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Supersingular prime (algebraic number theory)

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showed that every elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero (if
328: 287: 1312: 544: 65: 1823: 997: 920: 343: 834: 537: 1813: 505: 467: 433: 87: 1818: 1171: 219:, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open. 1252: 456:
The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25–July 20, 1979
530: 493: 459: 1374: 1032: 148: 1399: 865: 1307: 48: 185: 365: 58: 52: 44: 940: 101: 17: 1457: 586: 69: 1794: 1384: 1037: 945: 1364: 373: 309: 268: 1359: 1017: 485: 1467: 1404: 1394: 1379: 1012: 870: 397: 791: 1436: 1411: 1389: 1369: 992: 964: 657: 501: 463: 429: 359:(1987). "The existence of infinitely many supersingular primes for every elliptic curve over 1346: 1336: 1331: 1268: 1115: 982: 885: 511: 473: 439: 381: 231: 393: 1047: 1007: 890: 855: 819: 774: 627: 615: 515: 497: 477: 443: 425: 413: 389: 331: 256: 121: 377: 1452: 1426: 1323: 1191: 1042: 1002: 987: 859: 750: 715: 670: 595: 577: 136: 109: 1807: 1462: 1227: 1091: 1064: 900: 765: 703: 694: 679: 642: 568: 401: 152: 1783: 1778: 1773: 1768: 1763: 1758: 1753: 1748: 1743: 1738: 1733: 1728: 1723: 1718: 1713: 1708: 1703: 1698: 1693: 1688: 1683: 1678: 1673: 1668: 1663: 1658: 1653: 1648: 1643: 1638: 1633: 1628: 1623: 1618: 1613: 1416: 1139: 1022: 905: 895: 880: 875: 839: 553: 295: 227: 113: 1608: 1603: 1598: 1593: 1588: 1583: 1578: 1573: 1568: 1563: 1558: 1553: 1548: 1543: 1538: 1533: 1528: 1523: 1518: 1513: 1508: 1354: 1027: 935: 930: 910: 824: 727: 603: 356: 167: 1431: 1247: 1155: 1075: 925: 829: 451: 409: 1472: 1421: 1302: 454:(1980). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey (eds.). 974: 522: 385: 178:
conjectured that the number of supersingular primes less than a bound
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Prime number with a certain relationship to an elliptic curve
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with a certain relationship to that curve. If the curve
458:. Proc. Symp. Pure Math. Vol. 37. Providence, RI: 312: 271: 188: 1481: 1445: 1345: 1322: 1296: 1063: 1056: 954: 848: 812: 561: 322: 281: 211: 1185: = 0, 1, 2, 3, ... 57:but its sources remain unclear because it lacks 538: 8: 175: 18:Supersingular prime (for an elliptic curve) 1060: 545: 531: 523: 212:{\displaystyle {\frac {\sqrt {X}}{\ln X}}} 314: 313: 311: 273: 272: 270: 189: 187: 88:Learn how and when to remove this message 174:does not have complex multiplication). 344:Supersingular prime (moonshine theory) 7: 315: 274: 1824:Unsolved problems in number theory 25: 490:The Arithmetic of Elliptic Curves 182:is within a constant multiple of 921:Supersingular (moonshine theory) 34: 323:{\displaystyle {\mathfrak {p}}} 282:{\displaystyle {\mathfrak {p}}} 916:Supersingular (elliptic curve) 1: 697:2 ± 2 ± 1 494:Graduate Texts in Mathematics 460:American Mathematical Society 418:Frobenius distributions in GL 149:supersingular elliptic curve 496:. Vol. 106. New York: 302:such that the reduction of 1840: 1792: 176:Lang & Trotter (1976) 1814:Classes of prime numbers 1303:Mega (1,000,000+ digits) 1172:Arithmetic progression ( 43:This article includes a 1819:Algebraic number theory 102:algebraic number theory 72:more precise citations. 1458:Industrial-grade prime 835:Newman–Shanks–Williams 324: 283: 213: 1795:List of prime numbers 1253:Sophie Germain/Safe ( 325: 284: 214: 977:(10 − 1)/9 486:Silverman, Joseph H. 462:. pp. 521–532. 310: 269: 265:supersingular prime 186: 120:is defined over the 1286: ± 7, ... 813:By integer sequence 598:(2 + 1)/3 378:1987InMat..89..561E 330:is a supersingular 222:More generally, if 106:supersingular prime 1468:Formula for primes 1101: + 2 or 1033:Smarandache–Wellin 386:10.1007/BF01388985 320: 279: 209: 130:supersingular for 45:list of references 1801: 1800: 1412:Carmichael number 1347:Composite numbers 1282: ± 3, 8 1278: ± 1, 4 1241: ± 1, … 1237: ± 1, 4 1233: ± 1, 2 1223: 1222: 768:3·2 − 1 673:2·3 + 1 587:Double Mersenne ( 207: 195: 98: 97: 90: 16:(Redirected from 1831: 1332:Eisenstein prime 1287: 1263: 1242: 1214: 1186: 1166: 1150: 1134: 1129: + 6, 1125: + 2, 1110: 1105: + 4, 1086: 1061: 978: 941:Highly cototient 803: 802: 796: 786: 769: 760: 745: 722: 721:·2 − 1 710: 709:·2 + 1 698: 689: 674: 665: 652: 637: 622: 610: 609:·2 + 1 599: 590: 581: 572: 547: 540: 533: 524: 519: 481: 447: 414:Trotter, Hale F. 405: 329: 327: 326: 321: 319: 318: 288: 286: 285: 280: 278: 277: 232:finite extension 218: 216: 215: 210: 208: 206: 191: 190: 122:rational numbers 93: 86: 82: 79: 73: 68:this article by 59:inline citations 38: 37: 30: 21: 1839: 1838: 1834: 1833: 1832: 1830: 1829: 1828: 1804: 1803: 1802: 1797: 1788: 1482:First 60 primes 1477: 1441: 1341: 1324:Complex numbers 1318: 1292: 1270: 1254: 1229: 1228:Bi-twin chain ( 1219: 1193: 1173: 1157: 1141: 1117: 1093: 1077: 1052: 1038:Strobogrammatic 976: 950: 844: 808: 800: 794: 793: 776: 767: 752: 729: 717: 705: 696: 681: 672: 659: 651:# + 1 649: 644: 636:# ± 1 634: 629: 621:! ± 1 617: 605: 597: 589:2 − 1 588: 580:2 − 1 579: 571:2 + 1 570: 557: 551: 508: 498:Springer-Verlag 484: 470: 450: 436: 426:Springer-Verlag 421: 408: 357:Elkies, Noam D. 355: 352: 340: 332:abelian variety 308: 307: 267: 266: 257:abelian variety 246: 196: 184: 183: 163: 124:, then a prime 94: 83: 77: 74: 63: 49:related reading 39: 35: 28: 23: 22: 15: 12: 11: 5: 1837: 1835: 1827: 1826: 1821: 1816: 1806: 1805: 1799: 1798: 1793: 1790: 1789: 1787: 1786: 1781: 1776: 1771: 1766: 1761: 1756: 1751: 1746: 1741: 1736: 1731: 1726: 1721: 1716: 1711: 1706: 1701: 1696: 1691: 1686: 1681: 1676: 1671: 1666: 1661: 1656: 1651: 1646: 1641: 1636: 1631: 1626: 1621: 1616: 1611: 1606: 1601: 1596: 1591: 1586: 1581: 1576: 1571: 1566: 1561: 1556: 1551: 1546: 1541: 1536: 1531: 1526: 1521: 1516: 1511: 1506: 1501: 1496: 1491: 1485: 1483: 1479: 1478: 1476: 1475: 1470: 1465: 1460: 1455: 1453:Probable prime 1449: 1447: 1446:Related topics 1443: 1442: 1440: 1439: 1434: 1429: 1427:Sphenic number 1424: 1419: 1414: 1409: 1408: 1407: 1402: 1397: 1392: 1387: 1382: 1377: 1372: 1367: 1362: 1351: 1349: 1343: 1342: 1340: 1339: 1337:Gaussian prime 1334: 1328: 1326: 1320: 1319: 1317: 1316: 1315: 1305: 1300: 1298: 1294: 1293: 1291: 1290: 1266: 1262: + 1 1250: 1245: 1224: 1221: 1220: 1218: 1217: 1189: 1169: 1165: + 6 1153: 1149: + 4 1137: 1133: + 8 1113: 1109: + 6 1089: 1085: + 2 1072: 1070: 1058: 1054: 1053: 1051: 1050: 1045: 1040: 1035: 1030: 1025: 1020: 1015: 1010: 1005: 1000: 995: 990: 985: 980: 972: 967: 961: 959: 952: 951: 949: 948: 943: 938: 933: 928: 923: 918: 913: 908: 903: 898: 893: 888: 883: 878: 873: 868: 863: 852: 850: 846: 845: 843: 842: 837: 832: 827: 822: 816: 814: 810: 809: 807: 806: 789: 785: − 1 772: 763: 748: 725: 713: 701: 692: 677: 668: 664: + 1 655: 647: 640: 632: 625: 613: 601: 593: 584: 575: 565: 563: 559: 558: 552: 550: 549: 542: 535: 527: 521: 520: 506: 482: 468: 448: 434: 419: 406: 372:(3): 561–567. 351: 348: 347: 346: 339: 336: 317: 276: 242: 205: 202: 199: 194: 159: 110:elliptic curve 96: 95: 53:external links 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 1836: 1825: 1822: 1820: 1817: 1815: 1812: 1811: 1809: 1796: 1791: 1785: 1782: 1780: 1777: 1775: 1772: 1770: 1767: 1765: 1762: 1760: 1757: 1755: 1752: 1750: 1747: 1745: 1742: 1740: 1737: 1735: 1732: 1730: 1727: 1725: 1722: 1720: 1717: 1715: 1712: 1710: 1707: 1705: 1702: 1700: 1697: 1695: 1692: 1690: 1687: 1685: 1682: 1680: 1677: 1675: 1672: 1670: 1667: 1665: 1662: 1660: 1657: 1655: 1652: 1650: 1647: 1645: 1642: 1640: 1637: 1635: 1632: 1630: 1627: 1625: 1622: 1620: 1617: 1615: 1612: 1610: 1607: 1605: 1602: 1600: 1597: 1595: 1592: 1590: 1587: 1585: 1582: 1580: 1577: 1575: 1572: 1570: 1567: 1565: 1562: 1560: 1557: 1555: 1552: 1550: 1547: 1545: 1542: 1540: 1537: 1535: 1532: 1530: 1527: 1525: 1522: 1520: 1517: 1515: 1512: 1510: 1507: 1505: 1502: 1500: 1497: 1495: 1492: 1490: 1487: 1486: 1484: 1480: 1474: 1471: 1469: 1466: 1464: 1463:Illegal prime 1461: 1459: 1456: 1454: 1451: 1450: 1448: 1444: 1438: 1435: 1433: 1430: 1428: 1425: 1423: 1420: 1418: 1415: 1413: 1410: 1406: 1403: 1401: 1398: 1396: 1393: 1391: 1388: 1386: 1383: 1381: 1378: 1376: 1373: 1371: 1368: 1366: 1363: 1361: 1358: 1357: 1356: 1353: 1352: 1350: 1348: 1344: 1338: 1335: 1333: 1330: 1329: 1327: 1325: 1321: 1314: 1311: 1310: 1309: 1308:Largest known 1306: 1304: 1301: 1299: 1295: 1289: 1285: 1281: 1277: 1273: 1267: 1265: 1261: 1257: 1251: 1249: 1246: 1244: 1240: 1236: 1232: 1226: 1225: 1216: 1213: 1210: +  1209: 1205: 1201: 1198: −  1197: 1190: 1188: 1184: 1180: 1177: +  1176: 1170: 1168: 1164: 1160: 1154: 1152: 1148: 1144: 1138: 1136: 1132: 1128: 1124: 1120: 1114: 1112: 1108: 1104: 1100: 1096: 1090: 1088: 1084: 1080: 1074: 1073: 1071: 1069: 1067: 1062: 1059: 1055: 1049: 1046: 1044: 1041: 1039: 1036: 1034: 1031: 1029: 1026: 1024: 1021: 1019: 1016: 1014: 1011: 1009: 1006: 1004: 1001: 999: 996: 994: 991: 989: 986: 984: 981: 979: 973: 971: 968: 966: 963: 962: 960: 957: 953: 947: 944: 942: 939: 937: 934: 932: 929: 927: 924: 922: 919: 917: 914: 912: 909: 907: 904: 902: 899: 897: 894: 892: 889: 887: 884: 882: 879: 877: 874: 872: 869: 867: 864: 861: 857: 854: 853: 851: 847: 841: 838: 836: 833: 831: 828: 826: 823: 821: 818: 817: 815: 811: 805: 799: 790: 788: 784: 780: 773: 771: 764: 762: 759: 756: +  755: 749: 747: 744: 741: −  740: 736: 733: −  732: 726: 724: 720: 714: 712: 708: 702: 700: 693: 691: 688: 685: +  684: 678: 676: 669: 667: 663: 658:Pythagorean ( 656: 654: 650: 641: 639: 635: 626: 624: 620: 614: 612: 608: 602: 600: 594: 592: 585: 583: 576: 574: 567: 566: 564: 560: 555: 548: 543: 541: 536: 534: 529: 528: 525: 517: 513: 509: 507:0-387-96203-4 503: 499: 495: 491: 487: 483: 479: 475: 471: 469:0-8218-1440-0 465: 461: 457: 453: 449: 445: 441: 437: 435:0-387-07550-X 431: 427: 423: 415: 411: 407: 403: 399: 395: 391: 387: 383: 379: 375: 371: 368: 367: 366:Invent. Math. 362: 358: 354: 353: 349: 345: 342: 341: 337: 335: 333: 305: 301: 297: 293: 292: 262: 259:defined over 258: 254: 250: 245: 241: 237: 233: 229: 225: 220: 203: 200: 197: 192: 181: 177: 173: 169: 165: 162: 158: 154: 153:residue field 150: 146: 142: 138: 134: 133: 127: 123: 119: 115: 111: 107: 103: 92: 89: 81: 71: 67: 61: 60: 54: 50: 46: 41: 32: 31: 19: 1417:Almost prime 1375:Euler–Jacobi 1283: 1279: 1275: 1271: 1269:Cunningham ( 1259: 1255: 1238: 1234: 1230: 1211: 1207: 1203: 1199: 1195: 1194:consecutive 1182: 1178: 1174: 1162: 1158: 1146: 1142: 1130: 1126: 1122: 1118: 1116:Quadruplet ( 1106: 1102: 1098: 1094: 1082: 1078: 1065: 1013:Full reptend 915: 871:Wolstenholme 866:Wall–Sun–Sun 797: 782: 778: 757: 753: 742: 738: 734: 730: 718: 706: 686: 682: 661: 645: 630: 618: 606: 554:Prime number 489: 455: 417: 369: 364: 360: 303: 299: 296:finite place 290: 264: 260: 252: 248: 243: 239: 235: 228:global field 223: 221: 179: 171: 166: 160: 156: 144: 143:modulo  140: 131: 129: 125: 117: 114:prime number 108:for a given 105: 99: 84: 75: 64:Please help 56: 1400:Somer–Lucas 1355:Pseudoprime 993:Truncatable 965:Palindromic 849:By property 628:Primorial ( 616:Factorial ( 422:-extensions 410:Lang, Serge 168:Noam Elkies 78:August 2019 70:introducing 1808:Categories 1437:Pernicious 1432:Interprime 1192:Balanced ( 983:Permutable 958:-dependent 775:Williams ( 671:Pierpont ( 596:Wagstaff 578:Mersenne ( 562:By formula 516:0585.14026 478:0448.10021 452:Ogg, A. P. 444:0329.12015 350:References 234:either of 1473:Prime gap 1422:Semiprime 1385:Frobenius 1092:Triplet ( 891:Ramanujan 886:Fortunate 856:Wieferich 820:Fibonacci 751:Leyland ( 716:Woodall ( 695:Solinas ( 680:Quartan ( 402:123646933 263:, then a 230:—i.e., a 201:⁡ 151:over the 137:reduction 1365:Elliptic 1140:Cousin ( 1057:Patterns 1048:Tetradic 1043:Dihedral 1008:Primeval 1003:Delicate 988:Circular 975:Repunit 766:Thabit ( 704:Cullen ( 643:Euclid ( 569:Fermat ( 488:(1986). 416:(1976). 338:See also 1360:Catalan 1297:By size 1068:-tuples 998:Minimal 901:Regular 792:Mills ( 728:Cuban ( 604:Proth ( 556:classes 394:0903384 374:Bibcode 306:modulo 226:is any 135:if the 66:improve 1405:Strong 1395:Perrin 1380:Fermat 1156:Sexy ( 1076:Twin ( 1018:Unique 946:Unique 906:Strong 896:Pillai 876:Wilson 840:Perrin 514:  504:  476:  466:  442:  432:  400:  392:  255:is an 251:)—and 238:or of 155:  1390:Lucas 1370:Euler 1023:Happy 970:Emirp 936:Higgs 931:Super 911:Stern 881:Lucky 825:Lucas 398:S2CID 294:is a 147:is a 112:is a 51:, or 1313:list 1248:Chen 1028:Self 956:Base 926:Good 860:pair 830:Pell 781:−1)· 502:ISBN 464:ISBN 430:ISBN 289:for 104:, a 1784:281 1779:277 1774:271 1769:269 1764:263 1759:257 1754:251 1749:241 1744:239 1739:233 1734:229 1729:227 1724:223 1719:211 1714:199 1709:197 1704:193 1699:191 1694:181 1689:179 1684:173 1679:167 1674:163 1669:157 1664:151 1659:149 1654:139 1649:137 1644:131 1639:127 1634:113 1629:109 1624:107 1619:103 1614:101 1274:, 2 1258:, 2 1179:a·n 737:)/( 512:Zbl 474:Zbl 440:Zbl 382:doi 363:". 298:of 139:of 128:is 100:In 1810:: 1609:97 1604:89 1599:83 1594:79 1589:73 1584:71 1579:67 1574:61 1569:59 1564:53 1559:47 1554:43 1549:41 1544:37 1539:31 1534:29 1529:23 1524:19 1519:17 1514:13 1509:11 1206:, 1202:, 1181:, 1161:, 1145:, 1121:, 1097:, 1081:, 510:. 500:. 492:. 472:. 438:. 428:. 412:; 396:. 390:MR 388:. 380:. 370:89 334:. 198:ln 164:. 55:, 47:, 1504:7 1499:5 1494:3 1489:2 1288:) 1284:p 1280:p 1276:p 1272:p 1264:) 1260:p 1256:p 1243:) 1239:n 1235:n 1231:n 1215:) 1212:n 1208:p 1204:p 1200:n 1196:p 1187:) 1183:n 1175:p 1167:) 1163:p 1159:p 1151:) 1147:p 1143:p 1135:) 1131:p 1127:p 1123:p 1119:p 1111:) 1107:p 1103:p 1099:p 1095:p 1087:) 1083:p 1079:p 1066:k 862:) 858:( 804:) 801:⌋ 798:A 795:⌊ 787:) 783:b 779:b 777:( 770:) 761:) 758:y 754:x 746:) 743:y 739:x 735:y 731:x 723:) 719:n 711:) 707:n 699:) 690:) 687:y 683:x 675:) 666:) 662:n 660:4 653:) 648:n 646:p 638:) 633:n 631:p 623:) 619:n 611:) 607:k 591:) 582:) 573:) 546:e 539:t 532:v 518:. 480:. 446:. 420:2 404:. 384:: 376:: 361:Q 316:p 304:A 300:K 291:A 275:p 261:K 253:A 249:t 247:( 244:p 240:F 236:Q 224:K 204:X 193:X 180:X 172:E 161:p 157:F 145:p 141:E 132:E 126:p 118:E 91:) 85:( 80:) 76:( 62:. 20:)

Index

Supersingular prime (for an elliptic curve)
list of references
related reading
external links
inline citations
improve
introducing
Learn how and when to remove this message
algebraic number theory
elliptic curve
prime number
rational numbers
reduction
supersingular elliptic curve
residue field
Noam Elkies
Lang & Trotter (1976)
global field
finite extension
abelian variety
finite place
abelian variety
Supersingular prime (moonshine theory)
Elkies, Noam D.
Invent. Math.
Bibcode
1987InMat..89..561E
doi
10.1007/BF01388985
MR

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