1358:
This is slightly oversimplified. The Tate curve is really a curve over a formal power series ring rather than a curve over C. Intuitively, it is a family of curves depending on a formal parameter. When that formal parameter is zero it degenerates to a pinched torus, and when it is nonzero it is a
1255:
But the annulus does not correspond to the circle minus a point: the annulus is the set of complex numbers between two consecutive powers of q; say all complex numbers with magnitude between 1 and q. That gives us two circles, i.e., the inner and outer edges of an annulus.
817:
490:
329:
642:
1142:
942:
1353:
1055:
91:) in a 1959 manuscript originally titled "Rational Points on Elliptic Curves Over Complete Fields"; he did not publish his results until many years later, and his work first appeared in
335:
186:
1314:
896:
636:
1250:
1176:
1084:
971:
1007:
58:
1282:
1221:
1199:
197:
1675:
1550:
1503:
1466:
1631:
1586:
812:{\displaystyle y(w)=\sum _{m\in \mathbf {Z} }{\frac {(q^{m}w)^{2}}{(1-q^{m}w)^{3}}}+\sum _{m\geq 1}{\frac {q^{m}}{(1-q^{m})^{2}}}}
1395:
of multiplicative type. Conversely, every semistable elliptic curve over a local field is isomorphic to a Tate curve (up to
107:] of formal power series with integer coefficients given (in an affine open subset of the projective plane) by the equation
1619:
1259:
The image of the torus given here is a bunch of inlaid circles getting narrower and narrower as they approach the origin.
1089:
901:
21:
485:{\displaystyle -a_{6}=\sum _{n}{\frac {7n^{5}+5n^{3}}{12}}\times {\frac {q^{n}}{1-q^{n}}}=q+23q^{2}+154q^{3}+\cdots }
1392:
1319:
1012:
1649:
113:
84:
1701:
1287:
857:
1661:
569:
1226:
1147:
1060:
947:
976:
1666:
1611:
26:
1265:
1204:
1181:
To see why the Tate curve morally corresponds to a torus when the field is C with the usual norm,
1581:, Hamburger Mathematische Einzelschriften (N.F.), Heft 1, Göttingen: Vandenhoeck & Ruprecht,
1671:
1627:
1582:
1546:
1509:
1499:
1462:
1637:
1600:
1564:
1538:
1517:
1480:
1454:
1685:
1596:
1560:
1476:
1252:, which is a torus. In other words, we have an annulus, and we glue inner and outer edges.
1681:
1641:
1623:
1604:
1592:
1568:
1556:
1534:
1521:
1484:
1472:
1450:
1396:
324:{\displaystyle -a_{4}=5\sum _{n}{\frac {n^{3}q^{n}}{1-q^{n}}}=5q+45q^{2}+140q^{3}+\cdots }
1262:
This is slightly different from the usual method beginning with a flat sheet of paper,
1184:
73:
65:
1695:
1660:, Series in Number Theory, vol. I, Int. Press, Cambridge, MA, pp. 162–184,
1491:
1201:
is already singly periodic; modding out by q's integral powers you are modding out
77:
1576:
1449:, Graduate Texts in Mathematics, vol. 112 (2nd ed.), Berlin, New York:
1384:
1368:
1653:
1542:
1458:
1442:
519:|<1. Then the series above all converge, and define an elliptic curve over
1658:
Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993)
1513:
1498:. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.).
1316:, and then gluing together the edges of the cylinder to make a torus,
1533:, Lecture Notes in Mathematics, vol. 326, Berlin, New York:
973:
is the discrete subgroup generated by one multiplicative period
826:
to the point at infinity of the elliptic curve. The series
103:
The Tate curve is the projective plane curve over the ring
507:
is complete with respect to some absolute value | |, and
60:
with integer coefficients. Over the open subscheme where
527:
is non-zero then there is an isomorphism of groups from
1578:
Analytic theory of elliptic functions over local fields
1322:
1290:
1268:
1229:
1207:
1187:
1150:
1092:
1063:
1015:
979:
950:
904:
860:
645:
572:
338:
200:
116:
29:
1616:
Advanced Topics in the
Arithmetic of Elliptic Curves
1284:, and gluing together the sides to make a cylinder
1137:{\displaystyle {(\mathbb {C} ,+)}/(\mathbb {Z} ,+)}
1347:
1308:
1276:
1244:
1215:
1193:
1170:
1136:
1078:
1049:
1001:
965:
936:
890:
854:In the case of the curve over the complete field,
811:
630:
484:
323:
180:
52:
937:{\displaystyle \mathbb {C} ^{*}/q^{\mathbb {Z} }}
1654:"A review of non-Archimedean elliptic functions"
1371:of the Tate curve is given by a power series in
76:of norm less than 1, in which case the formal
1348:{\displaystyle \mathbb {C} /\mathbb {Z} ^{2}}
1050:{\displaystyle \tau =\omega _{1}/\omega _{2}}
8:
495:are power series with integer coefficients.
1656:, in Coates, John; Yau, Shing-Tung (eds.),
181:{\displaystyle y^{2}+xy=x^{3}+a_{4}x+a_{6}}
1665:
1339:
1335:
1334:
1328:
1324:
1323:
1321:
1309:{\displaystyle \mathbb {C} /\mathbb {Z} }
1302:
1301:
1296:
1292:
1291:
1289:
1270:
1269:
1267:
1236:
1232:
1231:
1228:
1209:
1208:
1206:
1186:
1155:
1154:
1149:
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1120:
1112:
1098:
1097:
1093:
1091:
1070:
1066:
1065:
1062:
1041:
1032:
1026:
1014:
984:
978:
957:
956:
955:
949:
928:
927:
926:
917:
911:
907:
906:
903:
882:
881:
880:
871:
865:
859:
800:
790:
770:
764:
752:
736:
723:
702:
689:
679:
672:
665:
644:
616:
571:
470:
454:
429:
412:
406:
391:
375:
365:
359:
346:
337:
309:
293:
265:
247:
237:
230:
224:
208:
199:
172:
156:
143:
121:
115:
68:. The Tate curve can also be defined for
31:
30:
28:
92:
1408:
1391:is non-integral and the Tate curve has
1178:is the complex numbers under addition.
891:{\displaystyle k^{*}/q^{\mathbb {Z} }}
7:
1496:Introduction to Modern Number Theory
1415:Manin & Panchishkin (2007) p.220
631:{\displaystyle x(w)=-y(w)-y(w^{-1})}
499:The Tate curve over a complete field
88:
64:is invertible, the Tate curve is an
1433:Manin & Panchiskin (2007) p.300
898:, the easiest case to visualize is
511:is a non-zero element of the field
14:
842:) are not formal power series in
83:The Tate curve was introduced by
1245:{\displaystyle \mathbb {Z} ^{2}}
1171:{\displaystyle (\mathbb {C} ,+)}
1079:{\displaystyle \mathbb {C} ^{*}}
966:{\displaystyle q^{\mathbb {Z} }}
673:
1002:{\displaystyle e^{2\pi i\tau }}
535:to this elliptic curve, taking
1165:
1151:
1131:
1117:
1108:
1094:
797:
777:
733:
710:
699:
682:
655:
649:
625:
609:
600:
594:
582:
576:
47:
44:
38:
35:
1:
1620:Graduate Texts in Mathematics
1494:; Panchishkin, A. A. (2007).
53:{\displaystyle \mathbb {Z} ]}
1277:{\displaystyle \mathbb {C} }
1216:{\displaystyle \mathbb {C} }
20:is a curve defined over the
22:ring of formal power series
1718:
1543:10.1007/978-3-540-46916-2
1459:10.1007/978-1-4612-4752-4
1575:Roquette, Peter (1970),
503:Suppose that the field
1529:Robert, Alain (1973),
1424:Silverman (1994) p.423
1349:
1310:
1278:
1246:
1217:
1195:
1172:
1138:
1080:
1051:
1003:
967:
938:
892:
813:
632:
486:
325:
182:
54:
1350:
1311:
1279:
1247:
1218:
1196:
1173:
1139:
1081:
1052:
1004:
968:
939:
893:
822:and taking powers of
814:
633:
487:
326:
183:
55:
1612:Silverman, Joseph H.
1393:semistable reduction
1320:
1288:
1266:
1227:
1205:
1185:
1148:
1090:
1061:
1013:
977:
948:
902:
858:
643:
570:
336:
198:
114:
27:
16:In mathematics, the
1009:, where the period
72:as an element of a
1447:Elliptic functions
1375:with leading term
1345:
1306:
1274:
1242:
1213:
1191:
1168:
1134:
1076:
1047:
999:
963:
934:
888:
809:
763:
678:
628:
482:
364:
321:
229:
178:
50:
1677:978-1-57146-026-4
1622:. Vol. 151.
1552:978-3-540-06309-4
1505:978-3-540-20364-3
1468:978-0-387-96508-6
1194:{\displaystyle q}
1086:is isomorphic to
850:Intuitive example
807:
748:
743:
661:
523:. If in addition
436:
401:
355:
272:
220:
1709:
1688:
1669:
1645:
1607:
1571:
1525:
1487:
1434:
1431:
1425:
1422:
1416:
1413:
1354:
1352:
1351:
1346:
1344:
1343:
1338:
1332:
1327:
1315:
1313:
1312:
1307:
1305:
1300:
1295:
1283:
1281:
1280:
1275:
1273:
1251:
1249:
1248:
1243:
1241:
1240:
1235:
1222:
1220:
1219:
1214:
1212:
1200:
1198:
1197:
1192:
1177:
1175:
1174:
1169:
1158:
1143:
1141:
1140:
1135:
1124:
1116:
1111:
1101:
1085:
1083:
1082:
1077:
1075:
1074:
1069:
1056:
1054:
1053:
1048:
1046:
1045:
1036:
1031:
1030:
1008:
1006:
1005:
1000:
998:
997:
972:
970:
969:
964:
962:
961:
960:
943:
941:
940:
935:
933:
932:
931:
921:
916:
915:
910:
897:
895:
894:
889:
887:
886:
885:
875:
870:
869:
818:
816:
815:
810:
808:
806:
805:
804:
795:
794:
775:
774:
765:
762:
744:
742:
741:
740:
728:
727:
708:
707:
706:
694:
693:
680:
677:
676:
637:
635:
634:
629:
624:
623:
491:
489:
488:
483:
475:
474:
459:
458:
437:
435:
434:
433:
417:
416:
407:
402:
397:
396:
395:
380:
379:
366:
363:
351:
350:
330:
328:
327:
322:
314:
313:
298:
297:
273:
271:
270:
269:
253:
252:
251:
242:
241:
231:
228:
213:
212:
187:
185:
184:
179:
177:
176:
161:
160:
148:
147:
126:
125:
59:
57:
56:
51:
34:
1717:
1716:
1712:
1711:
1710:
1708:
1707:
1706:
1702:Elliptic curves
1692:
1691:
1678:
1667:10.1.1.367.7205
1648:
1634:
1624:Springer-Verlag
1610:
1589:
1574:
1553:
1535:Springer-Verlag
1531:Elliptic curves
1528:
1506:
1490:
1469:
1451:Springer-Verlag
1441:
1438:
1437:
1432:
1428:
1423:
1419:
1414:
1410:
1405:
1397:quadratic twist
1365:
1333:
1318:
1317:
1286:
1285:
1264:
1263:
1230:
1225:
1224:
1203:
1202:
1183:
1182:
1146:
1145:
1088:
1087:
1064:
1059:
1058:
1037:
1022:
1011:
1010:
980:
975:
974:
951:
946:
945:
922:
905:
900:
899:
876:
861:
856:
855:
852:
796:
786:
776:
766:
732:
719:
709:
698:
685:
681:
641:
640:
612:
568:
567:
559:not a power of
501:
466:
450:
425:
418:
408:
387:
371:
367:
342:
334:
333:
305:
289:
261:
254:
243:
233:
232:
204:
196:
195:
168:
152:
139:
117:
112:
111:
101:
93:Roquette (1970)
25:
24:
12:
11:
5:
1715:
1713:
1705:
1704:
1694:
1693:
1690:
1689:
1676:
1646:
1632:
1608:
1587:
1572:
1551:
1526:
1504:
1488:
1467:
1436:
1435:
1426:
1417:
1407:
1406:
1404:
1401:
1364:
1361:
1342:
1337:
1331:
1326:
1304:
1299:
1294:
1272:
1239:
1234:
1211:
1190:
1167:
1164:
1161:
1157:
1153:
1133:
1130:
1127:
1123:
1119:
1115:
1110:
1107:
1104:
1100:
1096:
1073:
1068:
1044:
1040:
1035:
1029:
1025:
1021:
1018:
996:
993:
990:
987:
983:
959:
954:
930:
925:
920:
914:
909:
884:
879:
874:
868:
864:
851:
848:
820:
819:
803:
799:
793:
789:
785:
782:
779:
773:
769:
761:
758:
755:
751:
747:
739:
735:
731:
726:
722:
718:
715:
712:
705:
701:
697:
692:
688:
684:
675:
671:
668:
664:
660:
657:
654:
651:
648:
638:
627:
622:
619:
615:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
500:
497:
493:
492:
481:
478:
473:
469:
465:
462:
457:
453:
449:
446:
443:
440:
432:
428:
424:
421:
415:
411:
405:
400:
394:
390:
386:
383:
378:
374:
370:
362:
358:
354:
349:
345:
341:
331:
320:
317:
312:
308:
304:
301:
296:
292:
288:
285:
282:
279:
276:
268:
264:
260:
257:
250:
246:
240:
236:
227:
223:
219:
216:
211:
207:
203:
189:
188:
175:
171:
167:
164:
159:
155:
151:
146:
142:
138:
135:
132:
129:
124:
120:
100:
97:
74:complete field
66:elliptic curve
49:
46:
43:
40:
37:
33:
13:
10:
9:
6:
4:
3:
2:
1714:
1703:
1700:
1699:
1697:
1687:
1683:
1679:
1673:
1668:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1633:0-387-94328-5
1629:
1625:
1621:
1617:
1613:
1609:
1606:
1602:
1598:
1594:
1590:
1588:9783525403013
1584:
1580:
1579:
1573:
1570:
1566:
1562:
1558:
1554:
1548:
1544:
1540:
1536:
1532:
1527:
1523:
1519:
1515:
1511:
1507:
1501:
1497:
1493:
1492:Manin, Yu. I.
1489:
1486:
1482:
1478:
1474:
1470:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1439:
1430:
1427:
1421:
1418:
1412:
1409:
1402:
1400:
1398:
1394:
1390:
1387:, therefore,
1386:
1382:
1378:
1374:
1370:
1362:
1360:
1356:
1340:
1329:
1297:
1260:
1257:
1253:
1237:
1188:
1179:
1162:
1159:
1128:
1125:
1113:
1105:
1102:
1071:
1042:
1038:
1033:
1027:
1023:
1019:
1016:
994:
991:
988:
985:
981:
952:
923:
918:
912:
877:
872:
866:
862:
849:
847:
845:
841:
837:
833:
829:
825:
801:
791:
787:
783:
780:
771:
767:
759:
756:
753:
749:
745:
737:
729:
724:
720:
716:
713:
703:
695:
690:
686:
669:
666:
662:
658:
652:
646:
639:
620:
617:
613:
606:
603:
597:
591:
588:
585:
579:
573:
566:
565:
564:
562:
558:
554:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
498:
496:
479:
476:
471:
467:
463:
460:
455:
451:
447:
444:
441:
438:
430:
426:
422:
419:
413:
409:
403:
398:
392:
388:
384:
381:
376:
372:
368:
360:
356:
352:
347:
343:
339:
332:
318:
315:
310:
306:
302:
299:
294:
290:
286:
283:
280:
277:
274:
266:
262:
258:
255:
248:
244:
238:
234:
225:
221:
217:
214:
209:
205:
201:
194:
193:
192:
173:
169:
165:
162:
157:
153:
149:
144:
140:
136:
133:
130:
127:
122:
118:
110:
109:
108:
106:
98:
96:
94:
90:
86:
85:John Tate
81:
79:
75:
71:
67:
63:
41:
23:
19:
1657:
1615:
1577:
1530:
1495:
1446:
1429:
1420:
1411:
1388:
1380:
1376:
1372:
1366:
1357:
1261:
1258:
1254:
1180:
1057:. Note that
853:
843:
839:
835:
831:
827:
823:
821:
560:
556:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
508:
504:
502:
494:
190:
104:
102:
82:
78:power series
69:
61:
17:
15:
1443:Lang, Serge
1385:local field
1369:j-invariant
1650:Tate, John
1642:0911.14015
1605:0194.52002
1569:0256.14013
1522:1079.11002
1485:0615.14018
1403:References
1379:. Over a
1363:Properties
99:Definition
80:converge.
18:Tate curve
1662:CiteSeerX
1652:(1995) ,
1514:0938-0396
1072:∗
1039:ω
1024:ω
1017:τ
995:τ
989:π
913:∗
867:∗
784:−
757:≥
750:∑
717:−
670:∈
663:∑
618:−
604:−
589:−
480:⋯
423:−
404:×
357:∑
340:−
319:⋯
259:−
222:∑
202:−
1696:Category
1614:(1994).
1445:(1987),
1359:torus).
1144:, where
944:, where
563:, where
1686:1363501
1597:0260753
1561:0352107
1477:0890960
555:)) for
87: (
1684:
1674:
1664:
1640:
1630:
1603:
1595:
1585:
1567:
1559:
1549:
1520:
1512:
1502:
1483:
1475:
1465:
1383:-adic
834:) and
515:with |
191:where
1672:ISBN
1628:ISBN
1583:ISBN
1547:ISBN
1510:ISSN
1500:ISBN
1463:ISBN
1367:The
539:to (
89:1995
1638:Zbl
1601:Zbl
1565:Zbl
1539:doi
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