376:
757:
113:
641:
285:
240:
159:
811:
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is the Hasse–Weil bound. This provides a bound on the number of points on a curve over a finite field. If the number of points on the curve
585:
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The Hasse–Weil bound reduces to the usual Hasse bound when applied to elliptic curves, which have genus
167:
in his thesis. It was proven by Hasse in 1933, with the proof published in a series of papers in 1936.
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137:
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958:
847:
433:
1387:
1259:
918:
783:
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704:
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616:
500:
438:
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563:
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25:, also referred to as the Hasse bound, provides an estimate of the number of points on an
8:
1224:
1102:
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1024:
1004:
712:
674:
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461:(1924), "Quadratische Körper im Gebiete der höheren Kongruenzen. II. Analytischer Teil",
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175:
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524:(1936), "Zur Theorie der abstrakten elliptischen Funktionenkörper. I, II & III",
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842:
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663:
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612:
496:
199:
127:
1042:
867:
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829:
382:
171:
131:
26:
539:
1411:
1374:
1155:
1135:
1062:
857:
789:
608:
547:
484:
576:
416:
1321:
1295:
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1275:
1077:
897:
521:
30:
20:
1196:
1034:
1191:
476:
458:
164:
16:
Estimates the number of points on an elliptic curve over a finite field
1052:
765:
1364:
1349:
371:{\displaystyle |\#C(\mathbb {F} _{q})-(q+1)|\leq 2g{\sqrt {q}}.}
1344:
645:
130:
over the same field, by an 'error term' that is the sum of two
381:
This result is again equivalent to the determination of the
638:
Many
Rational Points. Coding Theory and Algebraic Geometry
170:
Hasse's theorem is equivalent to the determination of the
182:. In this form it can be seen to be the analogue of the
419:
in 1949 and proved by André Weil in the case of curves.
754:
296:
252:
219:
140:
57:
683:
Algebraic
Geometry in Coding Theory and Cryptography
581:"Numbers of solutions of equations in finite fields"
370:
279:
234:
153:
107:
1409:
673:
163:This result had originally been conjectured by
819:
198:A generalization of the Hasse bound to higher
40:is the number of points on the elliptic curve
805:
586:Bulletin of the American Mathematical Society
411:The Hasse–Weil bound is a consequence of the
108:{\displaystyle |N-(q+1)|\leq 2{\sqrt {q}}.}
33:, bounding the value both above and below.
812:
798:
748:
48:elements, then Hasse's result states that
758:Discrete Mathematics and its Applications
711:
598:
313:
264:
222:
1410:
1235:Clifford's theorem on special divisors
793:
520:
457:
280:{\displaystyle \#C(\mathbb {F} _{q})}
632:
575:
190:associated with the elliptic curve.
1428:Theorems in algebraic number theory
193:
13:
1393:Vector bundles on algebraic curves
1327:Weber's theorem (Algebraic curves)
924:Hasse's theorem on elliptic curves
914:Counting points on elliptic curves
302:
253:
14:
1439:
717:The arithmetic of elliptic curves
126:+ 1, the number of points of the
642:Mathematics and its Applications
235:{\displaystyle \mathbb {F} _{q}}
1015:Hurwitz's automorphisms theorem
600:10.1090/S0002-9904-1949-09219-4
1240:Gonality of an algebraic curve
1151:Differential of the first kind
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308:
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259:
85:
81:
69:
59:
1:
1383:Birkhoff–Grothendieck theorem
1093:Nagata's conjecture on curves
964:Schoof–Elkies–Atkin algorithm
838:Five points determine a conic
721:Graduate Texts in Mathematics
626:
393:, and is the analogue of the
23:'s theorem on elliptic curves
954:Supersingular elliptic curve
644:, vol. 564, Dordrecht:
154:{\displaystyle {\sqrt {q}}.}
7:
1161:Riemann's existence theorem
1088:Hilbert's sixteenth problem
980:Elliptic curve cryptography
893:Fundamental pair of periods
723:, vol. 106, New York:
422:
401:associated with the curve.
10:
1444:
1291:Moduli of algebraic curves
687:Princeton University Press
44:over a finite field with
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1335:
1304:
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997:
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828:
540:10.1515/crll.1936.175.193
464:Mathematische Zeitschrift
415:, originally proposed by
134:, each of absolute value
1058:Cayley–Bacharach theorem
985:Elliptic curve primality
444:
1317:Riemann–Hurwitz formula
1281:Gromov–Witten invariant
1141:Compact Riemann surface
929:Mazur's torsion theorem
750:Washington, Lawrence C.
934:Modular elliptic curve
372:
281:
236:
213:over the finite field
155:
109:
848:Rational normal curve
373:
282:
237:
156:
110:
1388:Stable vector bundle
1260:Weil reciprocity law
1250:Riemann–Roch theorem
1230:Brill–Noether theory
1166:Riemann–Roch theorem
1083:Genus–degree formula
944:Mordell–Weil theorem
919:Division polynomials
713:Silverman, Joseph H.
675:Niederreiter, Harald
429:Sato–Tate conjecture
385:of the roots of the
294:
250:
217:
174:of the roots of the
138:
55:
1211:Structure of curves
1103:Quartic plane curve
1025:Hyperelliptic curve
1005:De Franchis theorem
949:Nagell–Lutz theorem
387:local zeta-function
176:local zeta-function
118:The reason is that
1218:Divisors on curves
1010:Faltings's theorem
959:Schoof's algorithm
939:Modularity theorem
762:Chapman & Hall
477:10.1007/BF01181075
434:Schoof's algorithm
395:Riemann hypothesis
368:
277:
232:
184:Riemann hypothesis
151:
105:
1405:
1404:
1401:
1400:
1312:Hasse–Witt matrix
1255:Weierstrass point
1202:Smooth completion
1171:TeichmĂĽller space
1073:Cubic plane curve
993:
992:
907:Arithmetic theory
888:Elliptic integral
883:Elliptic function
775:978-1-4200-7146-7
734:978-0-387-96203-0
696:978-0-6911-0288-7
363:
146:
100:
1435:
1245:Jacobian variety
1215:
1214:
1118:Riemann surfaces
1108:Real plane curve
1068:Cramer's paradox
1048:BĂ©zout's theorem
873:
872:
822:algebraic curves
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620:
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527:Crelle's Journal
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413:Weil conjectures
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203:algebraic curves
194:Hasse–Weil Bound
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1418:Elliptic curves
1408:
1407:
1406:
1397:
1369:
1360:Delta invariant
1331:
1300:
1264:
1225:Abel–Jacobi map
1206:
1180:
1176:Torelli theorem
1146:Dessin d'enfant
1126:Belyi's theorem
1112:
1098:PlĂĽcker formula
1029:
1020:Hurwitz surface
989:
968:
902:
876:Analytic theory
868:Elliptic curves
862:
843:Projective line
830:Rational curves
824:
818:
776:
735:
725:Springer-Verlag
697:
660:
650:Springer-Verlag
634:Hurt, Norman E.
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132:complex numbers
128:projective line
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17:
12:
11:
5:
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1375:Vector bundles
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1114:
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853:Riemann sphere
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809:
802:
794:
788:
787:
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760:, Boca Raton:
746:
733:
708:
695:
679:Xing, Chaoping
671:
658:
628:
625:
622:
621:
593:(5): 497–508,
568:
513:
471:(1): 207–246,
449:
448:
446:
443:
442:
441:
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431:
424:
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399:function field
383:absolute value
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188:function field
172:absolute value
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27:elliptic curve
15:
9:
6:
4:
3:
2:
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1424:
1423:Finite fields
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1337:Singularities
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1185:Constructions
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1177:
1174:
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1167:
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1159:
1157:
1156:Klein quartic
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1136:Bolza surface
1134:
1132:
1131:Bring's curve
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1123:
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1064:
1063:Conic section
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1043:AF+BG theorem
1041:
1040:
1038:
1036:
1032:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1002:
1000:
996:
986:
983:
981:
978:
977:
975:
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962:
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950:
947:
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909:
905:
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894:
891:
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871:
869:
865:
859:
858:Twisted cubic
856:
854:
851:
849:
846:
844:
841:
839:
836:
835:
833:
831:
827:
823:
815:
810:
808:
803:
801:
796:
795:
792:
785:
781:
777:
771:
767:
763:
759:
755:
751:
747:
744:
740:
736:
730:
726:
722:
718:
714:
710:Chapter V of
709:
706:
702:
698:
692:
688:
685:, Princeton:
684:
680:
676:
672:
669:
665:
661:
659:1-4020-1766-9
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651:
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639:
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528:
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522:Hasse, Helmut
517:
510:
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502:
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460:
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392:
388:
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365:
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335:
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318:
305:
290:
289:
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245:
227:
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208:
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201:
191:
189:
185:
181:
177:
173:
168:
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122:differs from
121:
102:
97:
92:
89:
78:
75:
72:
66:
63:
51:
50:
49:
47:
43:
39:
34:
32:
28:
24:
22:
1322:Prym variety
1296:Stable curve
1286:Hodge bundle
1276:ELSV formula
1078:Fermat curve
1035:Plane curves
998:Higher genus
973:Applications
923:
898:Modular form
753:
716:
682:
637:
590:
584:
571:
531:
525:
516:
468:
462:
453:
439:Weil's bound
410:
405:
403:
390:
380:
243:
210:
206:
197:
179:
169:
162:
123:
119:
117:
45:
41:
37:
35:
31:finite field
19:
18:
1197:Polar curve
577:Weil, André
459:Artin, Emil
1412:Categories
1192:Dual curve
820:Topics in
627:References
564:0014.14903
493:51.0144.05
417:André Weil
165:Emil Artin
1305:Morphisms
1053:Bitangent
766:CRC Press
609:0002-9904
556:118733025
548:0075-4102
509:117936362
485:0025-5874
350:≤
327:−
303:#
254:#
242:of order
209:of genus
90:≤
67:−
752:(2008),
715:(1994),
681:(2009),
636:(2003),
579:(1949),
423:See also
397:for the
186:for the
1365:Tacnode
1350:Crunode
784:2404461
743:1329092
705:2573098
668:2042828
617:0029393
534:(175),
501:1544652
287:, then
29:over a
1345:Acnode
1269:Moduli
782:
772:
741:
731:
703:
693:
666:
656:
646:Kluwer
615:
607:
562:
554:
546:
507:
499:
491:
483:
552:S2CID
505:S2CID
445:Notes
200:genus
21:Hasse
1355:Cusp
770:ISBN
729:ISBN
691:ISBN
654:ISBN
605:ISSN
544:ISSN
532:1936
481:ISSN
595:doi
560:Zbl
536:doi
489:JFM
473:doi
406:g=1
389:of
246:is
178:of
36:If
1414::
780:MR
778:,
768:,
756:,
739:MR
737:,
727:,
719:,
701:MR
699:,
689:,
677:;
664:MR
662:,
652:,
640:,
613:MR
611:,
603:,
591:55
589:,
583:,
558:,
550:,
542:,
530:,
503:,
497:MR
495:,
487:,
479:,
469:19
467:,
408:.
813:e
806:t
799:v
764:/
648:/
597::
538::
475::
391:C
366:.
361:q
356:g
353:2
346:|
342:)
339:1
336:+
333:q
330:(
324:)
319:q
314:F
309:(
306:C
299:|
275:)
270:q
265:F
260:(
257:C
244:q
228:q
223:F
211:g
207:C
180:E
149:.
144:q
124:q
120:N
103:.
98:q
93:2
86:|
82:)
79:1
76:+
73:q
70:(
64:N
60:|
46:q
42:E
38:N
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.