437:
1604:
2287:
2758:
2000:
274:
1412:
2084:
2567:
1186:
1816:
997:
432:{\displaystyle \operatorname {Rep} _{\mathrm {crys} }(K)\subsetneq \operatorname {Rep} _{ss}(K)\subsetneq \operatorname {Rep} _{dR}(K)\subsetneq \operatorname {Rep} _{HT}(K)\subsetneq \operatorname {Rep} _{\mathbf {Q} _{p}}(K)}
1401:
1599:{\displaystyle B_{\mathrm {HT} }\otimes _{K}\mathrm {gr} H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {HT} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})}
2282:{\displaystyle B_{\mathrm {cris} }\otimes _{K_{0}}H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {cris} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})}
2753:{\displaystyle B_{\mathrm {st} }\otimes _{K_{0}}H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {st} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})}
877:
1074:
1995:{\displaystyle B_{\mathrm {dR} }\otimes _{K}H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {dR} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})}
680:
2493:
1668:
250:
2355:
891:
1223:
can be said to contain all the periods necessary to compare algebraic de Rham cohomology with singular cohomology, and could hence be called a period ring in this situation.
746:
2906:
3327:
1332:
3521:
3423:
3206:
3138:
1181:{\displaystyle H_{\mathrm {dR} }^{\ast }(X/\mathbf {C} )\cong H^{\ast }(X(\mathbf {C} ),\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C} .}
798:
2557:
has semi-stable reduction, the de Rham cohomology can be equipped with the φ-action and a monodromy operator by its comparison with the
1779:
To improve the Hodge–Tate conjecture to one involving the de Rham cohomology (not just its associated graded), Fontaine constructed a
601:
2412:
3511:
2027:-adic étale cohomology, just as the complex numbers above were used with the comparison with singular cohomology. This is where
1628:
463:
73:
475:
202:
1723:
of the special fiber (with the
Frobenius endomorphism on this group and the Hodge filtration on this group tensored with
1009:
This formalism (and the name period ring) grew out of a few results and conjectures regarding comparison isomorphisms in
3531:
3099:
1042:
992:{\displaystyle \alpha _{V}:B_{\ast }\otimes _{E}D_{B_{\ast }}(V)\longrightarrow B_{\ast }\otimes _{\mathbf {Q} _{p}}V}
883:
775:
749:
2306:
582:
564:
61:
2558:
573:
555:
447:
3341:
2801:
1781:
451:
156:
35:
2041:
Similarly, to formulate a conjecture explaining
Grothendieck's mysterious functor, Fontaine introduced a ring
1753:
523:
455:
2514:
1241:
3516:
3289:
3268:
1696:
1692:
546:
3526:
3232:
3094:
1767:-adic étale cohomology to crystalline cohomology (and back), for all varieties with good reduction over
148:
28:
2498:
In other words, the
Dieudonné modules should be thought of as giving the other cohomologies related to
2505:
In the late eighties, Fontaine and Uwe
Jannsen formulated another comparison isomorphism conjecture, C
1228:
In the mid sixties, Tate conjectured that a similar isomorphism should hold for proper smooth schemes
3411:
3350:
1319:
1212:
1032:
1674:
in the late eighties after partial results by several other mathematicians (including Tate himself).
2798:
1611:
1258:
1054:
1010:
462:-adic representations. In addition, two other categories of representations can be introduced, the
97:
85:
711:
3463:
3401:
3374:
3249:
3223:
3171:
105:
2877:
3419:
3321:
3309:
3202:
3134:
1262:
1200:
101:
57:
3358:
3241:
3163:
3126:
2543:
2364:-vector space with φ-action given by its comparison with crystalline cohomology). Both the C
1622:
1014:
443:
261:
196:
3495:
3475:
3433:
3370:
3302:
3280:
3261:
3230:-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti–Tate",
3216:
3183:
3491:
3490:, Astérisque, vol. 189–190, Paris: Société Mathématique de France, pp. 325–374,
3471:
3429:
3366:
3298:
3276:
3257:
3212:
3179:
3089:
1208:
1204:
1029:
1026:
69:
1763:, up to isogeny. Grothendieck conjectured that there should be a way to go directly from
3415:
3354:
486:). The latter strictly contains the former which in turn generally strictly contains Rep
2946:
1037:
265:
3339:-adic étale cohomology and crystalline cohomology in the semi-stable reduction case",
3505:
3378:
3287:
Hyodo, Osamu (1991), "On the de Rham–Witt complex attached to a semi-stable family",
3190:
3147:
2805:
1671:
1211:
and is generally a complex number. This explains why the singular cohomology must be
46:
3445:
3441:
3084:
175:
704:-action, but are endowed with linear algebraic structures inherited from the ring
3130:
1396:{\displaystyle B_{\mathrm {HT} }:=\oplus _{i\in \mathbf {Z} }\mathbf {C} _{K}(i)}
2950:
1003:
65:
38:
17:
541:-adic Hodge theory, introduced by Fontaine, is to construct certain so-called
81:
3362:
1207:
in the singular cohomology. The result of such an integration is called a
268:
objects that are easier to study. The basic classification is as follows:
1197:
2055:-action, a "Frobenius" φ, and a filtration after extending scalars from
92:-adic Hodge theory. Further developments were inspired by properties of
3253:
3175:
1193:
755:
introduced by
Fontaine. For a period ring like the aforementioned ones
446:
properly contained in the next. In order, these are the categories of
76:. Hodge–Tate representations are related to certain decompositions of
3406:
2023:-adic) periods required to compare algebraic de Rham cohomology with
3470:, Astérisque, vol. 223, Paris: Société Mathématique de France,
3245:
3167:
2833:
in question, but this relation is usually dropped from the notation.
2778:. This conjecture was proved in the late nineties by Takeshi Tsuji.
2531:-action, a "Frobenius" φ, a filtration after extending scalars from
260:-adic representations based on how nice they are, and also provides
872:{\displaystyle \dim _{E}D_{B_{\ast }}(V)=\dim _{\mathbf {Q} _{p}}V}
3482:
Illusie, Luc (1990), "Cohomologie de de Rham et cohomologie étale
3201:, Baltimore, MD: Johns Hopkins University Press, pp. 25–80,
2561:
first introduced by Osamu Hyodo. The conjecture then states that
3486:-adique (d'après G. Faltings, J.-M. Fontaine et al.) Exp. 726",
675:{\displaystyle D_{B}(V)=(B\otimes _{\mathbf {Q} _{p}}V)^{G_{K}}}
533:
Period rings and comparison isomorphisms in arithmetic geometry
3273:
592:
and some linear algebraic structure and to consider so-called
1752:) contained the same information. Both are equivalent to the
708:. In particular, they are vector spaces over the fixed field
3400:, vol. I, Berlin: Walter de Gruyter GmbH & Co. KG,
2488:{\displaystyle D_{B_{\ast }}(V)=H_{\mathrm {dR} }^{i}(X/K).}
3392:
Berger, Laurent (2004), "An introduction to the theory of
1771:-adic fields. This suggested relation became known as the
1041:, there is a classical comparison isomorphism between the
3197:-adic Galois representations", in Igusa, Jun-Ichi (ed.),
1670:
is its associated graded). This conjecture was proved by
2375:
Upon comparing these two conjectures with the notion of
2382:-admissible representations above, it is seen that if
1663:{\displaystyle \mathrm {gr} H_{\mathrm {dR} }^{\ast }}
27:
is a theory that provides a way to classify and study
2880:
2570:
2415:
2309:
2087:
1819:
1631:
1621:-action (the de Rham cohomology is equipped with the
1415:
1335:
1077:
894:
801:
714:
604:
277:
205:
245:{\displaystyle \mathrm {Rep} _{\mathbf {Q} _{p}}(K)}
108:
introduced many of the basic concepts of the field.
2299:-action, and filtration after extending scalars to
2900:
2752:
2487:
2349:
2281:
1994:
1662:
1598:
1395:
1192:This isomorphism can be obtained by considering a
1180:
991:
871:
740:
674:
431:
244:
3488:Séminaire Bourbaki. Vol. 1989/90. Exposés 715–729
1695:reformulated a theorem of Tate's to say that the
3271:(1971), "Groupes de Barsotti–Tate et cristaux",
3226:(1982), "Sur certains types de représentations
3199:Algebraic analysis, geometry, and number theory
2770:-action, filtration after extending scalars to
2398:-adic Galois representation obtained as is its
748:. This construction fits into the formalism of
3447:CMI Summer School notes on p-adic Hodge theory
2350:{\displaystyle H_{\mathrm {dR} }^{\ast }(X/K)}
256:-adic Hodge theory provides subcollections of
96:-adic Galois representations arising from the
697:-adic representation) which no longer have a
8:
3156:Journal of the American Mathematical Society
2989:
762:(for ∗ = HT, dR, st, cris), the category of
3123:Proceedings of a Conference on Local Fields
518:) (with equality when the residue field of
3326:: CS1 maint: location missing publisher (
1748:) (with the action of the Galois group of
3405:
2890:
2885:
2879:
2741:
2736:
2722:
2716:
2700:
2684:
2683:
2682:
2681:
2669:
2664:
2662:
2648:
2647:
2629:
2617:
2608:
2607:
2595:
2590:
2576:
2575:
2569:
2471:
2459:
2450:
2449:
2425:
2420:
2414:
2336:
2324:
2315:
2314:
2308:
2270:
2265:
2251:
2245:
2229:
2213:
2212:
2211:
2210:
2198:
2193:
2191:
2171:
2170:
2152:
2140:
2131:
2130:
2118:
2113:
2093:
2092:
2086:
1983:
1978:
1964:
1958:
1942:
1926:
1925:
1924:
1923:
1911:
1906:
1904:
1890:
1889:
1871:
1859:
1850:
1849:
1839:
1825:
1824:
1818:
1654:
1645:
1644:
1632:
1630:
1587:
1582:
1568:
1562:
1546:
1530:
1529:
1528:
1527:
1515:
1510:
1508:
1494:
1493:
1475:
1463:
1454:
1453:
1441:
1435:
1421:
1420:
1414:
1378:
1373:
1365:
1358:
1341:
1340:
1334:
1236:between algebraic de Rham cohomology and
1203:in the algebraic de Rham cohomology over
1170:
1163:
1162:
1150:
1139:
1124:
1109:
1104:
1092:
1083:
1082:
1076:
978:
973:
971:
961:
937:
932:
922:
912:
899:
893:
855:
850:
848:
824:
819:
806:
800:
730:
725:
713:
664:
659:
644:
639:
637:
609:
603:
409:
404:
402:
374:
346:
318:
283:
282:
276:
225:
220:
218:
207:
204:
3047:
3036:
3024:
3012:
3001:
2978:
2817:
2066:. He conjectured the following (called C
1403:. Then there is a functorial isomorphism
2786:
1799:and conjectured the following (called C
464:potentially crystalline representations
3319:
2829:These rings depend on the local field
476:potentially semistable representations
458:, Hodge–Tate representations, and all
3312:(1967), "Résumé des cours, 1965–66",
3070:
3059:
2967:
2372:conjectures were proved by Faltings.
774:) mentioned above is the category of
7:
3193:(1989), "Crystalline cohomology and
124:be a local field with residue field
56:). The theory has its beginnings in
2406:-adic étale cohomology group, then
2695:
2686:
2652:
2649:
2612:
2609:
2580:
2577:
2454:
2451:
2319:
2316:
2224:
2215:
2181:
2178:
2175:
2172:
2135:
2132:
2103:
2100:
2097:
2094:
1937:
1928:
1894:
1891:
1854:
1851:
1829:
1826:
1649:
1646:
1636:
1633:
1541:
1532:
1498:
1495:
1458:
1455:
1445:
1442:
1425:
1422:
1345:
1342:
1087:
1084:
522:is finite, a statement called the
293:
290:
287:
284:
214:
211:
208:
14:
3398:Geometric aspects of Dwork theory
3275:, vol. 1, pp. 431–436,
502:) generally strictly contains Rep
2763:as vector spaces with φ-action,
2737:
2665:
2542:(and fixing an extension of the
2292:as vector spaces with φ-action,
2266:
2194:
1979:
1907:
1583:
1511:
1374:
1366:
1171:
1164:
1151:
1140:
1110:
974:
851:
640:
405:
221:
3522:Representation theory of groups
2386:is a proper smooth scheme over
2070:) for any smooth proper scheme
2005:as filtered vector spaces with
1803:) for any smooth proper scheme
1219:, and from this point of view,
3125:, Springer, pp. 158–183,
2747:
2706:
2637:
2623:
2549:), and a "monodromy operator"
2517:. Fontaine constructed a ring
2479:
2465:
2439:
2433:
2344:
2330:
2276:
2235:
2160:
2146:
2019:could be said to contain all (
1989:
1948:
1879:
1865:
1593:
1552:
1483:
1469:
1390:
1384:
1155:
1144:
1136:
1130:
1114:
1098:
954:
951:
945:
838:
832:
656:
627:
621:
615:
426:
420:
392:
386:
364:
358:
336:
330:
308:
302:
239:
233:
1:
3314:Annuaire du Collège de France
159:representation ρ :
41:with residual characteristic
3131:10.1007/978-3-642-87942-5_12
2727:
2357:is given its structure as a
2256:
1969:
1573:
1240:-adic étale cohomology (the
1043:algebraic de Rham cohomology
741:{\displaystyle E:=B^{G_{K}}}
32:-adic Galois representations
1792:whose associated graded is
1683:with good reduction over a
753:-admissible representations
448:crystalline representations
442:where each collection is a
3548:
2559:log-crystalline cohomology
2390:(with good reduction) and
1323:-adic cyclotomic character
510:), and is contained in Rep
452:semistable representations
112:General classification of
84:theories analogous to the
3100:p-adic Teichmüller theory
2901:{\displaystyle B^{G_{K}}}
2774:, and monodromy operator
766:-adic representations Rep
191:-adic representations of
74:Hodge–Tate representation
3396:-adic representations",
3342:Inventiones Mathematicae
3335:Tsuji, Takeshi (1999), "
2802:discrete valuation field
1329:is an integer), and let
537:The general strategy of
187:. The collection of all
174:is a finite-dimensional
3512:Algebraic number theory
3316:, Paris, pp. 49–58
3269:Grothendieck, Alexander
2804:whose residue field is
1731:-adic étale cohomology
1679:For an abelian variety
527:-adic monodromy theorem
456:de Rham representations
3290:Compositio Mathematica
3050:, Exposé II, section 3
2930:, respectively, where
2902:
2754:
2489:
2351:
2283:
2036:ring of p-adic periods
2012:-action. In this way,
1996:
1697:crystalline cohomology
1693:Alexander Grothendieck
1664:
1600:
1397:
1182:
993:
882:or, equivalently, the
873:
788:-adic representations
742:
689:is a period ring, and
676:
433:
246:
3363:10.1007/s002220050330
3233:Annals of Mathematics
3154:-adic Hodge theory",
3121:-Divisible Groups"",
3095:Hodge-Arakelov theory
2903:
2755:
2515:semi-stable reduction
2509:, this time allowing
2490:
2352:
2284:
1997:
1665:
1601:
1398:
1248:). Specifically, let
1242:Hodge–Tate conjecture
1183:
994:
874:
743:
677:
434:
247:
149:absolute Galois group
134:p-adic representation
132:. In this article, a
116:-adic representations
3117:Tate, John (1967), "
2878:
2568:
2413:
2307:
2085:
2078:with good reduction
2034:obtains its name of
1817:
1629:
1612:graded vector spaces
1413:
1333:
1291:where the action of
1075:
892:
799:
712:
602:
494:); additionally, Rep
275:
203:
3532:Arithmetic geometry
3464:Fontaine, Jean-Marc
3416:2002math.....10184B
3355:1999InMat.137..233T
3224:Fontaine, Jean-Marc
2793:In this article, a
2705:
2622:
2464:
2329:
2234:
2145:
1947:
1864:
1659:
1551:
1468:
1097:
1055:singular cohomology
884:comparison morphism
581:which have both an
86:Hodge decomposition
3468:Périodes p-adiques
3310:Serre, Jean-Pierre
2898:
2750:
2677:
2603:
2485:
2445:
2347:
2310:
2279:
2206:
2126:
1992:
1919:
1845:
1773:mysterious functor
1660:
1640:
1596:
1523:
1449:
1393:
1201:differential forms
1178:
1078:
989:
869:
738:
672:
429:
242:
128:of characteristic
106:Jean-Marc Fontaine
72:and the notion of
25:-adic Hodge theory
3440:Brinon, Olivier;
3425:978-3-11-017478-6
3386:Secondary sources
3208:978-0-8018-3841-5
3140:978-3-642-87942-5
3027:, Conjecture A.11
2990:Grothendieck 1971
2730:
2692:
2259:
2221:
1972:
1934:
1576:
1538:
1263:algebraic closure
784:ones, i.e. those
594:Dieudonné modules
264:to categories of
262:faithful functors
252:in this article.
88:, hence the name
70:abelian varieties
58:Jean-Pierre Serre
3539:
3498:
3478:
3459:
3458:
3457:
3452:
3436:
3409:
3381:
3331:
3325:
3317:
3305:
3283:
3264:
3219:
3186:
3143:
3073:
3068:
3062:
3057:
3051:
3045:
3039:
3034:
3028:
3022:
3016:
3015:, Conjecture A.6
3010:
3004:
2999:
2993:
2987:
2981:
2976:
2970:
2964:
2958:
2907:
2905:
2904:
2899:
2897:
2896:
2895:
2894:
2840:
2834:
2827:
2821:
2815:
2809:
2791:
2759:
2757:
2756:
2751:
2746:
2745:
2740:
2731:
2723:
2721:
2720:
2704:
2699:
2698:
2694:
2693:
2685:
2676:
2675:
2674:
2673:
2668:
2657:
2656:
2655:
2633:
2621:
2616:
2615:
2602:
2601:
2600:
2599:
2585:
2584:
2583:
2494:
2492:
2491:
2486:
2475:
2463:
2458:
2457:
2432:
2431:
2430:
2429:
2356:
2354:
2353:
2348:
2340:
2328:
2323:
2322:
2288:
2286:
2285:
2280:
2275:
2274:
2269:
2260:
2252:
2250:
2249:
2233:
2228:
2227:
2223:
2222:
2214:
2205:
2204:
2203:
2202:
2197:
2186:
2185:
2184:
2156:
2144:
2139:
2138:
2125:
2124:
2123:
2122:
2108:
2107:
2106:
2001:
1999:
1998:
1993:
1988:
1987:
1982:
1973:
1965:
1963:
1962:
1946:
1941:
1940:
1936:
1935:
1927:
1918:
1917:
1916:
1915:
1910:
1899:
1898:
1897:
1875:
1863:
1858:
1857:
1844:
1843:
1834:
1833:
1832:
1757:-divisible group
1669:
1667:
1666:
1661:
1658:
1653:
1652:
1639:
1623:Hodge filtration
1605:
1603:
1602:
1597:
1592:
1591:
1586:
1577:
1569:
1567:
1566:
1550:
1545:
1544:
1540:
1539:
1531:
1522:
1521:
1520:
1519:
1514:
1503:
1502:
1501:
1479:
1467:
1462:
1461:
1448:
1440:
1439:
1430:
1429:
1428:
1402:
1400:
1399:
1394:
1383:
1382:
1377:
1371:
1370:
1369:
1350:
1349:
1348:
1318:(where χ is the
1187:
1185:
1184:
1179:
1174:
1169:
1168:
1167:
1154:
1143:
1129:
1128:
1113:
1108:
1096:
1091:
1090:
1015:complex geometry
998:
996:
995:
990:
985:
984:
983:
982:
977:
966:
965:
944:
943:
942:
941:
927:
926:
917:
916:
904:
903:
878:
876:
875:
870:
862:
861:
860:
859:
854:
831:
830:
829:
828:
811:
810:
747:
745:
744:
739:
737:
736:
735:
734:
681:
679:
678:
673:
671:
670:
669:
668:
651:
650:
649:
648:
643:
614:
613:
444:full subcategory
438:
436:
435:
430:
416:
415:
414:
413:
408:
382:
381:
354:
353:
326:
325:
298:
297:
296:
266:linear algebraic
251:
249:
248:
243:
232:
231:
230:
229:
224:
217:
197:abelian category
98:étale cohomology
36:characteristic 0
3547:
3546:
3542:
3541:
3540:
3538:
3537:
3536:
3502:
3501:
3481:
3462:
3455:
3453:
3450:
3439:
3426:
3391:
3388:
3334:
3318:
3308:
3286:
3267:
3246:10.2307/2007012
3222:
3209:
3189:
3168:10.2307/1990970
3146:
3141:
3116:
3113:
3111:Primary sources
3108:
3090:Arakelov theory
3081:
3076:
3069:
3065:
3058:
3054:
3046:
3042:
3035:
3031:
3023:
3019:
3011:
3007:
3000:
2996:
2988:
2984:
2977:
2973:
2965:
2961:
2936:
2929:
2922:
2886:
2881:
2876:
2875:
2873:
2866:
2859:
2852:
2841:
2837:
2828:
2824:
2816:
2812:
2792:
2788:
2784:
2768:
2735:
2712:
2663:
2658:
2643:
2591:
2586:
2571:
2566:
2565:
2547:-adic logarithm
2537:
2529:
2523:
2508:
2421:
2416:
2411:
2410:
2381:
2371:
2367:
2363:
2305:
2304:
2297:
2264:
2241:
2192:
2187:
2166:
2114:
2109:
2088:
2083:
2082:
2069:
2061:
2053:
2047:
2033:
2018:
2010:
1977:
1954:
1905:
1900:
1885:
1835:
1820:
1815:
1814:
1802:
1798:
1791:
1747:
1722:
1627:
1626:
1619:
1581:
1558:
1509:
1504:
1489:
1431:
1416:
1411:
1410:
1372:
1354:
1336:
1331:
1330:
1296:
1290:
1277:
1256:
1247:
1244:, also called C
1158:
1120:
1073:
1072:
972:
967:
957:
933:
928:
918:
908:
895:
890:
889:
849:
844:
820:
815:
802:
797:
796:
781:
769:
761:
726:
721:
710:
709:
702:
660:
655:
638:
633:
605:
600:
599:
590:
579:
570:
561:
552:
535:
513:
505:
497:
489:
481:
469:
403:
398:
370:
342:
314:
278:
273:
272:
219:
206:
201:
200:
186:
164:
145:
118:
54:
12:
11:
5:
3545:
3543:
3535:
3534:
3529:
3524:
3519:
3514:
3504:
3503:
3500:
3499:
3479:
3466:, ed. (1994),
3460:
3437:
3424:
3387:
3384:
3383:
3382:
3349:(2): 233–411,
3332:
3306:
3297:(3): 241–260,
3284:
3265:
3240:(3): 529–577,
3220:
3207:
3191:Faltings, Gerd
3187:
3162:(1): 255–299,
3148:Faltings, Gerd
3144:
3139:
3112:
3109:
3107:
3104:
3103:
3102:
3097:
3092:
3087:
3080:
3077:
3075:
3074:
3063:
3052:
3040:
3029:
3017:
3005:
2994:
2982:
2971:
2959:
2947:fraction field
2934:
2927:
2920:
2893:
2889:
2884:
2871:
2864:
2857:
2850:
2835:
2822:
2810:
2785:
2783:
2780:
2766:
2761:
2760:
2749:
2744:
2739:
2734:
2729:
2726:
2719:
2715:
2711:
2708:
2703:
2697:
2691:
2688:
2680:
2672:
2667:
2661:
2654:
2651:
2646:
2642:
2639:
2636:
2632:
2628:
2625:
2620:
2614:
2611:
2606:
2598:
2594:
2589:
2582:
2579:
2574:
2535:
2527:
2521:
2506:
2496:
2495:
2484:
2481:
2478:
2474:
2470:
2467:
2462:
2456:
2453:
2448:
2444:
2441:
2438:
2435:
2428:
2424:
2419:
2379:
2369:
2365:
2361:
2346:
2343:
2339:
2335:
2332:
2327:
2321:
2318:
2313:
2295:
2290:
2289:
2278:
2273:
2268:
2263:
2258:
2255:
2248:
2244:
2240:
2237:
2232:
2226:
2220:
2217:
2209:
2201:
2196:
2190:
2183:
2180:
2177:
2174:
2169:
2165:
2162:
2159:
2155:
2151:
2148:
2143:
2137:
2134:
2129:
2121:
2117:
2112:
2105:
2102:
2099:
2096:
2091:
2067:
2059:
2051:
2045:
2031:
2016:
2008:
2003:
2002:
1991:
1986:
1981:
1976:
1971:
1968:
1961:
1957:
1953:
1950:
1945:
1939:
1933:
1930:
1922:
1914:
1909:
1903:
1896:
1893:
1888:
1884:
1881:
1878:
1874:
1870:
1867:
1862:
1856:
1853:
1848:
1842:
1838:
1831:
1828:
1823:
1800:
1796:
1789:
1777:
1776:
1759:associated to
1743:
1718:
1676:
1675:
1657:
1651:
1648:
1643:
1638:
1635:
1617:
1608:
1607:
1606:
1595:
1590:
1585:
1580:
1575:
1572:
1565:
1561:
1557:
1554:
1549:
1543:
1537:
1534:
1526:
1518:
1513:
1507:
1500:
1497:
1492:
1488:
1485:
1482:
1478:
1474:
1471:
1466:
1460:
1457:
1452:
1447:
1444:
1438:
1434:
1427:
1424:
1419:
1405:
1404:
1392:
1389:
1386:
1381:
1376:
1368:
1364:
1361:
1357:
1353:
1347:
1344:
1339:
1294:
1286:
1273:
1252:
1245:
1225:
1224:
1190:
1189:
1188:
1177:
1173:
1166:
1161:
1157:
1153:
1149:
1146:
1142:
1138:
1135:
1132:
1127:
1123:
1119:
1116:
1112:
1107:
1103:
1100:
1095:
1089:
1086:
1081:
1067:
1066:
1000:
999:
988:
981:
976:
970:
964:
960:
956:
953:
950:
947:
940:
936:
931:
925:
921:
915:
911:
907:
902:
898:
880:
879:
868:
865:
858:
853:
847:
843:
840:
837:
834:
827:
823:
818:
814:
809:
805:
779:
767:
759:
733:
729:
724:
720:
717:
700:
683:
682:
667:
663:
658:
654:
647:
642:
636:
632:
629:
626:
623:
620:
617:
612:
608:
588:
577:
568:
559:
550:
534:
531:
511:
503:
495:
487:
479:
467:
440:
439:
428:
425:
422:
419:
412:
407:
401:
397:
394:
391:
388:
385:
380:
377:
373:
369:
366:
363:
360:
357:
352:
349:
345:
341:
338:
335:
332:
329:
324:
321:
317:
313:
310:
307:
304:
301:
295:
292:
289:
286:
281:
241:
238:
235:
228:
223:
216:
213:
210:
182:
162:
143:
117:
110:
50:
13:
10:
9:
6:
4:
3:
2:
3544:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3517:Galois theory
3515:
3513:
3510:
3509:
3507:
3497:
3493:
3489:
3485:
3480:
3477:
3473:
3469:
3465:
3461:
3449:
3448:
3443:
3442:Conrad, Brian
3438:
3435:
3431:
3427:
3421:
3417:
3413:
3408:
3403:
3399:
3395:
3390:
3389:
3385:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3343:
3338:
3333:
3329:
3323:
3315:
3311:
3307:
3304:
3300:
3296:
3292:
3291:
3285:
3282:
3278:
3274:
3270:
3266:
3263:
3259:
3255:
3251:
3247:
3243:
3239:
3235:
3234:
3229:
3225:
3221:
3218:
3214:
3210:
3204:
3200:
3196:
3192:
3188:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3142:
3136:
3132:
3128:
3124:
3120:
3115:
3114:
3110:
3105:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3082:
3078:
3072:
3067:
3064:
3061:
3056:
3053:
3049:
3048:Fontaine 1994
3044:
3041:
3038:
3037:Faltings 1989
3033:
3030:
3026:
3025:Fontaine 1982
3021:
3018:
3014:
3013:Fontaine 1982
3009:
3006:
3003:
3002:Fontaine 1982
2998:
2995:
2992:, p. 435
2991:
2986:
2983:
2980:
2979:Faltings 1988
2975:
2972:
2969:
2963:
2960:
2956:
2952:
2948:
2944:
2940:
2933:
2926:
2919:
2915:
2911:
2891:
2887:
2882:
2870:
2863:
2856:
2849:
2845:
2839:
2836:
2832:
2826:
2823:
2820:, p. 114
2819:
2818:Fontaine 1994
2814:
2811:
2807:
2803:
2800:
2796:
2790:
2787:
2781:
2779:
2777:
2773:
2769:
2742:
2732:
2724:
2717:
2713:
2709:
2701:
2689:
2678:
2670:
2659:
2644:
2640:
2634:
2630:
2626:
2618:
2604:
2596:
2592:
2587:
2572:
2564:
2563:
2562:
2560:
2556:
2552:
2548:
2546:
2541:
2534:
2530:
2520:
2516:
2512:
2503:
2501:
2482:
2476:
2472:
2468:
2460:
2446:
2442:
2436:
2426:
2422:
2417:
2409:
2408:
2407:
2405:
2401:
2397:
2393:
2389:
2385:
2378:
2373:
2360:
2341:
2337:
2333:
2325:
2311:
2302:
2298:
2271:
2261:
2253:
2246:
2242:
2238:
2230:
2218:
2207:
2199:
2188:
2167:
2163:
2157:
2153:
2149:
2141:
2127:
2119:
2115:
2110:
2089:
2081:
2080:
2079:
2077:
2073:
2065:
2058:
2054:
2044:
2039:
2037:
2030:
2026:
2022:
2015:
2011:
1984:
1974:
1966:
1959:
1955:
1951:
1943:
1931:
1920:
1912:
1901:
1886:
1882:
1876:
1872:
1868:
1860:
1846:
1840:
1836:
1821:
1813:
1812:
1811:
1810:
1806:
1795:
1788:
1784:
1783:
1774:
1770:
1766:
1762:
1758:
1756:
1751:
1746:
1742:
1738:
1734:
1730:
1726:
1721:
1717:
1713:
1709:
1705:
1701:
1698:
1694:
1690:
1686:
1682:
1678:
1677:
1673:
1672:Gerd Faltings
1655:
1641:
1624:
1620:
1613:
1609:
1588:
1578:
1570:
1563:
1559:
1555:
1547:
1535:
1524:
1516:
1505:
1490:
1486:
1480:
1476:
1472:
1464:
1450:
1436:
1432:
1417:
1409:
1408:
1407:
1406:
1387:
1379:
1362:
1359:
1355:
1351:
1337:
1328:
1324:
1322:
1317:
1313:
1309:
1305:
1301:
1297:
1289:
1285:
1281:
1276:
1272:
1268:
1264:
1260:
1255:
1251:
1243:
1239:
1235:
1231:
1227:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1199:
1195:
1191:
1175:
1159:
1147:
1133:
1125:
1121:
1117:
1105:
1101:
1093:
1079:
1071:
1070:
1069:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1039:
1034:
1031:
1028:
1024:
1020:
1019:
1018:
1016:
1012:
1007:
1005:
986:
979:
968:
962:
958:
948:
938:
934:
929:
923:
919:
913:
909:
905:
900:
896:
888:
887:
886:
885:
866:
863:
856:
845:
841:
835:
825:
821:
816:
812:
807:
803:
795:
794:
793:
791:
787:
783:
778:
773:
765:
758:
754:
752:
731:
727:
722:
718:
715:
707:
703:
696:
692:
688:
665:
661:
652:
645:
634:
630:
624:
618:
610:
606:
598:
597:
596:
595:
591:
584:
580:
576:
571:
567:
562:
558:
553:
549:
544:
540:
532:
530:
528:
526:
521:
517:
509:
501:
493:
485:
477:
473:
465:
461:
457:
453:
449:
445:
423:
417:
410:
399:
395:
389:
383:
378:
375:
371:
367:
361:
355:
350:
347:
343:
339:
333:
327:
322:
319:
315:
311:
305:
299:
279:
271:
270:
269:
267:
263:
259:
255:
236:
226:
198:
194:
190:
185:
181:
177:
173:
169:
165:
158:
154:
150:
146:
139:
135:
131:
127:
123:
115:
111:
109:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
53:
49:
44:
40:
37:
33:
31:
26:
24:
19:
3527:Hodge theory
3487:
3483:
3467:
3454:, retrieved
3446:
3407:math/0210184
3397:
3393:
3346:
3340:
3336:
3313:
3294:
3288:
3272:
3237:
3231:
3227:
3198:
3194:
3159:
3155:
3151:
3122:
3118:
3085:Hodge theory
3066:
3055:
3043:
3032:
3020:
3008:
2997:
2985:
2974:
2962:
2954:
2951:Witt vectors
2942:
2938:
2931:
2924:
2917:
2913:
2909:
2868:
2861:
2854:
2847:
2843:
2838:
2830:
2825:
2813:
2794:
2789:
2775:
2771:
2764:
2762:
2554:
2550:
2544:
2539:
2532:
2525:
2518:
2510:
2504:
2499:
2497:
2403:
2399:
2395:
2391:
2387:
2383:
2376:
2374:
2358:
2300:
2293:
2291:
2075:
2071:
2063:
2056:
2049:
2042:
2040:
2035:
2028:
2024:
2020:
2013:
2006:
2004:
1808:
1804:
1793:
1786:
1780:
1778:
1772:
1768:
1764:
1760:
1754:
1749:
1744:
1740:
1736:
1732:
1728:
1724:
1719:
1715:
1711:
1707:
1703:
1699:
1688:
1687:-adic field
1684:
1680:
1615:
1326:
1320:
1315:
1311:
1307:
1303:
1299:
1292:
1287:
1283:
1279:
1274:
1270:
1266:
1253:
1249:
1237:
1233:
1229:
1220:
1216:
1196:obtained by
1062:
1058:
1050:
1046:
1036:
1022:
1008:
1001:
881:
789:
785:
776:
771:
763:
756:
750:
705:
698:
694:
690:
686:
684:
593:
586:
574:
565:
556:
547:
543:period rings
542:
538:
536:
524:
519:
515:
507:
499:
491:
483:
471:
459:
441:
257:
253:
192:
188:
183:
179:
176:vector space
171:
167:
160:
155:) will be a
152:
141:
137:
133:
129:
125:
121:
119:
113:
93:
89:
77:
66:Tate modules
64:'s study of
51:
47:
42:
39:local fields
29:
22:
21:
15:
2795:local field
1198:integrating
1004:isomorphism
782:-admissible
18:mathematics
3506:Categories
3456:2010-02-05
3106:References
3071:Tsuji 1999
3060:Hyodo 1991
2968:Serre 1967
1727:) and the
1259:completion
1011:arithmetic
792:for which
474:) and the
157:continuous
82:cohomology
3379:121547567
3150:(1988), "
2728:¯
2714:×
2702:∗
2690:´
2660:⊗
2641:≅
2619:∗
2588:⊗
2427:∗
2368:and the C
2326:∗
2257:¯
2243:×
2231:∗
2219:´
2189:⊗
2164:≅
2142:∗
2111:⊗
1970:¯
1956:×
1944:∗
1932:´
1902:⊗
1883:≅
1861:∗
1837:⊗
1656:∗
1574:¯
1560:×
1548:∗
1536:´
1506:⊗
1487:≅
1465:∗
1433:⊗
1363:∈
1356:⊕
1282:) denote
1160:⊗
1126:∗
1118:≅
1094:∗
969:⊗
963:∗
955:⟶
939:∗
920:⊗
914:∗
897:α
864:
826:∗
813:
635:⊗
418:
396:⊊
384:
368:⊊
356:
340:⊊
328:
312:⊊
300:
170:), where
102:varieties
62:John Tate
45:(such as
3444:(2009),
3322:citation
3079:See also
2945:)), the
2799:complete
2513:to have
1782:filtered
1213:tensored
1053:and the
545:such as
199:denoted
195:form an
3496:1099881
3476:1293969
3434:2023292
3412:Bibcode
3371:1705837
3351:Bibcode
3303:1106296
3281:0578496
3262:0657238
3254:2007012
3217:1463696
3184:0924705
3176:1990970
2949:of the
2937:= Frac(
2806:perfect
2553:. When
2394:is the
1298:is via
1257:be the
1194:pairing
685:(where
140:(or of
3494:
3474:
3432:
3422:
3377:
3369:
3301:
3279:
3260:
3252:
3215:
3205:
3182:
3174:
3137:
2923:, and
2867:, and
2303:(here
1625:, and
1325:, and
1269:, let
1261:of an
1209:period
1205:cycles
1033:scheme
1030:smooth
1027:proper
1002:is an
583:action
572:, and
147:, the
80:-adic
3451:(PDF)
3402:arXiv
3375:S2CID
3250:JSTOR
3172:JSTOR
2782:Notes
2524:with
2074:over
2048:with
1807:over
1785:ring
1714:)) ⊗
1614:with
1232:over
1049:over
1035:over
1025:is a
693:is a
468:pcris
178:over
166:→ GL(
3420:ISBN
3328:link
3203:ISBN
3135:ISBN
2966:See
2872:cris
2842:For
2370:cris
2068:cris
2046:cris
1306:= χ(
1013:and
569:cris
488:cris
120:Let
60:and
3359:doi
3347:137
3242:doi
3238:115
3164:doi
3127:doi
2953:of
2908:is
2797:is
2538:to
2402:th
2062:to
1610:of
1265:of
1215:to
1057:of
1045:of
1021:If
846:dim
804:dim
585:by
529:).
496:pst
480:pst
478:Rep
466:Rep
400:Rep
372:Rep
344:Rep
316:Rep
280:Rep
151:of
136:of
100:of
68:of
34:of
16:In
3508::
3492:MR
3472:MR
3430:MR
3428:,
3418:,
3410:,
3373:,
3367:MR
3365:,
3357:,
3345:,
3324:}}
3320:{{
3299:MR
3295:78
3293:,
3277:MR
3258:MR
3256:,
3248:,
3236:,
3213:MR
3211:,
3180:MR
3178:,
3170:,
3158:,
3133:,
2916:,
2912:,
2874:,
2865:st
2860:,
2858:dR
2853:,
2851:HT
2846:=
2522:st
2507:st
2502:.
2366:dR
2038:.
2032:dR
2017:dR
1801:dR
1797:HT
1790:dR
1691:,
1352::=
1246:HT
1017::
1006:.
719::=
578:HT
563:,
560:st
554:,
551:dR
512:dR
504:st
454:,
450:,
104:.
20:,
3484:p
3414::
3404::
3394:p
3361::
3353::
3337:p
3330:)
3244::
3228:p
3195:p
3166::
3160:1
3152:p
3129::
3119:p
2957:.
2955:k
2943:k
2941:(
2939:W
2935:0
2932:K
2928:0
2925:K
2921:0
2918:K
2914:K
2910:K
2892:K
2888:G
2883:B
2869:B
2862:B
2855:B
2848:B
2844:B
2831:K
2808:.
2776:N
2772:K
2767:K
2765:G
2748:)
2743:p
2738:Q
2733:,
2725:K
2718:K
2710:X
2707:(
2696:t
2687:e
2679:H
2671:p
2666:Q
2653:t
2650:s
2645:B
2638:)
2635:K
2631:/
2627:X
2624:(
2613:R
2610:d
2605:H
2597:0
2593:K
2581:t
2578:s
2573:B
2555:X
2551:N
2545:p
2540:K
2536:0
2533:K
2528:K
2526:G
2519:B
2511:X
2500:V
2483:.
2480:)
2477:K
2473:/
2469:X
2466:(
2461:i
2455:R
2452:d
2447:H
2443:=
2440:)
2437:V
2434:(
2423:B
2418:D
2404:p
2400:i
2396:p
2392:V
2388:K
2384:X
2380:∗
2377:B
2362:0
2359:K
2345:)
2342:K
2338:/
2334:X
2331:(
2320:R
2317:d
2312:H
2301:K
2296:K
2294:G
2277:)
2272:p
2267:Q
2262:,
2254:K
2247:K
2239:X
2236:(
2225:t
2216:e
2208:H
2200:p
2195:Q
2182:s
2179:i
2176:r
2173:c
2168:B
2161:)
2158:K
2154:/
2150:X
2147:(
2136:R
2133:d
2128:H
2120:0
2116:K
2104:s
2101:i
2098:r
2095:c
2090:B
2076:K
2072:X
2064:K
2060:0
2057:K
2052:K
2050:G
2043:B
2029:B
2025:p
2021:p
2014:B
2009:K
2007:G
1990:)
1985:p
1980:Q
1975:,
1967:K
1960:K
1952:X
1949:(
1938:t
1929:e
1921:H
1913:p
1908:Q
1895:R
1892:d
1887:B
1880:)
1877:K
1873:/
1869:X
1866:(
1855:R
1852:d
1847:H
1841:K
1830:R
1827:d
1822:B
1809:K
1805:X
1794:B
1787:B
1775:.
1769:p
1765:p
1761:X
1755:p
1750:K
1745:p
1741:Q
1739:,
1737:X
1735:(
1733:H
1729:p
1725:K
1720:p
1716:Q
1712:k
1710:(
1708:W
1706:/
1704:X
1702:(
1700:H
1689:K
1685:p
1681:X
1650:R
1647:d
1642:H
1637:r
1634:g
1618:K
1616:G
1594:)
1589:p
1584:Q
1579:,
1571:K
1564:K
1556:X
1553:(
1542:t
1533:e
1525:H
1517:p
1512:Q
1499:T
1496:H
1491:B
1484:)
1481:K
1477:/
1473:X
1470:(
1459:R
1456:d
1451:H
1446:r
1443:g
1437:K
1426:T
1423:H
1418:B
1391:)
1388:i
1385:(
1380:K
1375:C
1367:Z
1360:i
1346:T
1343:H
1338:B
1327:i
1321:p
1316:z
1314:·
1312:g
1310:)
1308:g
1304:z
1302:·
1300:g
1295:K
1293:G
1288:K
1284:C
1280:i
1278:(
1275:K
1271:C
1267:K
1254:K
1250:C
1238:p
1234:K
1230:X
1221:C
1217:C
1176:.
1172:C
1165:Q
1156:)
1152:Q
1148:,
1145:)
1141:C
1137:(
1134:X
1131:(
1122:H
1115:)
1111:C
1106:/
1102:X
1099:(
1088:R
1085:d
1080:H
1065:)
1063:C
1061:(
1059:X
1051:C
1047:X
1038:C
1023:X
987:V
980:p
975:Q
959:B
952:)
949:V
946:(
935:B
930:D
924:E
910:B
906::
901:V
867:V
857:p
852:Q
842:=
839:)
836:V
833:(
822:B
817:D
808:E
790:V
786:p
780:∗
777:B
772:K
770:(
768:∗
764:p
760:∗
757:B
751:B
732:K
728:G
723:B
716:E
706:B
701:K
699:G
695:p
691:V
687:B
666:K
662:G
657:)
653:V
646:p
641:Q
631:B
628:(
625:=
622:)
619:V
616:(
611:B
607:D
589:K
587:G
575:B
566:B
557:B
548:B
539:p
525:p
520:K
516:K
514:(
508:K
506:(
500:K
498:(
492:K
490:(
484:K
482:(
472:K
470:(
460:p
427:)
424:K
421:(
411:p
406:Q
393:)
390:K
387:(
379:T
376:H
365:)
362:K
359:(
351:R
348:d
337:)
334:K
331:(
323:s
320:s
309:)
306:K
303:(
294:s
291:y
288:r
285:c
258:p
254:p
240:)
237:K
234:(
227:p
222:Q
215:p
212:e
209:R
193:K
189:p
184:p
180:Q
172:V
168:V
163:K
161:G
153:K
144:K
142:G
138:K
130:p
126:k
122:K
114:p
94:p
90:p
78:p
52:p
48:Q
43:p
30:p
23:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.