25:
225:
414:
365:
54:
105:
symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an
147:
528:"Über einige allgemeine Eigenschaften der Gleichung, von welcher die Teilung der ganzen Lemniskate abhängt, nebst Anwendungen derselben auf die Zahlentheorie"
625:
585:
76:
651:
137:, 9.3) gives the following example of an elliptic Gauss sum, for the case of an elliptic curve with complex multiplication by
37:
47:
41:
33:
58:
656:
488:
295:
613:
467:
617:
607:
110:
388:
339:
558:
457:
621:
581:
550:
507:
106:
102:
542:
497:
635:
595:
519:
479:
631:
591:
577:
515:
475:
220:{\displaystyle -\sum _{t}\chi (t)\varphi \left({\frac {t}{\pi }}\right)^{\frac {p-1}{m}}}
117:), at least in the lemniscate case when the elliptic curve has complex multiplication by
471:
98:
527:
502:
645:
562:
486:
Cassou-Noguès, Ph.; Taylor, M. J. (1991), "Un élément de
Stickelberger quadratique",
456:, RIMS Kôkyûroku Bessatsu, B4, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 79–121,
603:
571:
554:
546:
511:
94:
454:
Proceedings of the
Symposium on Algebraic Number Theory and Related Topics
462:
121:, but seem to have been forgotten or ignored until the paper (
18:
576:, Springer Monographs in Mathematics, Berlin, New York:
612:, Inst. Math. Appl. Conf. Ser. New Ser., vol. 14,
444:
Asai, Tetsuya (2007), "Elliptic Gauss sums and Hecke
391:
342:
150:
609:Computers in mathematical research (Cardiff, 1986)
408:
359:
219:
606:, in Stephens, Nelson M.; Thorne., M. P. (eds.),
46:but its sources remain unclear because it lacks
604:"Galois module structure of elliptic functions"
535:Journal für die Reine und Angewandte Mathematik
8:
385:is a primary prime in the Gaussian integers
238:whose representatives are Gaussian integers
134:
114:
501:
461:
393:
392:
390:
344:
343:
341:
198:
184:
158:
149:
77:Learn how and when to remove this message
269:is a rational prime congruent to 1 mod 4
565:, Reprinted in Math. Werke II, 556–619
425:is a prime in the ring of integers of
122:
7:
14:
101:with complex multiplication. The
23:
250:is a positive integer dividing
403:
397:
354:
348:
173:
167:
16:Gauss sum on an elliptic curve
1:
526:Eisenstein, Gotthold (1850),
503:10.1016/S0022-314X(05)80046-0
234:The sum is over residues mod
409:{\displaystyle \mathbb {Z} }
360:{\displaystyle \mathbb {Q} }
570:Lemmermeyer, Franz (2000),
308:th power residue symbol in
673:
312:with respect to the prime
109:. They were introduced by
547:10.1515/crll.1850.39.224
489:Journal of Number Theory
296:sine lemniscate function
32:This article includes a
652:Algebraic number theory
614:Oxford University Press
298:, an elliptic function.
61:more precise citations.
410:
361:
221:
433:with inertia degree 1
411:
362:
244:is a positive integer
222:
389:
340:
148:
472:2007arXiv0707.3711A
89:In mathematics, an
602:Pinch, R. (1988),
406:
357:
217:
163:
93:is an analog of a
91:elliptic Gauss sum
34:list of references
627:978-0-19-853620-8
587:978-3-540-66957-9
452: = 1",
214:
192:
154:
107:elliptic function
103:quadratic residue
87:
86:
79:
664:
638:
598:
573:Reciprocity laws
566:
532:
522:
505:
482:
465:
432:
428:
424:
419:
415:
413:
412:
407:
396:
384:
378:
371:
366:
364:
363:
358:
347:
335:
330:
324:
319:
315:
311:
307:
303:
293:
289:
268:
256:
249:
243:
237:
226:
224:
223:
218:
216:
215:
210:
199:
197:
193:
185:
162:
140:
135:Lemmermeyer 2000
120:
97:depending on an
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
672:
671:
667:
666:
665:
663:
662:
661:
657:Elliptic curves
642:
641:
628:
601:
588:
578:Springer-Verlag
569:
541:(39): 224–287,
530:
525:
485:
443:
440:
430:
426:
422:
417:
387:
386:
382:
373:
372:is a primitive
369:
338:
337:
333:
326:
322:
317:
313:
309:
305:
301:
291:
272:
259:
251:
247:
241:
235:
200:
180:
179:
146:
145:
138:
131:
118:
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
670:
668:
660:
659:
654:
644:
643:
640:
639:
626:
599:
586:
567:
523:
496:(3): 307–342,
483:
439:
436:
435:
434:
420:
405:
402:
399:
395:
380:
367:
356:
353:
350:
346:
331:
320:
299:
270:
257:
245:
239:
228:
227:
213:
209:
206:
203:
196:
191:
188:
183:
178:
175:
172:
169:
166:
161:
157:
153:
130:
127:
99:elliptic curve
85:
84:
42:external links
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
669:
658:
655:
653:
650:
649:
647:
637:
633:
629:
623:
619:
615:
611:
610:
605:
600:
597:
593:
589:
583:
579:
575:
574:
568:
564:
560:
556:
552:
548:
544:
540:
536:
529:
524:
521:
517:
513:
509:
504:
499:
495:
491:
490:
484:
481:
477:
473:
469:
464:
459:
455:
451:
447:
442:
441:
437:
421:
400:
381:
377:
368:
351:
336:is the field
332:
329:
325:is the field
321:
300:
297:
287:
283:
279:
275:
271:
266:
262:
258:
255:
246:
240:
233:
232:
231:
211:
207:
204:
201:
194:
189:
186:
181:
176:
170:
164:
159:
155:
151:
144:
143:
142:
136:
128:
126:
124:
116:
112:
108:
104:
100:
96:
92:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
608:
572:
538:
534:
493:
487:
453:
449:
445:
429:lying above
379:th root of 1
375:
327:
285:
281:
280:) = sl((1 –
277:
273:
264:
260:
253:
229:
132:
90:
88:
73:
64:
53:Please help
45:
616:, pp.
448:-values at
59:introducing
646:Categories
438:References
416:with norm
123:Pinch 1988
111:Eisenstein
563:123157985
555:0075-4102
512:0022-314X
463:0707.3711
205:−
190:π
177:φ
165:χ
156:∑
152:−
95:Gauss sum
67:June 2020
636:0960495
596:1761696
520:1096447
480:2402004
468:Bibcode
304:is the
294:is the
129:Example
113: (
55:improve
634:
624:
594:
584:
561:
553:
518:
510:
478:
290:where
230:where
618:69–91
559:S2CID
531:(PDF)
458:arXiv
40:, or
622:ISBN
582:ISBN
551:ISSN
539:1850
508:ISSN
115:1850
543:doi
498:doi
316:of
267:+ 1
263:= 4
125:).
648::
632:MR
630:,
620:,
592:MR
590:,
580:,
557:,
549:,
537:,
533:,
516:MR
514:,
506:,
494:37
492:,
476:MR
474:,
466:,
292:sl
286:ωz
141:.
44:,
36:,
545::
500::
470::
460::
450:s
446:L
431:π
427:K
423:P
418:p
404:]
401:i
398:[
394:Z
383:π
376:n
374:4
370:ζ
355:]
352:i
349:[
345:Q
334:k
328:k
323:K
318:K
314:P
310:K
306:m
302:χ
288:)
284:)
282:i
278:z
276:(
274:φ
265:n
261:p
254:n
252:4
248:m
242:n
236:P
212:m
208:1
202:p
195:)
187:t
182:(
174:)
171:t
168:(
160:t
139:i
133:(
119:i
80:)
74:(
69:)
65:(
51:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.