277:
Restricting to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0, thus trivially establishing the Hasse principle for this class of forms. It is known that Heath-Brown's result is best
105:-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and
515:
282:
showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables. Davenport, Heath-Brown and Hooley all used the
274:. Since every cubic form over the p-adic numbers with at least ten variables represents 0, the local–global principle holds trivially for cubic forms over the rationals in at least 14 variables.
177:
150:
366:
580:
221:
554:
1092:
283:
278:
possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that do not represent zero. However,
695:
1056:
1042:
1005:
980:
433:
97:
Given a polynomial equation with rational coefficients, if it has a rational solution, then this also yields a real solution and a
1097:
1082:
1087:
531:
The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.
1032:
941:
295:
558:
193:
1027:
270:
showed that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of
252: = 0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial solution in which
537:
and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group
575:
40:
70:. A more formal version of the Hasse principle states that certain types of equations have a rational solution
52:
109:-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution?
370:
298:, which accounts completely for the failure of the Hasse principle for some classes of variety. However,
287:
1022:
839:
704:
659:
290:, the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the
158:
131:
36:
302:
has shown that the Brauer–Manin obstruction cannot explain all the failures of the Hasse principle.
240:
shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3
197:
299:
877:
855:
829:
798:
720:
675:
315:
279:
44:
330:
972:
224:
states that the local–global principle applies to the condition of being a relative norm for a
1001:
997:
991:
976:
873:
738:
647:
311:
267:
209:
996:. Cambridge Tracts in Mathematics. Vol. 144. Cambridge: Cambridge Univ. Press. pp.
938:
Algebraic Groups and
Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)
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225:
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117:
56:
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201:
113:
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64:
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421:
291:
213:
121:
48:
24:
397:
385:
217:
153:
75:
60:
20:
784:
754:
956:
671:
318:
show that the Hasse–Minkowski theorem is not extensible to forms of degree 10
936:
Kneser, Martin (1966), "Hasse principle for HÂą of simply connected groups",
898:
881:
716:
851:
834:
916:
Chernousov, V. I. (1989), "The Hasse principle for groups of type E8",
629:
600:
553:
The Hasse principle for algebraic groups was used in the proofs of the
771:(1937). "A remark on indeterminate equations in several variables".
196:
states that the local–global principle holds for the problem of
820:
Alexei N. Skorobogatov (1999). "Beyond the Manin obstruction".
357:) variables represents 0: the Hasse principle holds trivially.
882:"Some forms of odd degree for which the Hasse principle fails"
510:{\displaystyle H^{1}(k,G)\rightarrow \prod _{s}H^{1}(k_{s},G)}
369:
establishes a local–global principle for the splitting of a
693:
H. Davenport (1963). "Cubic forms in sixteen variables".
216:(as proved by Hasse), when one uses all the appropriate
101:-adic solution, as the rationals embed in the reals and
420:
is a simply-connected algebraic group defined over the
436:
161:
134:
16:
Solving integer equations from all modular solutions
520:is injective, where the product is over all places
51:. This is handled by examining the equation in the
509:
337:is any odd natural number, then there is a number
171:
144:
803:Journal fĂĽr die reine und angewandte Mathematik
743:Proceedings of the London Mathematical Society
8:
1053:Diophantine Equations: Progress and Problems
124:. For number fields, rather than reals and
773:Journal of the London Mathematical Society
547:
897:
833:
628:
492:
479:
469:
441:
435:
286:in their proofs. According to an idea of
163:
162:
160:
136:
135:
133:
741:(1983). "Cubic forms in ten variables".
581:Grothendieck–Katz p-curvature conjecture
212:'s result); and more generally over any
128:-adics, one uses complex embeddings and
650:(2007). "Cubic forms in 14 variables".
591:
534:
7:
642:
640:
555:Weil conjecture for Tamagawa numbers
408:Hasse principle for algebraic groups
222:Hasse's theorem on cyclic extensions
367:Albert–Brauer–Hasse–Noether theorem
361:Albert–Brauer–Hasse–Noether theorem
164:
137:
35:, is the idea that one can find an
696:Proceedings of the Royal Society A
550:many years after the other cases.
14:
801:(1988). "On nonary cubic forms".
376:over an algebraic number field
345:) such that any form of degree
172:{\displaystyle {\mathfrak {p}}}
145:{\displaystyle {\mathfrak {p}}}
37:integer solution to an equation
963:Survey of Diophantine geometry
886:Pacific Journal of Mathematics
504:
485:
462:
459:
447:
284:Hardy–Littlewood circle method
1:
942:American Mathematical Society
120:: integers, for instance, or
85:-adic numbers for each prime
990:Alexei Skorobogatov (2001).
559:strong approximation theorem
546:which was only completed by
74:they have a solution in the
43:to piece together solutions
1028:Encyclopedia of Mathematics
993:Torsors and rational points
396:then it is isomorphic to a
326:is a non-negative integer.
112:One can ask this for other
1114:
1093:Localization (mathematics)
1067:Mathematical Intelligencer
601:"The Diophantine equation
264:are all rational numbers.
672:10.1007/s00222-007-0062-1
47:powers of each different
41:Chinese remainder theorem
785:10.1112/jlms/s1-12.1.127
755:10.1112/plms/s3-47.2.225
599:Ernst S. Selmer (1951).
412:The Hasse principle for
296:Brauer–Manin obstruction
1098:Mathematical principles
1083:Algebraic number theory
1069:36 (4) (Dec 2014), 4–9.
899:10.2140/pjm.1976.67.161
194:Hasse–Minkowski theorem
717:10.1098/rspa.1963.0054
511:
371:central simple algebra
306:Forms of higher degree
220:necessary conditions.
173:
146:
29:local–global principle
1088:Diophantine equations
852:10.1007/s002220050291
576:Grunwald–Wang theorem
512:
380:. It states that if
174:
147:
944:, pp. 159–163,
940:, Providence, R.I.:
434:
236:A counterexample by
183:Forms representing 0
159:
132:
31:, also known as the
844:1999InMat.135..399S
709:1963RSPSA.272..285D
664:2007InMat.170..199H
329:On the other hand,
310:Counterexamples by
1045:2004-03-13 at the
1040:PlanetMath article
918:Soviet Math. Dokl.
630:10.1007/BF02395746
507:
474:
384:splits over every
228:of number fields.
169:
142:
1051:Swinnerton-Dyer,
1023:"Hasse principle"
739:D. R. Heath-Brown
703:(1350): 285–303.
548:Chernousov (1989)
465:
268:Roger Heath-Brown
1105:
1063:Global and local
1036:
1011:
986:
966:
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835:alg-geom/9711006
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648:D.R. Heath-Brown
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617:Acta Mathematica
596:
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226:cyclic extension
206:rational numbers
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57:rational numbers
1113:
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1047:Wayback Machine
1021:
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989:
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969:Springer-Verlag
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792:
767:
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762:
737:
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692:
691:
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646:
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613: = 0"
598:
597:
593:
589:
567:
544:
488:
475:
437:
432:
431:
416:states that if
410:
395:
363:
331:Birch's theorem
308:
238:Ernst S. Selmer
234:
202:quadratic forms
190:
188:Quadratic forms
185:
157:
156:
130:
129:
95:
33:Hasse principle
17:
12:
11:
5:
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1090:
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1016:External links
1014:
1013:
1012:
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987:
981:
953:
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911:
908:
906:
905:
892:(1): 161–169.
865:
828:(2): 399–424.
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779:(2): 127–129.
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749:(2): 225–257.
730:
685:
658:(1): 199–230.
636:
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571:Local analysis
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398:matrix algebra
391:
362:
359:
333:shows that if
307:
304:
294:; this is the
248: + 5
244: + 4
233:
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198:representing 0
189:
186:
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166:
139:
94:
91:
72:if and only if
15:
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9:
6:
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1061:J. Franklin,
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1007:0-521-80237-7
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982:3-540-61223-8
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535:Kneser (1966)
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498:
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450:
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429:
428:
427:then the map
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349:in more than
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122:number fields
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92:
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88:
84:
80:
77:
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69:
68:-adic numbers
67:
62:
58:
54:
50:
46:
42:
39:by using the
38:
34:
30:
26:
22:
1066:
1057:online notes
1052:
1026:
992:
962:
937:
921:
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822:Invent. Math
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652:Invent. Math
651:
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422:global field
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300:Skorobogatov
292:Brauer group
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257:
253:
249:
245:
241:
235:
214:number field
191:
154:prime ideals
152:-adics, for
125:
111:
106:
102:
98:
96:
86:
82:
78:
76:real numbers
65:
61:real numbers
49:prime number
32:
28:
25:Helmut Hasse
18:
971:. pp.
924:: 592–596,
874:M. Fujiwara
623:: 203–362.
322:+ 5, where
232:Cubic forms
218:local field
53:completions
21:mathematics
1077:Categories
957:Serge Lang
910:References
386:completion
208:(which is
1033:EMS Press
799:C. Hooley
725:122443854
467:∏
463:→
272:Davenport
210:Minkowski
204:over the
93:Intuition
1043:Archived
998:1–7, 112
959:(1997).
880:(1976).
860:14285244
809:: 32–98.
680:16600794
565:See also
557:and the
312:Fujiwara
63:and the
1035:, 2001
950:0220736
930:1014762
878:M. Sudo
840:Bibcode
705:Bibcode
660:Bibcode
81:in the
55:of the
1004:
979:
975:–258.
948:
928:
858:
723:
678:
280:Hooley
260:, and
118:fields
59:: the
45:modulo
856:S2CID
830:arXiv
721:S2CID
676:S2CID
587:Notes
400:over
288:Manin
114:rings
1002:ISBN
977:ISBN
365:The
316:Sudo
314:and
192:The
973:250
894:doi
848:doi
826:135
807:386
781:doi
751:doi
713:doi
701:272
668:doi
656:170
625:doi
524:of
200:by
116:or
79:and
27:'s
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1065:,
1055:,
1031:,
1025:,
1000:.
967:.
946:MR
926:MR
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920:,
890:67
888:.
884:.
876:;
854:.
846:.
838:.
824:.
805:.
777:12
775:.
747:47
745:.
719:.
711:.
699:.
674:.
666:.
654:.
639:^
621:85
619:.
615:.
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1010:.
985:.
902:.
896::
862:.
850::
842::
832::
787:.
783::
757:.
753::
727:.
715::
707::
682:.
670::
662::
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627::
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320:n
262:z
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