1288:
646:
1029:
298:
2151:
to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the
Dedekind zeta function of a number field does not determine its class number.
500:
1283:{\displaystyle \Lambda _{K}(s)=\left|\Delta _{K}\right|^{s/2}\Gamma _{\mathbf {R} }(s)^{r_{1}}\Gamma _{\mathbf {C} }(s)^{r_{2}}\zeta _{K}(s)\qquad \Xi _{K}(s)={\tfrac {1}{2}}(s^{2}+{\tfrac {1}{4}})\Lambda _{K}({\tfrac {1}{2}}+is)}
1619:
1407:
2293:
1847:
929:
1010:
2005:
749:
2627:
2206:
684:
493:
186:
1674:
462:
2077:
2041:
1417:
Analogously to the
Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field
641:{\displaystyle \zeta _{K}(s)=\prod _{{\mathfrak {p}}\subseteq {\mathcal {O}}_{K}}{\frac {1}{1-N_{K/\mathbf {Q} }({\mathfrak {p}})^{-s}}},{\text{ for Re}}(s)>1.}
2174:
435:
2672:
1506:
2483:
Flach, Mathias (2004), "The equivariant
Tamagawa number conjecture: a survey", in Burns, David; Popescu, Christian; Sands, Jonathan; et al. (eds.),
1299:
817:
2219:
2869:
1445:
2549:
2501:
2474:
2436:
2394:
2143:
Two fields are called arithmetically equivalent if they have the same
Dedekind zeta function. Wieb Bosma and Bart de Smit (
2854:
2373:
Bosma, Wieb; de Smit, Bart (2002), "On arithmetically equivalent number fields of small degree", in Kohel, David R.; Fieker, Claus (eds.),
775:) has an analytic continuation to a meromorphic function that is analytic at all points of the complex plane except for one simple pole at
2716:
2833:
1787:
852:
940:
2899:
2523:
2797:
146:
2935:
2665:
79:
2838:
2823:
2080:
142:
1947:
689:
2930:
2859:
2493:
2466:
2420:
2412:
784:
2763:
2100:
1430:
2658:
2124:
110:
1923:
780:
2569:
1698:
1872:. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet
1488:
839:
293:{\displaystyle \zeta _{K}(s)=\sum _{I\subseteq {\mathcal {O}}_{K}}{\frac {1}{(N_{K/\mathbf {Q} }(I))^{s}}}}
2828:
2724:
1911:
162:
39:
2180:
2864:
2702:
1877:
658:
467:
83:
56:
2792:
2707:
2511:
1926:
1770:
1713:
1627:
1422:
443:
87:
2046:
2010:
1869:
1734:
353:
308:
2818:
2772:
2116:
2104:
1738:
1717:
1693:) vanishes at all negative even integers. It even vanishes at all negative odd integers unless
2884:
2874:
2545:
2519:
2497:
2470:
2458:
2432:
2390:
1762:
2802:
2754:
2638:
2529:
2424:
2382:
1903:
1888:
312:
138:
71:
2559:
2446:
2404:
2555:
2533:
2442:
2400:
2378:
2148:
1778:
652:
64:
2484:
2544:, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, Chapter 7,
1020:
420:
166:
17:
2924:
2733:
2688:
2643:
1865:
374:
345:
91:
75:
1614:{\displaystyle \lim _{s\rightarrow 0}s^{-r}\zeta _{K}(s)=-{\frac {h(K)R(K)}{w(K)}}.}
2909:
2904:
2159:
1896:
1402:{\displaystyle \Lambda _{K}(s)=\Lambda _{K}(1-s).\qquad \Xi _{K}(-s)=\Xi _{K}(s)\;}
802:
The
Dedekind zeta function satisfies a functional equation relating its values at
2169:
are arithmetically equivalent if and only if all but finitely many prime numbers
792:
759:
438:
370:
98:
31:
2650:
2681:
2428:
2288:{\displaystyle (\dim _{\mathbf {Z} /p}{\mathcal {O}}_{K}/{\mathfrak {p}}_{i})}
2128:
788:
2386:
394:
173:
2516:
Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975
787:
and is made up of important arithmetic data involving invariants of the
1737:
conjectured specific values for these rational numbers in terms of the
27:
Generalization of the
Riemann zeta function for algebraic number fields
1680:) is infinite at all integers less than or equal to zero yields that
2377:, Lecture Notes in Comput. Sci., vol. 2369, Berlin, New York:
2510:
Martinet, J. (1977), "Character theory and Artin L-functions", in
1842:{\displaystyle {\frac {\zeta _{K}(s)}{\zeta _{\mathbf {Q} }(s)}}}
924:{\displaystyle \Gamma _{\mathbf {R} }(s)=\pi ^{-s/2}\Gamma (s/2)}
2457:-functions of mixed motives", in Jannsen, Uwe; Kleiman, Steven;
1005:{\displaystyle \Gamma _{\mathbf {C} }(s)=(2\pi )^{-s}\Gamma (s)}
409:, this definition reduces to that of the Riemann zeta function.
2654:
437:
has an Euler product which is a product over all the non-zero
2465:, Proceedings of Symposia in Pure Mathematics, vol. 55,
2252:
1769:, its Dedekind zeta function can be written as a product of
665:
544:
474:
226:
2079:): for general extensions the result would follow from the
1733:) is a non-zero rational number at negative odd integers.
389:). This sum converges absolutely for all complex numbers
105: = 1, and its values encode arithmetic data of
1676:. Combining the functional equation and the fact that Γ(
2000:{\displaystyle {\frac {\zeta _{L}(s)}{\zeta _{K}(s)}}}
1257:
1229:
1201:
2572:
2222:
2183:
2049:
2013:
1950:
1790:
1630:
1509:
1302:
1032:
943:
855:
744:{\displaystyle \mathrm {Re} (s)>1,\ \zeta _{K}(s)}
692:
661:
503:
470:
446:
423:
189:
2883:
2847:
2811:
2785:
2742:
2695:
2542:
Elementary and analytic theory of algebraic numbers
2486:
Stark's conjectures: recent work and new directions
2417:
Number theory, Volume II: Analytic and modern tools
2621:
2287:
2200:
2071:
2035:
1999:
1936:The relation with Artin L-functions shows that if
1841:
1668:
1613:
1401:
1282:
1004:
923:
743:
678:
640:
487:
456:
429:
292:
165:. Its Dedekind zeta function is first defined for
1918:and hence has a factorization in terms of Artin
1511:
651:This is the expression in analytic terms of the
1483: = 0 where it has a zero whose order
1475:, and the number of real and complex places of
2666:
1624:It follows from the functional equation that
755:Analytic continuation and functional equation
8:
2622:{\displaystyle \zeta _{K}(s)=\zeta _{K'}(s)}
2144:
1876:-function is an analytic formulation of the
2492:, Contemporary Mathematics, vol. 358,
653:uniqueness of prime factorization of ideals
180:) > 1 by the Dirichlet series
2673:
2659:
2651:
2323:
1398:
2642:
2599:
2577:
2571:
2276:
2270:
2269:
2263:
2257:
2251:
2250:
2236:
2231:
2230:
2221:
2192:
2186:
2185:
2182:
2054:
2048:
2018:
2012:
1979:
1958:
1951:
1949:
1820:
1819:
1798:
1791:
1789:
1654:
1641:
1629:
1564:
1543:
1530:
1514:
1508:
1383:
1358:
1329:
1307:
1301:
1256:
1247:
1228:
1219:
1200:
1182:
1162:
1150:
1145:
1128:
1127:
1115:
1110:
1093:
1092:
1078:
1074:
1064:
1037:
1031:
981:
949:
948:
942:
910:
891:
884:
861:
860:
854:
726:
693:
691:
670:
664:
663:
660:
618:
603:
593:
592:
582:
577:
573:
557:
549:
543:
542:
532:
531:
530:
508:
502:
479:
473:
472:
469:
448:
447:
445:
422:
281:
261:
256:
252:
239:
231:
225:
224:
216:
194:
188:
2566:Perlis, Robert (1977), "On the equation
2375:Algorithmic number theory (Sydney, 2002)
2344:
137:The Dedekind zeta function is named for
126:) = 0 and 0 < Re(
2316:
141:who introduced it in his supplement to
2155:
2423:, vol. 240, New York: Springer,
2356:
59:(which is obtained in the case where
7:
2855:Birch and Swinnerton-Dyer conjecture
1902:, its Dedekind zeta function is the
2271:
2201:{\displaystyle {\mathfrak {p}}_{i}}
2187:
594:
533:
449:
1380:
1355:
1326:
1304:
1244:
1179:
1124:
1089:
1061:
1034:
990:
945:
901:
857:
697:
694:
679:{\displaystyle {\mathcal {O}}_{K}}
488:{\displaystyle {\mathcal {O}}_{K}}
25:
2900:Main conjecture of Iwasawa theory
2518:, Academic Press, pp. 1–87,
1500:and the leading term is given by
2232:
2139:Arithmetically equivalent fields
2081:Artin conjecture for L-functions
1821:
1293:satisfy the functional equation
1129:
1094:
950:
862:
583:
401:) > 1. In the case
262:
2453:Deninger, Christopher (1994), "
1669:{\displaystyle r=r_{1}+r_{2}-1}
1471:, the absolute discriminant of
1353:
1177:
457:{\displaystyle {\mathfrak {p}}}
153:Definition and basic properties
2834:Ramanujan–Petersson conjecture
2824:Generalized Riemann hypothesis
2720:-functions of Hecke characters
2616:
2610:
2589:
2583:
2540:Narkiewicz, Władysław (2004),
2282:
2223:
2066:
2060:
2030:
2024:
1991:
1985:
1970:
1964:
1833:
1827:
1810:
1804:
1602:
1596:
1588:
1582:
1576:
1570:
1555:
1549:
1518:
1395:
1389:
1373:
1364:
1347:
1335:
1319:
1313:
1277:
1253:
1240:
1212:
1194:
1188:
1174:
1168:
1142:
1135:
1107:
1100:
1049:
1043:
999:
993:
978:
968:
962:
956:
918:
904:
874:
868:
738:
732:
707:
701:
629:
623:
600:
589:
520:
514:
417:The Dedekind zeta function of
278:
274:
268:
245:
206:
200:
147:Vorlesungen ĂĽber Zahlentheorie
143:Peter Gustav Lejeune Dirichlet
55:), is a generalization of the
1:
2793:Analytic class number formula
2494:American Mathematical Society
2467:American Mathematical Society
2421:Graduate Texts in Mathematics
2072:{\displaystyle \zeta _{L}(s)}
2036:{\displaystyle \zeta _{K}(s)}
1716:). In the totally real case,
1423:analytic class number formula
785:analytic class number formula
783:at that pole is given by the
2798:Riemann–von Mangoldt formula
2644:10.1016/0022-314X(77)90070-1
2177:in the two fields, i.e., if
838:) denote the number of real
352:(which is equal to both the
307:ranges through the non-zero
130:) < 1, then Re(
1944:is a Galois extension then
111:extended Riemann hypothesis
2952:
1781:this shows that the ratio
842:(resp. complex places) of
78:expansion, it satisfies a
70:). It can be defined as a
2429:10.1007/978-0-387-49894-2
65:field of rational numbers
2631:Journal of Number Theory
2295:need to be the same for
2208:are the prime ideals in
2101:Hasse–Weil zeta function
1864:, χ), where χ is a
1479:. Another example is at
806:and 1 −
2936:Algebraic number theory
2750:Dedekind zeta functions
2387:10.1007/3-540-45455-1_6
1467:) of roots of unity in
1425:relates the residue at
2623:
2289:
2202:
2073:
2037:
2001:
1912:regular representation
1843:
1757:For the case in which
1670:
1615:
1429: = 1 to the
1403:
1284:
1023:. Then, the functions
1006:
925:
745:
680:
642:
489:
458:
431:
294:
163:algebraic number field
40:algebraic number field
36:Dedekind zeta function
18:Dedekind zeta-function
2870:Bloch–Kato conjecture
2865:Beilinson conjectures
2848:Algebraic conjectures
2703:Riemann zeta function
2624:
2290:
2203:
2074:
2038:
2002:
1927:Artin representations
1878:quadratic reciprocity
1844:
1771:Dirichlet L-functions
1708: = 0; e.g.
1671:
1616:
1491:of the unit group of
1404:
1285:
1007:
926:
810:. Specifically, let Δ
746:
681:
643:
490:
459:
432:
295:
84:analytic continuation
57:Riemann zeta function
45:, generally denoted ζ
2931:Zeta and L-functions
2875:Langlands conjecture
2860:Deligne's conjecture
2812:Analytic conjectures
2570:
2469:, pp. 517–525,
2335:Martinet (1977) p.19
2220:
2181:
2047:
2011:
1948:
1788:
1773:. For example, when
1714:real quadratic field
1628:
1507:
1300:
1030:
941:
853:
779: = 1. The
690:
659:
501:
468:
444:
421:
369:or equivalently the
187:
88:meromorphic function
2829:Lindelöf hypothesis
2496:, pp. 79–125,
2216:, then the tuples
1870:Dirichlet character
1749:Relations to other
1735:Stephen Lichtenbaum
1421:. For example, the
134:) = 1/2.
80:functional equation
2819:Riemann hypothesis
2743:Algebraic examples
2619:
2459:Serre, Jean-Pierre
2410:Section 10.5.1 of
2381:, pp. 67–79,
2285:
2198:
2069:
2033:
1997:
1839:
1739:algebraic K-theory
1718:Carl Ludwig Siegel
1666:
1611:
1525:
1399:
1280:
1266:
1238:
1210:
1002:
921:
762:first proved that
741:
676:
638:
556:
485:
454:
427:
290:
238:
2918:
2917:
2696:Analytic examples
2551:978-3-540-21902-6
2503:978-0-8218-3480-0
2476:978-0-8218-1635-6
2438:978-0-387-49893-5
2396:978-3-540-43863-2
1995:
1837:
1763:abelian extension
1606:
1510:
1265:
1237:
1209:
721:
621:
613:
526:
430:{\displaystyle K}
288:
212:
16:(Redirected from
2943:
2839:Artin conjecture
2803:Weil conjectures
2675:
2668:
2661:
2652:
2647:
2646:
2628:
2626:
2625:
2620:
2609:
2608:
2607:
2582:
2581:
2562:
2536:
2506:
2491:
2479:
2449:
2407:
2360:
2354:
2348:
2342:
2336:
2333:
2327:
2321:
2294:
2292:
2291:
2286:
2281:
2280:
2275:
2274:
2267:
2262:
2261:
2256:
2255:
2245:
2244:
2240:
2235:
2207:
2205:
2204:
2199:
2197:
2196:
2191:
2190:
2158:showed that two
2149:Gassmann triples
2127:coming from the
2078:
2076:
2075:
2070:
2059:
2058:
2042:
2040:
2039:
2034:
2023:
2022:
2007:is holomorphic (
2006:
2004:
2003:
1998:
1996:
1994:
1984:
1983:
1973:
1963:
1962:
1952:
1889:Galois extension
1848:
1846:
1845:
1840:
1838:
1836:
1826:
1825:
1824:
1813:
1803:
1802:
1792:
1675:
1673:
1672:
1667:
1659:
1658:
1646:
1645:
1620:
1618:
1617:
1612:
1607:
1605:
1591:
1565:
1548:
1547:
1538:
1537:
1524:
1487:is equal to the
1408:
1406:
1405:
1400:
1388:
1387:
1363:
1362:
1334:
1333:
1312:
1311:
1289:
1287:
1286:
1281:
1267:
1258:
1252:
1251:
1239:
1230:
1224:
1223:
1211:
1202:
1187:
1186:
1167:
1166:
1157:
1156:
1155:
1154:
1134:
1133:
1132:
1122:
1121:
1120:
1119:
1099:
1098:
1097:
1087:
1086:
1082:
1073:
1069:
1068:
1042:
1041:
1011:
1009:
1008:
1003:
989:
988:
955:
954:
953:
930:
928:
927:
922:
914:
900:
899:
895:
867:
866:
865:
750:
748:
747:
742:
731:
730:
719:
700:
685:
683:
682:
677:
675:
674:
669:
668:
647:
645:
644:
639:
622:
619:
614:
612:
611:
610:
598:
597:
588:
587:
586:
581:
558:
555:
554:
553:
548:
547:
537:
536:
513:
512:
494:
492:
491:
486:
484:
483:
478:
477:
463:
461:
460:
455:
453:
452:
436:
434:
433:
428:
313:ring of integers
299:
297:
296:
291:
289:
287:
286:
285:
267:
266:
265:
260:
240:
237:
236:
235:
230:
229:
199:
198:
139:Richard Dedekind
72:Dirichlet series
21:
2951:
2950:
2946:
2945:
2944:
2942:
2941:
2940:
2921:
2920:
2919:
2914:
2879:
2843:
2807:
2781:
2738:
2691:
2679:
2600:
2595:
2573:
2568:
2567:
2565:
2552:
2539:
2526:
2509:
2504:
2489:
2482:
2477:
2463:Motives, Part 1
2452:
2439:
2411:
2397:
2379:Springer-Verlag
2372:
2369:
2364:
2363:
2355:
2351:
2343:
2339:
2334:
2330:
2324:Narkiewicz 2004
2322:
2318:
2313:
2303:for almost all
2268:
2249:
2226:
2218:
2217:
2184:
2179:
2178:
2175:inertia degrees
2141:
2114:
2094:
2050:
2045:
2044:
2014:
2009:
2008:
1975:
1974:
1954:
1953:
1946:
1945:
1883:In general, if
1815:
1814:
1794:
1793:
1786:
1785:
1779:quadratic field
1755:
1728:
1707:
1688:
1650:
1637:
1626:
1625:
1592:
1566:
1539:
1526:
1505:
1504:
1499:
1415:
1379:
1354:
1325:
1303:
1298:
1297:
1243:
1215:
1178:
1158:
1146:
1141:
1123:
1111:
1106:
1088:
1060:
1056:
1055:
1033:
1028:
1027:
977:
944:
939:
938:
880:
856:
851:
850:
837:
830:
815:
770:
757:
722:
688:
687:
662:
657:
656:
599:
569:
562:
541:
504:
499:
498:
471:
466:
465:
442:
441:
419:
418:
415:
384:
368:
339:
322:
277:
248:
244:
223:
190:
185:
184:
167:complex numbers
155:
121:
113:states that if
50:
28:
23:
22:
15:
12:
11:
5:
2949:
2947:
2939:
2938:
2933:
2923:
2922:
2916:
2915:
2913:
2912:
2907:
2902:
2896:
2894:
2881:
2880:
2878:
2877:
2872:
2867:
2862:
2857:
2851:
2849:
2845:
2844:
2842:
2841:
2836:
2831:
2826:
2821:
2815:
2813:
2809:
2808:
2806:
2805:
2800:
2795:
2789:
2787:
2783:
2782:
2780:
2779:
2770:
2761:
2752:
2746:
2744:
2740:
2739:
2737:
2736:
2731:
2722:
2714:
2705:
2699:
2697:
2693:
2692:
2680:
2678:
2677:
2670:
2663:
2655:
2649:
2648:
2637:(3): 342–360,
2618:
2615:
2612:
2606:
2603:
2598:
2594:
2591:
2588:
2585:
2580:
2576:
2563:
2550:
2537:
2524:
2507:
2502:
2480:
2475:
2450:
2437:
2408:
2395:
2368:
2365:
2362:
2361:
2349:
2337:
2328:
2315:
2314:
2312:
2309:
2284:
2279:
2273:
2266:
2260:
2254:
2248:
2243:
2239:
2234:
2229:
2225:
2195:
2189:
2173:have the same
2140:
2137:
2110:
2090:
2086:Additionally,
2068:
2065:
2062:
2057:
2053:
2032:
2029:
2026:
2021:
2017:
1993:
1990:
1987:
1982:
1978:
1972:
1969:
1966:
1961:
1957:
1922:-functions of
1880:law of Gauss.
1850:
1849:
1835:
1832:
1829:
1823:
1818:
1812:
1809:
1806:
1801:
1797:
1754:
1747:
1724:
1705:
1684:
1665:
1662:
1657:
1653:
1649:
1644:
1640:
1636:
1633:
1622:
1621:
1610:
1604:
1601:
1598:
1595:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1563:
1560:
1557:
1554:
1551:
1546:
1542:
1536:
1533:
1529:
1523:
1520:
1517:
1513:
1495:
1414:
1413:Special values
1411:
1410:
1409:
1397:
1394:
1391:
1386:
1382:
1378:
1375:
1372:
1369:
1366:
1361:
1357:
1352:
1349:
1346:
1343:
1340:
1337:
1332:
1328:
1324:
1321:
1318:
1315:
1310:
1306:
1291:
1290:
1279:
1276:
1273:
1270:
1264:
1261:
1255:
1250:
1246:
1242:
1236:
1233:
1227:
1222:
1218:
1214:
1208:
1205:
1199:
1196:
1193:
1190:
1185:
1181:
1176:
1173:
1170:
1165:
1161:
1153:
1149:
1144:
1140:
1137:
1131:
1126:
1118:
1114:
1109:
1105:
1102:
1096:
1091:
1085:
1081:
1077:
1072:
1067:
1063:
1059:
1054:
1051:
1048:
1045:
1040:
1036:
1021:gamma function
1013:
1012:
1001:
998:
995:
992:
987:
984:
980:
976:
973:
970:
967:
964:
961:
958:
952:
947:
932:
931:
920:
917:
913:
909:
906:
903:
898:
894:
890:
887:
883:
879:
876:
873:
870:
864:
859:
835:
828:
811:
766:
756:
753:
740:
737:
734:
729:
725:
718:
715:
712:
709:
706:
703:
699:
696:
673:
667:
649:
648:
637:
634:
631:
628:
625:
617:
609:
606:
602:
596:
591:
585:
580:
576:
572:
568:
565:
561:
552:
546:
540:
535:
529:
525:
522:
519:
516:
511:
507:
482:
476:
451:
426:
414:
411:
380:
364:
344:) denotes the
331:
318:
301:
300:
284:
280:
276:
273:
270:
264:
259:
255:
251:
247:
243:
234:
228:
222:
219:
215:
211:
208:
205:
202:
197:
193:
154:
151:
117:
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2948:
2937:
2934:
2932:
2929:
2928:
2926:
2911:
2908:
2906:
2903:
2901:
2898:
2897:
2895:
2893:
2891:
2887:
2882:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2852:
2850:
2846:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2816:
2814:
2810:
2804:
2801:
2799:
2796:
2794:
2791:
2790:
2788:
2784:
2778:
2776:
2771:
2769:
2767:
2762:
2760:
2758:
2753:
2751:
2748:
2747:
2745:
2741:
2735:
2734:Selberg class
2732:
2730:
2728:
2723:
2721:
2719:
2715:
2713:
2711:
2706:
2704:
2701:
2700:
2698:
2694:
2690:
2689:number theory
2686:
2684:
2676:
2671:
2669:
2664:
2662:
2657:
2656:
2653:
2645:
2640:
2636:
2632:
2613:
2604:
2601:
2596:
2592:
2586:
2578:
2574:
2564:
2561:
2557:
2553:
2547:
2543:
2538:
2535:
2531:
2527:
2525:0-12-268960-7
2521:
2517:
2513:
2508:
2505:
2499:
2495:
2488:
2487:
2481:
2478:
2472:
2468:
2464:
2460:
2456:
2451:
2448:
2444:
2440:
2434:
2430:
2426:
2422:
2418:
2414:
2409:
2406:
2402:
2398:
2392:
2388:
2384:
2380:
2376:
2371:
2370:
2366:
2358:
2353:
2350:
2346:
2345:Deninger 1994
2341:
2338:
2332:
2329:
2325:
2320:
2317:
2310:
2308:
2306:
2302:
2298:
2277:
2264:
2258:
2246:
2241:
2237:
2227:
2215:
2211:
2193:
2176:
2172:
2168:
2164:
2161:
2160:number fields
2157:
2156:Perlis (1977)
2153:
2150:
2146:
2138:
2136:
2134:
2130:
2126:
2122:
2120:
2113:
2109:
2106:
2102:
2098:
2093:
2089:
2084:
2082:
2063:
2055:
2051:
2027:
2019:
2015:
1988:
1980:
1976:
1967:
1959:
1955:
1943:
1939:
1934:
1932:
1928:
1925:
1921:
1917:
1913:
1909:
1907:
1901:
1898:
1894:
1890:
1886:
1881:
1879:
1875:
1871:
1867:
1866:Jacobi symbol
1863:
1859:
1855:
1830:
1816:
1807:
1799:
1795:
1784:
1783:
1782:
1780:
1776:
1772:
1768:
1764:
1760:
1752:
1748:
1746:
1744:
1740:
1736:
1732:
1727:
1723:
1719:
1715:
1711:
1704:
1700:
1696:
1692:
1687:
1683:
1679:
1663:
1660:
1655:
1651:
1647:
1642:
1638:
1634:
1631:
1608:
1599:
1593:
1585:
1579:
1573:
1567:
1561:
1558:
1552:
1544:
1540:
1534:
1531:
1527:
1521:
1515:
1503:
1502:
1501:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1459:, the number
1458:
1454:
1450:
1447:
1443:
1439:
1435:
1432:
1428:
1424:
1420:
1412:
1392:
1384:
1376:
1370:
1367:
1359:
1350:
1344:
1341:
1338:
1330:
1322:
1316:
1308:
1296:
1295:
1294:
1274:
1271:
1268:
1262:
1259:
1248:
1234:
1231:
1225:
1220:
1216:
1206:
1203:
1197:
1191:
1183:
1171:
1163:
1159:
1151:
1147:
1138:
1116:
1112:
1103:
1083:
1079:
1075:
1070:
1065:
1057:
1052:
1046:
1038:
1026:
1025:
1024:
1022:
1018:
996:
985:
982:
974:
971:
965:
959:
937:
936:
935:
915:
911:
907:
896:
892:
888:
885:
881:
877:
871:
849:
848:
847:
845:
841:
834:
827:
823:
819:
814:
809:
805:
800:
798:
794:
790:
786:
782:
778:
774:
769:
765:
761:
754:
752:
751:is non-zero.
735:
727:
723:
716:
713:
710:
704:
671:
654:
635:
632:
626:
615:
607:
604:
578:
574:
570:
566:
563:
559:
550:
538:
527:
523:
517:
509:
505:
497:
496:
495:
480:
440:
424:
413:Euler product
412:
410:
408:
405: =
404:
400:
396:
392:
388:
385: /
383:
379:
376:
375:quotient ring
372:
367:
363:
359:
355:
351:
347:
346:absolute norm
343:
338:
334:
330:
326:
321:
317:
314:
310:
306:
282:
271:
257:
253:
249:
241:
232:
220:
217:
213:
209:
203:
195:
191:
183:
182:
181:
179:
175:
171:
168:
164:
160:
152:
150:
148:
144:
140:
135:
133:
129:
125:
120:
116:
112:
108:
104:
100:
96:
93:
92:complex plane
89:
85:
81:
77:
76:Euler product
73:
69:
66:
62:
58:
54:
49:
44:
41:
37:
33:
19:
2910:Euler system
2905:Selmer group
2889:
2885:
2774:
2765:
2756:
2749:
2726:
2725:Automorphic
2717:
2709:
2682:
2634:
2630:
2541:
2515:
2512:Fröhlich, A.
2485:
2462:
2454:
2416:
2413:Cohen, Henri
2374:
2352:
2340:
2331:
2319:
2304:
2300:
2296:
2213:
2209:
2170:
2166:
2162:
2154:
2142:
2132:
2118:
2111:
2107:
2096:
2091:
2087:
2085:
1941:
1937:
1935:
1930:
1919:
1915:
1905:
1899:
1897:Galois group
1892:
1884:
1882:
1873:
1861:
1857:
1853:
1851:
1774:
1766:
1758:
1756:
1750:
1742:
1730:
1725:
1721:
1720:showed that
1709:
1702:
1699:totally real
1694:
1690:
1685:
1681:
1677:
1623:
1496:
1492:
1484:
1480:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1441:
1437:
1433:
1431:class number
1426:
1418:
1416:
1292:
1016:
1014:
933:
843:
832:
825:
821:
818:discriminant
812:
807:
803:
801:
796:
776:
772:
767:
763:
758:
650:
620: for Re
439:prime ideals
416:
406:
402:
398:
390:
386:
381:
377:
365:
361:
357:
349:
341:
336:
332:
328:
324:
319:
315:
304:
302:
177:
169:
158:
156:
136:
131:
127:
123:
118:
114:
106:
102:
97:with only a
94:
82:, it has an
74:, it has an
67:
60:
52:
47:
42:
35:
29:
2764:Hasse–Weil
2212:lying over
1924:irreducible
816:denote the
793:class group
760:Erich Hecke
371:cardinality
99:simple pole
32:mathematics
2925:Categories
2892:-functions
2777:-functions
2768:-functions
2759:-functions
2729:-functions
2712:-functions
2708:Dirichlet
2685:-functions
2534:0359.12015
2367:References
2357:Flach 2004
2129:cohomology
2043:"divides"
1856:-function
1753:-functions
846:, and let
789:unit group
2597:ζ
2575:ζ
2247:
2121:-function
2099:) is the
2052:ζ
2016:ζ
1977:ζ
1956:ζ
1908:-function
1817:ζ
1796:ζ
1661:−
1562:−
1541:ζ
1532:−
1519:→
1446:regulator
1381:Ξ
1368:−
1356:Ξ
1342:−
1327:Λ
1305:Λ
1245:Λ
1180:Ξ
1160:ζ
1125:Γ
1090:Γ
1062:Δ
1035:Λ
1019:) is the
991:Γ
983:−
975:π
946:Γ
902:Γ
886:−
882:π
858:Γ
724:ζ
605:−
567:−
539:⊆
528:∏
506:ζ
395:real part
221:⊆
214:∑
192:ζ
174:real part
2786:Theorems
2773:Motivic
2605:′
2461:(eds.),
2415:(2007),
2326:, §7.4.1
2299:and for
2131:of Spec
2117:motivic
2115:and the
1868:used as
1015:where Γ(
2560:2078267
2514:(ed.),
2447:2312338
2405:2041074
2147:) used
2123:of the
1910:of the
1852:is the
831:(resp.
781:residue
311:of the
90:on the
63:is the
2888:-adic
2755:Artin
2558:
2548:
2532:
2522:
2500:
2473:
2445:
2435:
2403:
2393:
2359:, §1.1
2125:motive
1904:Artin
1761:is an
1701:(i.e.
1444:, the
840:places
824:, let
720:
686:. For
309:ideals
303:where
161:be an
109:. The
38:of an
34:, the
2490:(PDF)
2311:Notes
1895:with
1887:is a
1777:is a
1712:or a
1455:) of
1440:) of
393:with
354:index
172:with
86:to a
2546:ISBN
2520:ISBN
2498:ISBN
2471:ISBN
2433:ISBN
2391:ISBN
2347:, §1
2165:and
2145:2002
2105:Spec
1489:rank
934:and
791:and
711:>
633:>
327:and
157:Let
2687:in
2639:doi
2629:",
2530:Zbl
2425:doi
2383:doi
2228:dim
2103:of
1929:of
1914:of
1891:of
1765:of
1741:of
1697:is
1512:lim
820:of
795:of
655:in
464:of
397:Re(
373:of
360:in
356:of
348:of
323:of
176:Re(
145:'s
101:at
30:In
2927::
2633:,
2556:MR
2554:,
2528:,
2443:MR
2441:,
2431:,
2419:,
2401:MR
2399:,
2389:,
2307:.
2135:.
2083:.
1933:.
1745:.
799:.
636:1.
149:.
2890:L
2886:p
2775:L
2766:L
2757:L
2727:L
2718:L
2710:L
2683:L
2674:e
2667:t
2660:v
2641::
2635:9
2617:)
2614:s
2611:(
2602:K
2593:=
2590:)
2587:s
2584:(
2579:K
2455:L
2427::
2385::
2305:p
2301:L
2297:K
2283:)
2278:i
2272:p
2265:/
2259:K
2253:O
2242:p
2238:/
2233:Z
2224:(
2214:p
2210:K
2194:i
2188:p
2171:p
2167:L
2163:K
2133:K
2119:L
2112:K
2108:O
2097:s
2095:(
2092:K
2088:ζ
2067:)
2064:s
2061:(
2056:L
2031:)
2028:s
2025:(
2020:K
1992:)
1989:s
1986:(
1981:K
1971:)
1968:s
1965:(
1960:L
1942:K
1940:/
1938:L
1931:G
1920:L
1916:G
1906:L
1900:G
1893:Q
1885:K
1874:L
1862:s
1860:(
1858:L
1854:L
1834:)
1831:s
1828:(
1822:Q
1811:)
1808:s
1805:(
1800:K
1775:K
1767:Q
1759:K
1751:L
1743:K
1731:s
1729:(
1726:K
1722:ζ
1710:Q
1706:2
1703:r
1695:K
1691:s
1689:(
1686:K
1682:ζ
1678:s
1664:1
1656:2
1652:r
1648:+
1643:1
1639:r
1635:=
1632:r
1609:.
1603:)
1600:K
1597:(
1594:w
1589:)
1586:K
1583:(
1580:R
1577:)
1574:K
1571:(
1568:h
1559:=
1556:)
1553:s
1550:(
1545:K
1535:r
1528:s
1522:0
1516:s
1497:K
1493:O
1485:r
1481:s
1477:K
1473:K
1469:K
1465:K
1463:(
1461:w
1457:K
1453:K
1451:(
1449:R
1442:K
1438:K
1436:(
1434:h
1427:s
1419:K
1396:)
1393:s
1390:(
1385:K
1377:=
1374:)
1371:s
1365:(
1360:K
1351:.
1348:)
1345:s
1339:1
1336:(
1331:K
1323:=
1320:)
1317:s
1314:(
1309:K
1278:)
1275:s
1272:i
1269:+
1263:2
1260:1
1254:(
1249:K
1241:)
1235:4
1232:1
1226:+
1221:2
1217:s
1213:(
1207:2
1204:1
1198:=
1195:)
1192:s
1189:(
1184:K
1175:)
1172:s
1169:(
1164:K
1152:2
1148:r
1143:)
1139:s
1136:(
1130:C
1117:1
1113:r
1108:)
1104:s
1101:(
1095:R
1084:2
1080:/
1076:s
1071:|
1066:K
1058:|
1053:=
1050:)
1047:s
1044:(
1039:K
1017:s
1000:)
997:s
994:(
986:s
979:)
972:2
969:(
966:=
963:)
960:s
957:(
951:C
919:)
916:2
912:/
908:s
905:(
897:2
893:/
889:s
878:=
875:)
872:s
869:(
863:R
844:K
836:2
833:r
829:1
826:r
822:K
813:K
808:s
804:s
797:K
777:s
773:s
771:(
768:K
764:ζ
739:)
736:s
733:(
728:K
717:,
714:1
708:)
705:s
702:(
698:e
695:R
672:K
666:O
630:)
627:s
624:(
616:,
608:s
601:)
595:p
590:(
584:Q
579:/
575:K
571:N
564:1
560:1
551:K
545:O
534:p
524:=
521:)
518:s
515:(
510:K
481:K
475:O
450:p
425:K
407:Q
403:K
399:s
391:s
387:I
382:K
378:O
366:K
362:O
358:I
350:I
342:I
340:(
337:Q
335:/
333:K
329:N
325:K
320:K
316:O
305:I
283:s
279:)
275:)
272:I
269:(
263:Q
258:/
254:K
250:N
246:(
242:1
233:K
227:O
218:I
210:=
207:)
204:s
201:(
196:K
178:s
170:s
159:K
132:s
128:s
124:s
122:(
119:K
115:ζ
107:K
103:s
95:C
68:Q
61:K
53:s
51:(
48:K
43:K
20:)
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