ * * * 0 ] 

sage: Q.local_good_density_congruence_even(1, None, None) 

3/4 

sage: Q.local_good_density_congruence_even(2, None, None) 

1 

sage: Q.local_good_density_congruence_even(5, None, None) 

3/4 

 

""" 

n = self.dim() 

 

## Put the Zvec congruence condition in a standard form 

if Zvec is None: 

Zvec = [] 

 

 

## Sanity Check on Zvec and NZvec: 

## ------------------------------- 

Sn = Set(range(n)) 

if (Zvec is not None) and (len(Set(Zvec) + Sn) > n): 

raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.") 

if (NZvec is not None) and (len(Set(NZvec) + Sn) > n): 

raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.") 

 

 

 

## Find the indices of x for which the associated Jordan blocks are non-zero mod 8 TO DO: Move this to special Jordan block code separately! 

## ------------------------------------------------------------------------------- 

Not8vec = [] 

for i in range(n): 

 

## DIAGNOSTIC 

verbose(" i = " + str(i)) 

verbose(" n = " + str(n)) 

verbose(" Not8vec = " + str(Not8vec)) 

 

nz_flag = False 

 

## Check if the diagonal entry isn't divisible 8 

if ((self[i,i] % 8) != 0): 

nz_flag = True 

 

## Check appropriate off-diagonal entries aren't divisible by 8 

else: 

 

## Special check for first off-diagonal entry 

if ((i == 0) and ((self[i,i+1] % 8) != 0)): 

nz_flag = True 

 

## Special check for last off-diagonal entry 

elif ((i == n-1) and ((self[i-1,i] % 8) != 0)): 

nz_flag = True 

 

## Check for the middle off-diagonal entries 

else: 

if ( (i > 0) and (i < n-1) and (((self[i,i+1] % 8) != 0) or ((self[i-1,i] % 8) != 0)) ): 

nz_flag = True 

 

## Remember the (vector) index if it's not part of a Jordan block of norm divisible by 8 

if nz_flag: 

Not8vec += [i] 

 

 

 

## Compute the number of Good-type solutions mod 8: 

## ------------------------------------------------ 

 

## Setup the indexing sets for additional zero congruence solutions 

Q_Not8 = self.extract_variables(Not8vec) 

Not8 = Set(Not8vec) 

Is8 = Set(range(n)) - Not8 

 

Z = Set(Zvec) 

Z_Not8 = Not8.intersection(Z) 

Z_Is8 = Is8.intersection(Z) 

Is8_minus_Z = Is8 - Z_Is8 

 

 

## DIAGNOSTIC 

verbose("Z = " + str(Z)) 

verbose("Z_Not8 = " + str(Z_Not8)) 

verbose("Z_Is8 = " + str(Z_Is8)) 

verbose("Is8_minus_Z = " + str(Is8_minus_Z)) 

 

 

## Take cases on the existence of additional non-zero congruence conditions (mod 2) 

if NZvec is None: 

total = (4 ** len(Z_Is8)) * (8 ** len(Is8_minus_Z)) \ 

* Q_Not8.count_congruence_solutions__good_type(2, 3, m, list(Z_Not8), None) 

else: 

ZNZ = Z + Set(NZvec) 

ZNZ_Not8 = Not8.intersection(ZNZ) 

ZNZ_Is8 = Is8.intersection(ZNZ) 

Is8_minus_ZNZ = Is8 - ZNZ_Is8 

 

## DIAGNOSTIC 

verbose("ZNZ = " + str(ZNZ)) 

verbose("ZNZ_Not8 = " + str(ZNZ_Not8)) 

verbose("ZNZ_Is8 = " + str(ZNZ_Is8)) 

verbose("Is8_minus_ZNZ = " + str(Is8_minus_ZNZ)) 

 

total = (4 ** len(Z_Is8)) * (8 ** len(Is8_minus_Z)) \ 

* Q_Not8.count_congruence_solutions__good_type(2, 3, m, list(Z_Not8), None) \ 

- (4 ** len(ZNZ_Is8)) * (8 ** len(Is8_minus_ZNZ)) \ 

* Q_Not8.count_congruence_solutions__good_type(2, 3, m, list(ZNZ_Not8), None) 

 

 

## DIAGNOSTIC 

verbose("total = " + str(total)) 

 

 

## Return the associated Good-type representation density 

good_density = QQ(total) / 8**(n-1) 

return good_density 

 

 

 

 

 

 

 

 

def local_good_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the Good-type local density of Q representing `m` at `p`. 

(Front end routine for parity specific routines for p.) 

 

TO DO: Add Documentation about the additional congruence 

conditions Zvec and NZvec. 

 

 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and p-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_good_density_congruence(2, 1, None, None) 

1 

sage: Q.local_good_density_congruence(3, 1, None, None) 

2/3 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_good_density_congruence(2, 1, None, None) 

1 

sage: Q.local_good_density_congruence(3, 1, None, None) 

8/9 

 

""" 

## DIAGNOSTIC 

verbose(" In local_good_density_congruence with ") 

verbose(" Q is: \n" + str(self)) 

verbose(" p = " + str(p)) 

verbose(" m = " + str(m)) 

verbose(" Zvec = " + str(Zvec)) 

verbose(" NZvec = " + str(NZvec)) 

 

## Put the Zvec congruence condition in a standard form 

if Zvec is None: 

Zvec = [] 

 

 

n = self.dim() 

 

## Sanity Check on Zvec and NZvec: 

## ------------------------------- 

Sn = Set(range(n)) 

if (Zvec is not None) and (len(Set(Zvec) + Sn) > n): 

raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.") 

if (NZvec is not None) and (len(Set(NZvec) + Sn) > n): 

raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.") 

 

 

 

## Check that Q is in local normal form -- should replace this with a diagonalization check? 

## (it often may not be since the reduction procedure 

## often mixes up the order of the valuations...) 

# 

#if (self != self.local_normal_form(p)) 

# print "Warning in local_good_density_congruence: Q is not in local normal form! \n"; 

 

 

 

 

## Decide which routine to use to compute the Good-type density 

if (p > 2): 

return self.local_good_density_congruence_odd(p, m, Zvec, NZvec) 

 

if (p == 2): 

return self.local_good_density_congruence_even(m, Zvec, NZvec) 

 

raise RuntimeError("\n Error in Local_Good_Density: The 'prime' p = " + str(p) + " is < 2. \n") 

 

 

 

 

 

 

 

def local_zero_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the Zero-type local density of Q representing `m` at `p`, 

allowing certain congruence conditions mod p. 

 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and `p`-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_zero_density_congruence(2, 2, None, None) 

0 

sage: Q.local_zero_density_congruence(2, 4, None, None) 

1/2 

sage: Q.local_zero_density_congruence(3, 6, None, None) 

0 

sage: Q.local_zero_density_congruence(3, 9, None, None) 

2/9 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_zero_density_congruence(2, 2, None, None) 

0 

sage: Q.local_zero_density_congruence(2, 4, None, None) 

1/4 

sage: Q.local_zero_density_congruence(3, 6, None, None) 

0 

sage: Q.local_zero_density_congruence(3, 9, None, None) 

8/81 

 

""" 

## DIAGNOSTIC 

verbose(" In local_zero_density_congruence with ") 

verbose(" Q is: \n" + str(self)) 

verbose(" p = " + str(p)) 

verbose(" m = " + str(m)) 

verbose(" Zvec = " + str(Zvec)) 

verbose(" NZvec = " + str(NZvec)) 

 

## Put the Zvec congruence condition in a standard form 

if Zvec is None: 

Zvec = [] 

 

 

n = self.dim() 

 

## Sanity Check on Zvec and NZvec: 

## ------------------------------- 

Sn = Set(range(n)) 

if (Zvec is not None) and (len(Set(Zvec) + Sn) > n): 

raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.") 

if (NZvec is not None) and (len(Set(NZvec) + Sn) > n): 

raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.") 

 

 

p2 = p * p 

 

## Check some conditions for no zero-type solutions to exist 

if ((m % (p2) != 0) or (NZvec is not None)): 

return 0 

 

## Use the reduction procedure to return the result 

return self.local_density_congruence(p, m / p2, None, None) / p**(self.dim() - 2) 

 

 

 

 

 

 

 

def local_badI_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the Bad-type I local density of Q representing `m` at `p`. 

(Assuming that p > 2 and Q is given in local diagonal form.) 

 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and `p`-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_badI_density_congruence(2, 1, None, None) 

0 

sage: Q.local_badI_density_congruence(2, 2, None, None) 

1 

sage: Q.local_badI_density_congruence(2, 4, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 1, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 6, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 9, None, None) 

0 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_badI_density_congruence(2, 1, None, None) 

0 

sage: Q.local_badI_density_congruence(2, 2, None, None) 

0 

sage: Q.local_badI_density_congruence(2, 4, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 2, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 6, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 9, None, None) 

0 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9]) 

sage: Q.local_badI_density_congruence(3, 1, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 3, None, None) 

4/3 

sage: Q.local_badI_density_congruence(3, 6, None, None) 

4/3 

sage: Q.local_badI_density_congruence(3, 9, None, None) 

0 

sage: Q.local_badI_density_congruence(3, 18, None, None) 

0 

 

 

""" 

## DIAGNOSTIC 

verbose(" In local_badI_density_congruence with ") 

verbose(" Q is: \n" + str(self)) 

verbose(" p = " + str(p)) 

verbose(" m = " + str(m)) 

verbose(" Zvec = " + str(Zvec)) 

verbose(" NZvec = " + str(NZvec)) 

 

## Put the Zvec congruence condition in a standard form 

if Zvec is None: 

Zvec = [] 

 

 

n = self.dim() 

 

 

 

## Sanity Check on Zvec and NZvec: 

## ------------------------------- 

Sn = Set(range(n)) 

if (Zvec is not None) and (len(Set(Zvec) + Sn) > n): 

raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.") 

if (NZvec is not None) and (len(Set(NZvec) + Sn) > n): 

raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.") 

 

 

 

## Define the indexing set S_0, and determine if S_1 is empty: 

## ----------------------------------------------------------- 

S0 = [] 

S1_empty_flag = True ## This is used to check if we should be computing BI solutions at all! 

## (We should really to this earlier, but S1 must be non-zero to proceed.) 

 

## Find the valuation of each variable (which will be the same over 2x2 blocks), 

## remembering those of valuation 0 and if an entry of valuation 1 exists. 

for i in range(n): 

 

## Compute the valuation of each index, allowing for off-diagonal terms 

if (self[i,i] == 0): 

if (i == 0): 

val = valuation(self[i,i+1], p) ## Look at the term to the right 

else: 

if (i == n-1): 

val = valuation(self[i-1,i], p) ## Look at the term above 

else: 

val = valuation(self[i,i+1] + self[i-1,i], p) ## Finds the valuation of the off-diagonal term since only one isn't zero 

else: 

val = valuation(self[i,i], p) 

 

if (val == 0): 

S0 += [i] 

elif (val == 1): 

S1_empty_flag = False ## Need to have a non-empty S1 set to proceed with Bad-type I reduction... 

 

 

 

 

 

## Check that S1 is non-empty and p|m to proceed, otherwise return no solutions. 

if S1_empty_flag or m % p != 0: 

return 0 

 

## Check some conditions for no bad-type I solutions to exist 

if (NZvec is not None) and (len(Set(S0).intersection(Set(NZvec))) != 0): 

return 0 

 

 

 

## Check that the form is primitive... WHY DO WE NEED TO DO THIS?!? 

if (S0 == []): 

print(" Using Q = " + str(self)) 

print(" and p = " + str(p)) 

raise RuntimeError("Oops! The form is not primitive!") 

 

 

 

## DIAGNOSTIC 

verbose(" m = " + str(m) + " p = " + str(p)) 

verbose(" S0 = " + str(S0)) 

verbose(" len(S0) = " + str(len(S0))) 

 

 

 

## Make the form Qnew for the reduction procedure: 

## ----------------------------------------------- 

Qnew = deepcopy(self) ## TO DO: DO THIS WITHOUT A copy(). =) 

for i in range(n): 

if i in S0: 

Qnew[i,i] = p * Qnew[i,i] 

if ((p == 2) and (i < n-1)): 

Qnew[i,i+1] = p * Qnew[i,i+1] 

else: 

Qnew[i,i] = Qnew[i,i] / p 

if ((p == 2) and (i < n-1)): 

Qnew[i,i+1] = Qnew[i,i+1] / p 

 

 

 

## DIAGNOSTIC 

verbose("\n\n Check of Bad-type I reduction: \n") 

verbose(" Q is " + str(self)) 

verbose(" Qnew is " + str(Qnew)) 

verbose(" p = " + str(p)) 

verbose(" m / p = " + str(m/p)) 

verbose(" NZvec " + str(NZvec)) 

 

 

 

## Do the reduction 

Zvec_geq_1 = list(Set([i for i in Zvec if i not in S0])) 

if NZvec is None: 

NZvec_geq_1 = NZvec 

else: 

NZvec_geq_1 = list(Set([i for i in NZvec if i not in S0])) 

 

return QQ(p**(1 - len(S0))) * Qnew.local_good_density_congruence(p, m / p, Zvec_geq_1, NZvec_geq_1) 

 

 

 

 

 

 

 

def local_badII_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the Bad-type II local density of Q representing `m` at `p`. 

(Assuming that `p` > 2 and Q is given in local diagonal form.) 

 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and p-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_badII_density_congruence(2, 1, None, None) 

0 

sage: Q.local_badII_density_congruence(2, 2, None, None) 

0 

sage: Q.local_badII_density_congruence(2, 4, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 1, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 6, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 9, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 27, None, None) 

0 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9,9]) 

sage: Q.local_badII_density_congruence(3, 1, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 3, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 6, None, None) 

0 

sage: Q.local_badII_density_congruence(3, 9, None, None) 

4/27 

sage: Q.local_badII_density_congruence(3, 18, None, None) 

4/9 

 

""" 

## DIAGNOSTIC 

verbose(" In local_badII_density_congruence with ") 

verbose(" Q is: \n" + str(self)) 

verbose(" p = " + str(p)) 

verbose(" m = " + str(m)) 

verbose(" Zvec = " + str(Zvec)) 

verbose(" NZvec = " + str(NZvec)) 

 

## Put the Zvec congruence condition in a standard form 

if Zvec is None: 

Zvec = [] 

 

 

n = self.dim() 

 

 

## Sanity Check on Zvec and NZvec: 

## ------------------------------- 

Sn = Set(range(n)) 

if (Zvec is not None) and (len(Set(Zvec) + Sn) > n): 

raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.") 

if (NZvec is not None) and (len(Set(NZvec) + Sn) > n): 

raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.") 

 

 

## Define the indexing sets S_i: 

## ----------------------------- 

S0 = [] 

S1 = [] 

S2plus = [] 

 

for i in range(n): 

 

## Compute the valuation of each index, allowing for off-diagonal terms 

if (self[i,i] == 0): 

if (i == 0): 

val = valuation(self[i,i+1], p) ## Look at the term to the right 

elif (i == n-1): 

val = valuation(self[i-1,i], p) ## Look at the term above 

else: 

val = valuation(self[i,i+1] + self[i-1,i], p) ## Finds the valuation of the off-diagonal term since only one isn't zero 

else: 

val = valuation(self[i,i], p) 

 

## Sort the indices into disjoint sets by their valuation 

if (val == 0): 

S0 += [i] 

elif (val == 1): 

S1 += [i] 

elif (val >= 2): 

S2plus += [i] 

 

 

 

## Check that S2 is non-empty and p^2 divides m to proceed, otherwise return no solutions. 

p2 = p * p 

if (S2plus == []) or (m % p2 != 0): 

return 0 

 

 

 

 

## Check some conditions for no bad-type II solutions to exist 

if (NZvec is not None) and (len(Set(S2plus).intersection(Set(NZvec))) == 0): 

return 0 

 

 

 

## Check that the form is primitive... WHY IS THIS NECESSARY? 

if (S0 == []): 

print(" Using Q = " + str(self)) 

print(" and p = " + str(p)) 

raise RuntimeError("Oops! The form is not primitive!") 

 

 

 

 

## DIAGNOSTIC 

verbose("\n Entering BII routine ") 

verbose(" S0 is " + str(S0)) 

verbose(" S1 is " + str(S1)) 

verbose(" S2plus is " + str(S2plus)) 

verbose(" m = " + str(m) + " p = " + str(p)) 

 

 

 

 

 

## Make the form Qnew for the reduction procedure: 

## ----------------------------------------------- 

Qnew = deepcopy(self) ## TO DO: DO THIS WITHOUT A copy(). =) 

for i in range(n): 

if i in S2plus: 

Qnew[i,i] = Qnew[i,i] / p2 

if (p == 2) and (i < n-1): 

Qnew[i,i+1] = Qnew[i,i+1] / p2 

 

## DIAGNOSTIC 

verbose("\n\n Check of Bad-type II reduction: \n") 

verbose(" Q is " + str(self)) 

verbose(" Qnew is " + str(Qnew)) 

 

 

 

## Perform the reduction formula 

Zvec_geq_2 = list(Set([i for i in Zvec if i in S2plus])) 

if NZvec is None: 

NZvec_geq_2 = NZvec 

else: 

NZvec_geq_2 = list(Set([i for i in NZvec if i in S2plus])) 

 

return QQ(p**(len(S2plus) + 2 - n)) \ 

* (Qnew.local_density_congruence(p, m / p2, Zvec_geq_2, NZvec_geq_2) \ 

- Qnew.local_density_congruence(p, m / p2, S2plus , NZvec_geq_2)) 

 

 

 

 

 

 

def local_bad_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the Bad-type local density of Q representing 

`m` at `p`, allowing certain congruence conditions mod `p`. 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and p-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_bad_density_congruence(2, 1, None, None) 

0 

sage: Q.local_bad_density_congruence(2, 2, None, None) 

1 

sage: Q.local_bad_density_congruence(2, 4, None, None) 

0 

sage: Q.local_bad_density_congruence(3, 1, None, None) 

0 

sage: Q.local_bad_density_congruence(3, 6, None, None) 

0 

sage: Q.local_bad_density_congruence(3, 9, None, None) 

0 

sage: Q.local_bad_density_congruence(3, 27, None, None) 

0 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9,9]) 

sage: Q.local_bad_density_congruence(3, 1, None, None) 

0 

sage: Q.local_bad_density_congruence(3, 3, None, None) 

4/3 

sage: Q.local_bad_density_congruence(3, 6, None, None) 

4/3 

sage: Q.local_bad_density_congruence(3, 9, None, None) 

4/27 

sage: Q.local_bad_density_congruence(3, 18, None, None) 

4/9 

sage: Q.local_bad_density_congruence(3, 27, None, None) 

8/27 

 

""" 

return self.local_badI_density_congruence(p, m, Zvec, NZvec) + self.local_badII_density_congruence(p, m, Zvec, NZvec) 

 

 

 

######################################################### 

## local_density and local_density_congruence routines ## 

######################################################### 

 

def local_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the local density of Q representing `m` at `p`, 

allowing certain congruence conditions mod `p`. 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and p-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_density_congruence(p=2, m=1, Zvec=None, NZvec=None) 

1 

sage: Q.local_density_congruence(p=3, m=1, Zvec=None, NZvec=None) 

8/9 

sage: Q.local_density_congruence(p=5, m=1, Zvec=None, NZvec=None) 

24/25 

sage: Q.local_density_congruence(p=7, m=1, Zvec=None, NZvec=None) 

48/49 

sage: Q.local_density_congruence(p=11, m=1, Zvec=None, NZvec=None) 

120/121 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_density_congruence(2, 1, None, None) 

1 

sage: Q.local_density_congruence(2, 2, None, None) 

1 

sage: Q.local_density_congruence(2, 4, None, None) 

3/2 

sage: Q.local_density_congruence(3, 1, None, None) 

2/3 

sage: Q.local_density_congruence(3, 6, None, None) 

4/3 

sage: Q.local_density_congruence(3, 9, None, None) 

14/9 

sage: Q.local_density_congruence(3, 27, None, None) 

2 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9,9]) 

sage: Q.local_density_congruence(3, 1, None, None) 

2 

sage: Q.local_density_congruence(3, 3, None, None) 

4/3 

sage: Q.local_density_congruence(3, 6, None, None) 

4/3 

sage: Q.local_density_congruence(3, 9, None, None) 

2/9 

sage: Q.local_density_congruence(3, 18, None, None) 

4/9 

 

""" 

return self.local_good_density_congruence(p, m, Zvec, NZvec) \ 

+ self.local_zero_density_congruence(p, m, Zvec, NZvec) \ 

+ self.local_bad_density_congruence(p, m, Zvec, NZvec) 

 

 

 

def local_primitive_density_congruence(self, p, m, Zvec=None, NZvec=None): 

""" 

Finds the primitive local density of Q representing 

`m` at `p`, allowing certain congruence conditions mod `p`. 

 

Note: The following routine is not used internally, but is included for consistency. 

 

INPUT: 

 

Q -- quadratic form assumed to be block diagonal and p-integral 

 

`p` -- a prime number 

 

`m` -- an integer 

 

Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None 

 

OUTPUT: 

 

a rational number 

 

EXAMPLES:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) 

sage: Q.local_primitive_density_congruence(p=2, m=1, Zvec=None, NZvec=None) 

1 

sage: Q.local_primitive_density_congruence(p=3, m=1, Zvec=None, NZvec=None) 

8/9 

sage: Q.local_primitive_density_congruence(p=5, m=1, Zvec=None, NZvec=None) 

24/25 

sage: Q.local_primitive_density_congruence(p=7, m=1, Zvec=None, NZvec=None) 

48/49 

sage: Q.local_primitive_density_congruence(p=11, m=1, Zvec=None, NZvec=None) 

120/121 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) 

sage: Q.local_primitive_density_congruence(2, 1, None, None) 

1 

sage: Q.local_primitive_density_congruence(2, 2, None, None) 

1 

sage: Q.local_primitive_density_congruence(2, 4, None, None) 

1 

sage: Q.local_primitive_density_congruence(3, 1, None, None) 

2/3 

sage: Q.local_primitive_density_congruence(3, 6, None, None) 

4/3 

sage: Q.local_primitive_density_congruence(3, 9, None, None) 

4/3 

sage: Q.local_primitive_density_congruence(3, 27, None, None) 

4/3 

 

:: 

 

sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9,9]) 

sage: Q.local_primitive_density_congruence(3, 1, None, None) 

2 

sage: Q.local_primitive_density_congruence(3, 3, None, None) 

4/3 

sage: Q.local_primitive_density_congruence(3, 6, None, None) 

4/3 

sage: Q.local_primitive_density_congruence(3, 9, None, None) 

4/27 

sage: Q.local_primitive_density_congruence(3, 18, None, None) 

4/9 

sage: Q.local_primitive_density_congruence(3, 27, None, None) 

8/27 

sage: Q.local_primitive_density_congruence(3, 81, None, None) 

8/27 

sage: Q.local_primitive_density_congruence(3, 243, None, None) 

8/27 

 

""" 

return self.local_good_density_congruence(p, m, Zvec, NZvec) \ 

+ self.local_bad_density_congruence(p, m, Zvec, NZvec)