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One can extend the
Hilbert modular group by a group of order 2, acting on the Hilbert modular group via the Galois action, and exchanging the two copies of the upper half plane.
462:
819:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}(z_{1},z_{2})=\left({\frac {az_{1}+bz_{2}}{cz_{1}+dz_{2}}},{\frac {a'z_{1}+b'z_{2}}{c'z_{1}+d'z_{2}}}\right)}
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by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal.
1277:
1339:"Über vierdimensionale RIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen"
1177:
Baily, W. L.; Borel, A. (November 1966). "Compactification of
Arithmetic Quotients of Bounded Symmetric Domains".
58:
obtained by taking a quotient of a product of multiple copies of the upper half-plane by a
Hilbert modular group.
1071:. From the Bailey-Borel compactification theorem, there is an embedding of this surface into a projective space.
309:
149:); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups.
1646:
1553:; Zagier, Don (1977), "Classification of Hilbert modular surfaces", in Baily, W. L.; Shioda., T. (eds.),
1411:(1971), "The Hilbert modular group, resolution of the singularities at the cusps and related problems",
955:
1125:. London Mathematical Society Lecture Note Series (242). Cambridge University Press. pp. 127–138.
1113:; Nakamura, Hiroaki (1997). "Some illustrative examples for anabelian geometry in high dimensions". In
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1215:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin,
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The
Hilbert modular group may be replaced by some subgroup of finite index, such as a
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A short introduction to
Hilbert modular surfaces and Hirzebruch-Zagier cycles
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of the action. It is compact, and has not only the quotient singularities of
70:
61:
Hilbert modular surfaces were first described by Otto Blumenthal (
1144:
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Ven, Antonius (2004).
1123:
Geometric Galois
Actions 1: Around Grothendieck's Esquisse d'un Programme
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179:
by resolving the singularities in a minimal way. It is a compact smooth
1509:
1492:"Hilbert modular surfaces and the classification of algebraic surfaces"
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270:
942:{\displaystyle X(p)=G\backslash {\mathfrak {H}}\times {\mathfrak {H}}}
234:
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1451:
Hirzebruch, Friedrich E. P. (1973), "Hilbert modular surfaces",
532:{\displaystyle SL(2,{\mathcal {O}}_{K})/\{\pm {\text{Id}}_{2}\}}
295:
blown up at its 10 Eckardt points is a
Hilbert modular surface.
42:
obtained by taking a quotient of a product of two copies of the
1148:. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 231.
1491:
298:
121:
surfaces related to this action, any of which may be called
1417:, Lecture Notes in Math, vol. 244, Berlin, New York:
1210:
1414:
SĂ©minaire
Bourbaki, 23ème année (1970/1971), Exp. No. 396
160:
by adding a finite number of points corresponding to the
411:
obtained from compactifying a certain quotient variety
919:
588:
221:
showed how to resolve the quotient singularities, and
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566:{\displaystyle {\mathfrak {H}}\times {\mathfrak {H}}}
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198:There are several variations of this construction:
1293:"Über Modulfunktionen von mehreren Veränderlichen"
1250:"Über Modulfunktionen von mehreren Veränderlichen"
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1038:. Resolving its singularities gives the variety
254:
1031:{\displaystyle {\text{Cl}}({\mathcal {O}}_{K})}
382:there is an associated Hilbert modular variety
1069:Hilbert modular variety of the field extension
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240:
1578:
891:. The associated quotient variety is denoted
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340:{\displaystyle K=\mathbb {Q} ({\sqrt {p}})}
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69:) using some unpublished notes written by
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1557:, Tokyo: Iwanami Shoten, pp. 43–77,
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299:Associated to a quadratic field extension
1555:Complex analysis and algebraic geometry
14:
1639:
168:, but also singularities at its cusps.
113:of two copies of the upper half plane
952:and can be compactified to a variety
440:and resolving its singularities. Let
464:denote the upper half plane and let
259:classification of algebraic surfaces
233:Hilbert modular varieties cannot be
93:, then the Hilbert modular group SL
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924:
558:
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449:
24:
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992:, which are in bijection with the
981:{\displaystyle {\overline {X}}(p)}
489:
251:Hirzebruch & Van de Ven (1974)
25:
1663:
1623:
1617:
288:gives a long table of examples.
213:
183:, but is not in general minimal.
27:Algebraic surface in mathematics
1490:; Van de Ven, Antonius (1974),
457:{\displaystyle {\mathfrak {H}}}
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255:Hirzebruch & Zagier (1977)
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1579:van der Geer, Gerard (1988),
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257:identified their type in the
228:
1131:10.1017/CBO9780511758874.010
964:
7:
1453:L'Enseignement Mathématique
1074:
880:{\displaystyle a',b',c',d'}
280:
10:
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241:Classification of surfaces
36:Hilbert–Blumenthal surface
1593:10.1007/978-3-642-61553-5
1519:21.11116/0000-0004-39A4-3
1366:21.11116/0000-0004-3A47-C
1291:Blumenthal, Otto (1904),
1221:10.1007/978-3-642-57739-0
1179:The Annals of Mathematics
1154:10.1007/978-3-642-57739-0
305:quadratic field extension
1581:Hilbert modular surfaces
1500:(Submitted manuscript),
1497:Inventiones Mathematicae
1213:Compact Complex Surfaces
1146:Compact Complex Surfaces
263:surfaces of general type
123:Hilbert modular surfaces
119:birationally equivalent
73:about 10 years before.
52:Hilbert modular variety
32:Hilbert modular surface
18:Hilbert modular surface
1091:Siegel modular variety
1086:Picard modular surface
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375:{\displaystyle p=4k+1}
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1551:Hirzebruch, Friedrich
1488:Hirzebruch, Friedrich
1409:Hirzebruch, Friedrich
1344:Mathematische Annalen
1335:Hirzebruch, Friedrich
1297:Mathematische Annalen
1254:Mathematische Annalen
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48:Hilbert modular group
1421:, pp. 275–288,
1081:Hilbert modular form
1060:{\displaystyle Y(p)}
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433:{\displaystyle X(p)}
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117:. There are several
50:. More generally, a
1465:10.5169/seals-46292
286:van der Geer (1988)
261:. Most of them are
204:congruence subgroup
133:is the quotient of
1647:Algebraic surfaces
1510:10.1007/BF01405200
1427:10.1007/BFb0058707
1357:10.1007/BF01343146
1309:10.1007/BF01449486
1266:10.1007/BF01444306
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265:, but several are
30:In mathematics, a
1602:978-3-540-17601-5
1564:978-0-521-09334-7
1436:978-3-540-05720-8
1230:978-3-540-00832-3
1163:978-3-540-00832-3
1006:
967:
889:Galois conjugates
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275:elliptic surfaces
267:rational surfaces
247:Hirzebruch (1971)
223:Hirzebruch (1971)
219:Hirzebruch (1953)
190:is obtained from
181:algebraic surface
175:is obtained from
156:is obtained from
56:algebraic variety
40:algebraic surface
16:(Redirected from
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1652:Complex surfaces
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1397:on 2016-03-05
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994:ideal classes
991:
988:, called the
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71:David Hilbert
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1413:
1399:, retrieved
1395:the original
1348:
1342:
1325:, retrieved
1321:the original
1300:
1296:
1282:, retrieved
1278:the original
1257:
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1212:
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1178:
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1145:
1139:
1122:
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269:or blown up
245:The papers
244:
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197:
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187:
186:The surface
176:
172:
171:The surface
165:
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153:
152:The surface
146:
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129:The surface
114:
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51:
35:
31:
29:
1624:Ehlen, S.,
1504:(1): 1–29,
1459:: 183–281,
1351:(1): 1–22,
1067:called the
271:K3 surfaces
77:Definitions
1641:Categories
1401:2013-09-12
1327:2013-09-12
1284:2013-09-12
1185:(3): 442.
1097:References
829:where the
229:Properties
89:of a real
1528:0020-9910
1473:0013-8584
1391:122862268
1375:0025-5831
1317:179178108
1274:122293576
965:¯
930:×
920:∖
554:×
512:±
235:anabelian
1544:73577779
1337:(1953),
1248:(1903),
1121:(eds.).
1075:See also
887:are the
874:′
863:′
852:′
841:′
795:′
774:′
754:′
733:′
303:Given a
281:Examples
1611:0930101
1573:0480356
1536:0364262
1481:0393045
1445:0417187
1383:0062842
1239:2030225
1199:1970457
539:act on
85:is the
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