24:
754:
1008:. Namely, it would imply the Lefschetz standard conjecture (that the inverse of the Lefschetz isomorphism is defined by an algebraic correspondence); that the KĂĽnneth components of the diagonal are algebraic; and that numerical equivalence and homological equivalence of algebraic cycles are the same.
803:
over finitely generated fields of characteristic not 2. (On a surface, the nontrivial part of the conjecture is about divisors.) In characteristic zero, the Tate conjecture for K3 surfaces was proved by André and
Tankeev. For K3 surfaces over finite fields of characteristic not 2, the Tate conjecture
566:
953:
749:{\displaystyle {\text{Hom}}(A,B)\otimes _{\mathbf {Z} }\mathbf {Q} _{\ell }\to {\text{Hom}}_{G}\left(H_{1}\left(A_{k_{s}},\mathbf {Q} _{\ell }\right),H_{1}\left(B_{k_{s}},\mathbf {Q} _{\ell }\right)\right)}
351:
529:. Zarhin extended these results to any finitely generated base field. The Tate conjecture for divisors on abelian varieties implies the Tate conjecture for divisors on any product of curves
548:
The (known) Tate conjecture for divisors on abelian varieties is equivalent to a powerful statement about homomorphisms between abelian varieties. Namely, for any abelian varieties
871:
1401:
1205:
1416:
1005:
80:
467:
be a morphism from a smooth projective surface onto a smooth projective curve over a finite field. Suppose that the generic fiber
273:
476:
1341:
499:
124:. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the
511:
1304:
1024:
D. Ulmer. Arithmetic
Geometry over Global Function Fields (2014), 283-337. Proposition 5.1.2 and Theorem 6.3.1.
1406:
1299:, Proceedings of Symposia in Pure Mathematics, vol. 55, American Mathematical Society, pp. 71–83,
452:
1381:
510:. By contrast, the Hodge conjecture for divisors on any smooth complex projective variety is known (the
854:
428:
means a finite linear combination of subvarieties; so an equivalent statement is that every element of
1292:
1245:
1226:
105:
51:
1159:
971:
998:
178:
1411:
1314:
1277:
1257:
1238:
1178:
1142:
1122:
1095:
517:
Probably the most important known case is that the Tate conjecture is true for divisors on
387:
240:
1074:
André, Yves (1996), "On the
Shafarevich and Tate conjectures for hyper-Kähler varieties",
966:, Tate showed that the Tate conjecture plus the semisimplicity conjecture would imply the
8:
148:
1261:
1182:
1126:
121:
1281:
1213:
1194:
1168:
1157:
Madapusi Pera, K. (2013), "The Tate conjecture for K3 surfaces in odd characteristic",
1146:
1099:
526:
144:
93:
37:
1229:(1965), "Algebraic cycles and poles of zeta functions", in Schilling, O. F. G. (ed.),
948:{\displaystyle H^{i}\left(X\times _{k}{\overline {k}},\mathbf {Q} _{\ell }(n)\right).}
1337:
1300:
1198:
1150:
1103:
205:
166:
113:
70:
1362:
1329:
1265:
1186:
1130:
1083:
503:
125:
1285:
1004:
Like the Hodge conjecture, the Tate conjecture would imply most of
Grothendieck's
1328:, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, pp. 283–337,
1310:
1273:
1234:
1138:
1091:
518:
109:
1385:
865:
showed that the Tate conjecture (as stated above) implies the semisimplicity of
209:
1333:
1190:
455:(algebraic cycles of codimension 1) is a major open problem. For example, let
1395:
1321:
1110:
525:
for abelian varieties over number fields, part of
Faltings's solution of the
522:
521:. This is a theorem of Tate for abelian varieties over finite fields, and of
219:
141:
117:
89:
41:
197:
1350:
805:
768:
252:
155:
1367:
1269:
1134:
1087:
800:
375:
101:
1113:(1983), "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern",
1042:
J. Tate. Arithmetical
Algebraic Geometry (1965), 93-110. Equation (8).
1218:
993:
is equal to the rank of the group of algebraic cycles of codimension
1173:
1248:(1966), "Endomorphisms of abelian varieties over finite fields",
764:
1295:(1994), "Conjectures on algebraic cycles in â„“-adic cohomology",
826:
be a smooth projective variety over a finitely generated field
23:
378:, which means that this representation of the Galois group
346:{\displaystyle H^{2i}(V_{k_{s}},\mathbf {Q} _{\ell }(i))=W}
1051:
759:
is an isomorphism. In particular, an abelian variety
1324:(2014), "Curves and Jacobians over function fields",
874:
834:
predicts that the representation of the Galois group
569:
276:
1060:
J. Tate. Motives (1994), Part 1, 71-83. Theorem 2.9.
1033:
J. Tate. Motives (1994), Part 1, 71-83. Theorem 5.2.
1353:(2017), "Recent progress on the Tate conjecture",
947:
748:
345:
1386:The Tate conjecture over finite fields (AIM talk)
1393:
267:) determines an element of the cohomology group
1326:Arithmetic Geometry over Global Function Fields
131:
1355:Bulletin of the American Mathematical Society
1233:, New York: Harper and Row, pp. 93–110,
1156:
416:-vector space, by the classes of codimension-
116:in terms of a more computable invariant, the
814:surveys known cases of the Tate conjecture.
494:). Then the Tate conjecture for divisors on
970:, namely that the order of the pole of the
22:
1402:Topological methods of algebraic geometry
1366:
1217:
1172:
1109:
1006:standard conjectures on algebraic cycles
853:is semisimple (that is, a direct sum of
81:Standard conjectures on algebraic cycles
1394:
1349:
1204:
862:
817:
811:
808:, Charles, Madapusi Pera, and Maulik.
432:is the class of an algebraic cycle on
1320:
1073:
154:which is finitely generated over its
1291:
1244:
1225:
767:by the Galois representation on its
500:Birch and Swinnerton-Dyer conjecture
799:The Tate conjecture also holds for
13:
1417:Unsolved problems in number theory
14:
1428:
1375:
918:
726:
671:
604:
596:
556:over a finitely generated field
315:
1231:Arithmetical Algebraic Geometry
1210:A remark on the Tate conjecture
263:(understood to be defined over
218:, scalars then extended to the
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1045:
1036:
1027:
1018:
934:
928:
849:) on the â„“-adic cohomology of
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331:
325:
290:
1:
1067:
908:
475:, which is a curve over the
208:groups (coefficients in the
7:
855:irreducible representations
228:) of the base extension of
132:Statement of the conjecture
10:
1433:
405:fixed by the Galois group
1334:10.1007/978-3-0348-0853-8
1191:10.1007/s00222-014-0557-5
832:semisimplicity conjecture
397:states that the subspace
200:â„“ which is invertible in
76:
65:
57:
47:
33:
21:
1250:Inventiones Mathematicae
1160:Inventiones Mathematicae
1115:Inventiones Mathematicae
1011:
451:The Tate conjecture for
108:that would describe the
804:was proved by Nygaard,
512:Lefschetz (1,1)-theorem
968:strong Tate conjecture
949:
750:
347:
1076:Mathematische Annalen
999:numerical equivalence
950:
861:of characteristic 0,
751:
498:is equivalent to the
382:is tensored with the
371: ) denotes the
348:
179:absolute Galois group
118:Galois representation
1407:Diophantine geometry
872:
763:is determined up to
567:
388:cyclotomic character
274:
1262:1966InMat...2..134T
1183:2013arXiv1301.6326M
1127:1983InMat..73..349F
818:Related conjectures
239:; these groups are
18:
1270:10.1007/bf01404549
1135:10.1007/BF01388432
1088:10.1007/BF01444219
945:
746:
560:, the natural map
527:Mordell conjecture
486:), is smooth over
356:which is fixed by
343:
145:projective variety
94:algebraic geometry
38:Algebraic geometry
16:
1368:10.1090/bull/1588
1343:978-3-0348-0852-1
911:
621:
573:
519:abelian varieties
409:is spanned, as a
206:â„“-adic cohomology
167:separable closure
86:
85:
71:abelian varieties
28:John Tate in 1993
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122:Ă©tale cohomology
110:algebraic cycles
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426:algebraic cycle
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395:Tate conjecture
366:
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241:representations
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210:â„“-adic integers
204:. Consider the
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98:Tate conjecture
29:
17:Tate conjecture
12:
11:
5:
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1376:External links
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1373:
1372:
1361:(4): 575–590,
1357:, New Series,
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1342:
1322:Ulmer, Douglas
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1289:
1256:(2): 134–144,
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1167:(2): 625–668,
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1121:(3): 349–366,
1111:Faltings, Gerd
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58:Conjectured in
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48:Conjectured by
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972:zeta function
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863:Moonen (2017)
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812:Totaro (2017)
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90:number theory
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42:number theory
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174:
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159:
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137:
135:
97:
87:
77:Consequences
69:divisors on
1412:Conjectures
1382:James Milne
1206:Moonen, Ben
1082:: 205–248,
801:K3 surfaces
769:Tate module
447:Known cases
253:codimension
156:prime field
66:Known cases
1396:Categories
1293:Tate, John
1246:Tate, John
1227:Tate, John
1068:References
376:Tate twist
247:. For any
173:, and let
102:conjecture
100:is a 1963
1199:253746655
1174:1301.6326
1151:121049418
1104:122949797
924:ℓ
909:¯
895:×
732:ℓ
677:ℓ
615:→
610:ℓ
592:⊗
321:ℓ
106:John Tate
52:John Tate
1208:(2017),
536:Ă— ... Ă—
523:Faltings
502:for the
459: :
453:divisors
196:. Fix a
1315:1265523
1297:Motives
1278:0206004
1258:Bibcode
1239:0225778
1179:Bibcode
1143:0718935
1123:Bibcode
1096:1391213
997:modulo
857:). For
765:isogeny
360:. Here
251:≥ 0, a
177:be the
147:over a
114:variety
1340:
1313:
1303:
1286:245902
1284:
1276:
1237:
1197:
1149:
1141:
1102:
1094:
838:= Gal(
830:. The
158:. Let
142:smooth
96:, the
1282:S2CID
1214:arXiv
1195:S2CID
1169:arXiv
1147:S2CID
1100:S2CID
1012:Notes
985:) at
436:with
424:. An
192:) of
165:be a
149:field
140:be a
112:on a
34:Field
1338:ISBN
1301:ISBN
958:For
822:Let
806:Ogus
552:and
393:The
181:Gal(
136:Let
92:and
61:1963
40:and
1363:doi
1330:doi
1266:doi
1187:doi
1165:201
1131:doi
1084:doi
1080:305
796:).
620:Hom
572:Hom
514:).
506:of
471:of
401:of
243:of
232:to
169:of
120:on
104:of
88:In
1398::
1384:,
1359:54
1336:,
1311:MR
1309:,
1280:,
1274:MR
1272:,
1264:,
1252:,
1235:MR
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1193:,
1185:,
1177:,
1163:,
1145:,
1139:MR
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1129:,
1119:73
1117:,
1098:,
1092:MR
1090:,
1078:,
1001:.
989:=
981:,
789:,
545:.
463:→
390:.
128:.
1388:.
1365::
1332::
1268::
1260::
1254:2
1216::
1189::
1181::
1171::
1133::
1125::
1086::
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991:q
987:t
983:t
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975:Z
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939:)
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365:â„“
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138:V
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