931:
953:
49:
865:
909:
887:
4390:
385:
2935:. This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc.
631:
2926:
The classification scheme above is typically used by geometers. There is a different classification for
Riemann surfaces which is typically used by complex analysts. It employs a different definition for "parabolic" and "hyperbolic". In this alternative classification scheme, a Riemann surface is
2721:
with a number of punctures. With no punctures, it is the
Riemann sphere, which is elliptic. With one puncture, which can be placed at infinity, it is the complex plane, which is parabolic. With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. With
2617:-dimensional. A similar classification of Riemann surfaces of finite type (that is homeomorphic to a closed surface minus a finite number of points) can be given. However in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such a description.
2681:
but any holomorphic map from the sphere to the plane is constant, any holomorphic map from the plane into the unit disk is constant (Liouville's theorem), and in fact any holomorphic map from the plane into the plane minus two points is constant (Little Picard theorem)!
1311:. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of
488:
591:
2633:: maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained (indeed, generally constant!). There are inclusions of the disc in the plane in the sphere:
1319:. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see
2679:
1980:
1772:
1618:
The set of all
Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant
2337:
1077:
whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted). Two conformally equivalent
Riemann surfaces are for all practical purposes identical.
1909:
2459:
of "marked" Riemann surfaces (in addition to the
Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic
2946:. For example, the Riemann surface consisting of "all complex numbers but 0 and 1" is parabolic in the function-theoretic classification but it is hyperbolic in the geometric classification.
2175:
2120:
2028:
1813:
1569:
415:
2719:
2405:
2285:
2365:
1440:
2836:
fixing infinity, and of the punctured plane is the subgroup leaving invariant the set containing only infinity and zero: either fixing them both, or interchanging them (1/
2730:
at infinity, so not defined at infinity, and misses zero and infinity), but all maps from zero punctures to one or more, or one or two punctures to three or more are constant.
1277:
2780:
1382:
499:
2250:
2225:
2072:
2050:
1839:
3541:
2586:
2453:
2433:
2615:
1700:
which belongs to the conformal class of
Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of
1678:
4350:
2557:
2526:
2498:
2203:
1698:
1641:
2805:
of the sphere (they have non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant.
1223:), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see
4205:
4170:
2915:, with order 504; this is the second Hurwitz surface, and its automorphism group is isomorphic to PSL(2,8), the fourth-smallest non-abelian simple group.
3964:
2626:
4210:
1320:
195:
2636:
3693:
170:). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is
1917:
1204:
4056:
3653:
3534:
3461:
3385:
3343:
3291:
3251:
3212:
3176:
2997:
1726:
4325:
4315:
4300:
4220:
4133:
3744:
3643:
2992:
2866:
4360:
4123:
3139:
4375:
3505:
2293:
1850:
3822:
3527:
2465:
1127:
4370:
4365:
4340:
4195:
4163:
3969:
3880:
3086:
809:
3890:
3817:
3567:
3496:
1346:
794:
4190:
3787:
4305:
3683:
2844:
4330:
4046:
4010:
3007:
2787:
3709:
3622:
3491:
2723:
4355:
4335:
483:{\displaystyle \mathbf {C} \ {\stackrel {}{\hookrightarrow }}\ S^{2}\ {\stackrel {}{\hookleftarrow }}\ \mathbf {C} .}
2140:
2085:
1993:
1778:
1455:
4020:
3658:
3372:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 19, European Mathematical Society (EMS), ZĂŒrich,
3320:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 13, European Mathematical Society (EMS), ZĂŒrich,
3278:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 11, European Mathematical Society (EMS), ZĂŒrich,
3131:
3110:
2134:
2130:
2126:
42:
4421:
4416:
4394:
4156:
4066:
3199:
1280:
845:
4345:
3979:
3959:
3895:
3812:
3673:
3002:
199:
4235:
4200:
3714:
2693:
2370:
2258:
1308:
930:
864:
407:
272:
3678:
2415:. But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter
1291:
The existence of non-constant meromorphic functions can be used to show that any compact
Riemann surface is a
952:
3125:
2875:
It is known that every finite group can be realized as the full group of isometries of some
Riemann surface.
382:
is a
Riemann surface. More generally, every non-empty open subset of a Riemann surface is a Riemann surface.
350:
by means of the charts. Showing that a conformal structure determines a complex structure is more difficult.
4255:
3870:
2889:, with order 168; this is the first Hurwitz surface, and its automorphism group is isomorphic to the unique
2802:
1775:
1712:
1644:
1304:
3079:
Discontinuous Groups and
Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland
2342:
1047:) wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces
4270:
3663:
1708:
1701:
979:
493:
On the intersection of these two open sets, composing one embedding with the inverse of the other gives
331:
4041:
3777:
2854:), though the square lattice and hexagonal lattice have addition symmetries from rotation by 90° and 60°.
4295:
3577:
2783:
2727:
2630:
908:
886:
674:
187:
3739:
3688:
1250:
2893:
of order 168, which is the second-smallest non-abelian simple group. This group is isomorphic to both
1406:
586:{\displaystyle \mathbf {C} ^{\times }\ \to \ \mathbf {C} ^{\times }\ \ \ \,\ \ \ z\ \mapsto \ z^{-1}.}
4128:
3989:
3900:
3648:
3486:
2795:
2749:
2746:
genus, except as constant maps. This is because holomorphic and meromorphic maps behave locally like
2456:
1396:
1351:
1193:
1003:
288:
132:
3954:
1232:
4320:
4230:
4215:
3832:
3797:
3754:
3734:
2970:
2931:
if there are no non-constant negative subharmonic functions on the surface and is otherwise called
2560:
2123:
1648:
1620:
813:
314:, the space is homeomorphic to a subset of the real plane. The supplement "Riemann" signifies that
303:
128:
35:
3270:
2233:
2208:
2055:
2033:
1822:
4240:
4095:
3668:
3453:
3412:
3357:
3321:
3305:
2791:
1316:
1292:
299:
215:
159:
31:
3875:
3855:
3827:
2960:
2955:
3417:
Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse
2500:
is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a
4310:
4250:
3984:
3931:
3802:
3617:
3612:
3457:
3424:
3381:
3339:
3287:
3247:
3226:
3208:
3172:
3135:
3082:
3074:
2987:
1312:
1189:
327:
323:
295:
236:
3186:
2565:
4245:
4225:
4179:
3974:
3860:
3837:
3400:
3373:
3331:
3279:
3194:
2912:
2905:
2726:. One can map from one puncture to two, via the exponential map (which is entire and has an
1912:
1716:
1300:
1244:
1169:
Every non-compact Riemann surface admits non-constant holomorphic functions (with values in
1012:
343:
252:
248:
232:
211:
175:
167:
105:
101:
93:
3471:
3436:
3353:
3301:
3222:
3056:
2625:
The geometric classification is reflected in maps between Riemann surfaces, as detailed in
2438:
2418:
4265:
4100:
3905:
3749:
3572:
3551:
3467:
3432:
3404:
3349:
3297:
3243:
3218:
3204:
3168:
2965:
2851:
2591:
2435:
in the third case gives non-isomorphic Riemann surfaces. The description by the parameter
1987:
1654:
1212:
801:
723:
613:
244:
229:
191:
3447:
2826:
729:
Every elliptic curve is an algebraic curve, given by (the compactification of) the locus
179:
3772:
4280:
4275:
4074:
3597:
3582:
3559:
2938:
To avoid confusion, call the classification based on metrics of constant curvature the
2814:
2542:
2511:
2505:
2501:
2483:
2188:
1683:
1626:
1442:
1228:
1197:
1174:
1135:
1057:
722:(or we restrict to an open subset containing no such points). This is an example of an
402:
284:
256:
112:: locally near every point they look like patches of the complex plane, but the global
48:
4410:
4115:
3885:
3865:
3792:
3587:
3309:
2975:
2886:
2879:
2469:
361:
334:
if the angles they measure are the same. Choosing an equivalence class of metrics on
260:
109:
17:
3361:
4260:
4051:
4025:
4015:
4005:
3807:
3627:
3443:
3102:
2890:
2818:
2461:
280:
183:
2847:
is the real Möbius group; this is conjugate to the automorphism group of the disk.
342:
A complex structure gives rise to a conformal structure by choosing the standard
4285:
3926:
3764:
1986:
A Riemann surface is elliptic, parabolic or hyperbolic according to whether its
1596:
1161:
has positive determinant. Consequently, the complex atlas is an oriented atlas.
1146:
371:
89:
3415:(1955), "Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten",
76:
in the complex plane; the vertical coordinate represents the imaginary part of
3921:
2205:
is a Riemann surface whose universal cover is isomorphic to the complex plane
1296:
1087:
837:
600:
is holomorphic, so these two embeddings define a Riemann surface structure on
384:
171:
3519:
3428:
2674:{\displaystyle \Delta \subset \mathbf {C} \subset {\widehat {\mathbf {C} }},}
3782:
3230:
3160:
3045:, Ch. 3.11) for the construction of a corresponding complex structure.
2464:(forgetting the marking) one takes the quotient of TeichmĂŒller space by the
1845:
1227:. Meromorphic functions can be given fairly explicitly, in terms of Riemann
1066:
2690:
These statements are clarified by considering the type of a Riemann sphere
3264:, Graduate Studies in Mathematics, vol. 5, American Mathematical Soc.
2850:
genus 1 â the isometry group of a torus is in general translations (as an
2411:
Topologically there are only three types: the plane, the cylinder and the
1595:
in the sense of algebraic geometry. Reversing this is accomplished by the
718:
defines a Riemann surface provided there are no points on this locus with
2898:
2894:
2833:
2738:
Continuing in this vein, compact Riemann surfaces can map to surfaces of
1975:{\displaystyle \mathbf {H} :=\{z\in \mathbf {C} :\mathrm {Im} (z)>0\}}
597:
163:
113:
4105:
4090:
2861: â„ 2, the isometry group is finite, and has order at most 84(
1767:{\displaystyle {\widehat {\mathbf {C} }}:=\mathbf {C} \cup \{\infty \}}
1283:
and the fact there exists a positive line bundle on any complex curve.
1303:. Actually, it can be shown that every compact Riemann surface can be
406:. It has two open subsets which we identify with the complex plane by
4085:
3395:
Remmert, Reinhold (1998). "From Riemann Surfaces to Complex Spaces".
3326:
3316:
Lawton, Sean; Peterson, Elisha (2009), Papadopoulos, Athanase (ed.),
3057:"KODAIRA'S THEOREM AND COMPACTIFICATION OF MUMFORD'S MODULI SPACE Mg"
2533:
394:
117:
178:. Given this, the sphere and torus admit complex structures but the
4148:
1719:
Riemann surface is conformally equivalent to one of the following:
697:) is any complex polynomial in two variables, its vanishing locus
673:
Important examples of non-compact Riemann surfaces are provided by
2529:
2412:
2137:
and so any Riemann surface whose universal cover is isomorphic to
640:
629:
383:
319:
121:
47:
108:. Riemann surfaces can be thought of as deformed versions of the
3377:
3335:
3283:
2786:, and for compact Riemann surfaces these are constrained by the
2074:. The elements in each class admit a more precise description.
630:
4152:
3523:
646:
has many different Riemann surface structures, all of the form
220:
There are several equivalent definitions of a Riemann surface.
2817:
of a uniformized Riemann surface (equivalently, the conformal
2332:{\displaystyle \mathbf {C} /(\mathbf {Z} +\mathbf {Z} \tau )}
1904:{\displaystyle \mathbf {D} :=\{z\in \mathbf {C} :|z|<1\}}
1065:
to emphasize the conformal point of view) if there exists a
852:> 1, there are other Riemann surface structures of genus
104:. These surfaces were first studied by and are named after
1134:
is just the real linear map given by multiplication by the
2559:
is compact. Then its topological type is described by its
612:âȘ {â}. The Riemann sphere has another description, as the
116:
can be quite different. For example, they can look like a
2227:
then it is isomorphic to one of the following surfaces:
52:
The Riemann surface for the multivalued complex function
3081:. Ann. Math. Studies. Vol. 79. pp. 207â226.
2942:, and the one based on degeneracy of function spaces
2752:
2696:
2639:
2594:
2568:
2545:
2514:
2486:
2441:
2421:
2373:
2345:
2296:
2261:
2236:
2211:
2191:
2143:
2088:
2058:
2036:
1996:
1920:
1853:
1825:
1781:
1729:
1686:
1657:
1629:
1458:
1409:
1354:
1253:
502:
418:
318:
is endowed with an additional structure which allows
41:
For the Riemann surface of a subring of a field, see
2722:
three or more punctures, it is hyperbolic â compare
1707:
In complex analytic terms, the PoincarĂ©–Koebe
283:
to the open unit disk of the complex plane, and the
4114:
4065:
4034:
3998:
3947:
3940:
3914:
3846:
3763:
3727:
3702:
3636:
3605:
3596:
3558:
1173:). In fact, every non-compact Riemann surface is a
1086:Each Riemann surface, being a complex manifold, is
400:has a unique Riemann surface structure, called the
338:
is the additional datum of the conformal structure.
2825:genus 0 â the isometry group of the sphere is the
2774:
2713:
2673:
2609:
2580:
2551:
2520:
2492:
2447:
2427:
2399:
2359:
2331:
2279:
2244:
2219:
2197:
2169:
2114:
2066:
2044:
2022:
1974:
1903:
1833:
1807:
1766:
1692:
1672:
1635:
1563:
1434:
1376:
1271:
585:
482:
346:given on the complex plane and transporting it to
306:. Again, manifold means that locally at any point
287:between two overlapping charts are required to be
666:is any complex non-real number. These are called
166:, but it contains more structure (specifically a
3506:"Complex Analysis on Riemann Surfaces Math 213b"
388:The Riemann sphere and stereographic projection.
1118:can be considered as a map from an open set of
2869:; surfaces that realize this bound are called
2170:{\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}
2115:{\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}
2023:{\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}
1808:{\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}
1564:{\displaystyle ^{2}=4^{3}-g_{2}\wp (z)-g_{3},}
4164:
3535:
2829:of projective transforms of the complex line,
2801:For example, hyperbolic Riemann surfaces are
2588:. Its TeichmĂŒller space and moduli space are
978:As with any map between complex manifolds, a
8:
4206:GrothendieckâHirzebruchâRiemannâRoch theorem
1969:
1929:
1898:
1862:
1761:
1755:
190:do not. Every compact Riemann surface is a
1623:. That is, every connected Riemann surface
1588:depend on Ï, thus giving an elliptic curve
1192:. However, there always exist non-constant
4171:
4157:
4149:
3944:
3602:
3542:
3528:
3520:
2911:For genus 7 the order is maximized by the
2885:For genus 3 the order is maximized by the
2878:For genus 2 the order is maximized by the
2508:for the surface). The topological type of
1196:(holomorphic functions with values in the
1184:every holomorphic function with values in
1180:In contrast, on a compact Riemann surface
844: + 1 such that the above has no
4351:RiemannâRoch theorem for smooth manifolds
3325:
3026:
2763:
2751:
2714:{\displaystyle {\widehat {\mathbf {C} }}}
2700:
2698:
2697:
2695:
2657:
2655:
2654:
2646:
2638:
2593:
2567:
2544:
2513:
2485:
2440:
2420:
2400:{\displaystyle \mathrm {Im} (\tau )>0}
2374:
2372:
2352:
2344:
2318:
2310:
2302:
2297:
2295:
2280:{\displaystyle \mathbf {C} /\mathbf {Z} }
2272:
2267:
2262:
2260:
2237:
2235:
2212:
2210:
2190:
2159:
2150:
2145:
2142:
2104:
2095:
2090:
2087:
2059:
2057:
2037:
2035:
2012:
2003:
1998:
1995:
1946:
1938:
1921:
1919:
1887:
1879:
1871:
1854:
1852:
1826:
1824:
1797:
1788:
1783:
1780:
1747:
1733:
1731:
1730:
1728:
1685:
1656:
1628:
1552:
1527:
1514:
1483:
1457:
1414:
1408:
1359:
1353:
1263:
1259:
1256:
1255:
1252:
571:
545:
530:
525:
509:
504:
501:
472:
464:
459:
457:
456:
447:
435:
430:
428:
427:
419:
417:
84:), whose real part is represented by hue.
3370:Handbook of TeichmĂŒller theory. Vol. III
64:) in a neighborhood of the origin. The (
3318:Handbook of TeichmĂŒller theory. Vol. II
3130:(1st ed.), Princeton, New Jersey:
3109:(1st ed.), Princeton, New Jersey:
3030:
3019:
2832:the isometry group of the plane is the
2528:can be any orientable surface save the
2129:on it by biholomorphic transformations
1090:as a real manifold. For complex charts
860:
322:measurement on the manifold, namely an
3965:Clifford's theorem on special divisors
3272:Handbook of TeichmĂŒller theory. Vol. I
2539:A case of particular interest is when
1224:
1149:of multiplication by a complex number
298:of (real) dimension two â a two-sided
3262:Algebraic curves and Riemann surfaces
2998:Identity theorem for Riemann surfaces
2944:the function-theoretic classification
2360:{\displaystyle \tau \in \mathbf {C} }
1247:since they can be embedded into some
808:have Riemann surface structures, as (
127:Examples of Riemann surfaces include
72:) coordinates are the coordinates of
7:
3368:Papadopoulos, Athanase, ed. (2012),
3269:Papadopoulos, Athanase, ed. (2007),
3042:
2122:is the only example, as there is no
2982:Theorems regarding Riemann surfaces
1403:. This function and its derivative
1039:is holomorphic (as a function from
4316:Riemannian connection on a surface
4221:Measurable Riemann mapping theorem
4134:Vector bundles on algebraic curves
4057:Weber's theorem (Algebraic curves)
3654:Hasse's theorem on elliptic curves
3644:Counting points on elliptic curves
3167:(2nd ed.), Berlin, New York:
3124:Rodin, Burton; Sario, Leo (1968),
2640:
2378:
2375:
1950:
1947:
1758:
1614:Classification of Riemann surfaces
1606:), which can be used to determine
1533:
1498:
1463:
1411:
1356:
1326:As an example, consider the torus
974:Further definitions and properties
367:is the most basic Riemann surface.
330:. Two such metrics are considered
124:or several sheets glued together.
25:
2922:Function-theoretic classification
2798:of a space and a ramified cover.
2177:must itself be isomorphic to it.
1651:with constant curvature equal to
1272:{\displaystyle \mathbb {CP} ^{n}}
1243:All compact Riemann surfaces are
4389:
4388:
3449:The concept of a Riemann surface
2701:
2658:
2647:
2353:
2319:
2311:
2298:
2273:
2263:
2238:
2213:
2160:
2146:
2105:
2091:
2060:
2038:
2013:
1999:
1939:
1922:
1872:
1855:
1827:
1798:
1784:
1748:
1734:
1435:{\displaystyle \wp _{\tau }'(z)}
951:
929:
907:
885:
863:
526:
505:
473:
420:
27:One-dimensional complex manifold
4301:Riemann's differential equation
4211:HirzebruchâRiemannâRoch theorem
3745:Hurwitz's automorphisms theorem
3075:"Maximal groups and signatures"
2993:Hurwitz's automorphisms theorem
2867:Hurwitz's automorphisms theorem
2775:{\displaystyle z\mapsto z^{n},}
1377:{\displaystyle \wp _{\tau }(z)}
100:is a connected one-dimensional
4326:RiemannâHilbert correspondence
4196:Generalized Riemann hypothesis
3970:Gonality of an algebraic curve
3881:Differential of the first kind
2908:is a highly symmetric surface.
2809:Isometries of Riemann surfaces
2756:
2388:
2382:
2326:
2307:
2164:
2156:
2109:
2101:
2017:
2009:
1960:
1954:
1888:
1880:
1802:
1794:
1542:
1536:
1511:
1507:
1501:
1495:
1480:
1476:
1470:
1459:
1429:
1423:
1371:
1365:
561:
518:
460:
431:
410:from the North or South poles
1:
4361:RiemannâSiegel theta function
4124:BirkhoffâGrothendieck theorem
3823:Nagata's conjecture on curves
3694:SchoofâElkiesâAtkin algorithm
3568:Five points determine a conic
3132:D. Von Nostrand Company, Inc.
2621:Maps between Riemann surfaces
1774:, which is isomorphic to the
993:between two Riemann surfaces
795:Weierstrass elliptic function
4376:Riemannâvon Mangoldt formula
3684:Supersingular elliptic curve
2504:(this is sometimes called a
2245:{\displaystyle \mathbf {C} }
2220:{\displaystyle \mathbf {C} }
2067:{\displaystyle \mathbf {D} }
2045:{\displaystyle \mathbf {C} }
1834:{\displaystyle \mathbf {C} }
1203:âȘ {â}). More precisely, the
749:for certain complex numbers
408:stereographically projecting
143:, e.g. the subset of pairs (
3891:Riemann's existence theorem
3818:Hilbert's sixteenth problem
3710:Elliptic curve cryptography
3623:Fundamental pair of periods
3492:Encyclopedia of Mathematics
2476:Hyperbolic Riemann surfaces
1911:which is isomorphic to the
158:Every Riemann surface is a
4438:
4371:RiemannâStieltjes integral
4366:RiemannâSilberstein vector
4341:RiemannâLiouville integral
4021:Moduli of algebraic curves
3452:(3rd ed.), New York:
3111:Princeton University Press
2843:the isometry group of the
2181:Parabolic Riemann surfaces
1309:complex projective 3-space
1069:holomorphic function from
209:
40:
29:
4384:
4306:Riemann's minimal surface
4186:
2821:) reflects its geometry:
2782:so non-constant maps are
2468:. In this case it is the
2078:Elliptic Riemann surfaces
1711:(a generalization of the
1384:belonging to the lattice
1281:Kodaira embedding theorem
1098:with transition function
162:: a two-dimensional real
4331:RiemannâHilbert problems
4236:Riemann curvature tensor
4201:Grand Riemann hypothesis
4191:CauchyâRiemann equations
3788:CayleyâBacharach theorem
3715:Elliptic curve primality
3240:Compact Riemann Surfaces
2940:geometric classification
2803:ramified covering spaces
2734:Ramified covering spaces
2135:properly discontinuously
1279:. This follows from the
294:A Riemann surface is an
247:that is endowed with an
30:Not to be confused with
4256:Riemann mapping theorem
4047:RiemannâHurwitz formula
4011:GromovâWitten invariant
3871:Compact Riemann surface
3659:Mazur's torsion theorem
3008:RiemannâHurwitz formula
2788:RiemannâHurwitz formula
2581:{\displaystyle g\geq 2}
2480:In the remaining cases
1713:Riemann mapping theorem
1574:where the coefficients
1449:. There is an equation
1295:, i.e. can be given by
1188:is constant due to the
192:complex algebraic curve
4356:RiemannâSiegel formula
4336:RiemannâLebesgue lemma
4271:Riemann series theorem
3664:Modular elliptic curve
3260:Miranda, Rick (1995),
3073:Greenberg, L. (1974).
2784:ramified covering maps
2776:
2715:
2675:
2611:
2582:
2553:
2522:
2494:
2449:
2429:
2401:
2361:
2333:
2281:
2246:
2221:
2199:
2171:
2116:
2068:
2046:
2024:
1976:
1905:
1835:
1809:
1768:
1709:uniformization theorem
1702:isothermal coordinates
1694:
1674:
1637:
1565:
1445:the function field of
1436:
1378:
1287:Analytic vs. algebraic
1273:
1157:|, so the Jacobian of
1063:conformally equivalent
814:hyperelliptic surfaces
635:
587:
484:
389:
85:
4296:Riemann zeta function
3578:Rational normal curve
3515:. Harvard University.
3397:SĂ©minaires et CongrĂšs
3238:Jost, JĂŒrgen (2006),
3187:Riemann Surfaces Book
3105:; Sario, Leo (1960),
3027:Farkas & Kra 1980
2777:
2728:essential singularity
2716:
2676:
2631:Little Picard theorem
2612:
2583:
2554:
2523:
2495:
2450:
2448:{\displaystyle \tau }
2430:
2428:{\displaystyle \tau }
2402:
2362:
2334:
2282:
2247:
2222:
2200:
2172:
2117:
2069:
2047:
2025:
1977:
1906:
1836:
1810:
1769:
1695:
1675:
1638:
1566:
1437:
1379:
1274:
1194:meromorphic functions
1145:). However, the real
675:analytic continuation
633:
588:
485:
387:
374:of the complex plane
239:one. This means that
210:Further information:
188:real projective plane
133:multivalued functions
51:
43:ZariskiâRiemann space
18:Conformally invariant
4346:RiemannâRoch theorem
4129:Stable vector bundle
3990:Weil reciprocity law
3980:RiemannâRoch theorem
3960:BrillâNoether theory
3896:RiemannâRoch theorem
3813:Genusâdegree formula
3674:MordellâWeil theorem
3649:Division polynomials
3246:, pp. 208â219,
3242:, Berlin, New York:
3203:, Berlin, New York:
3184:Pablo Arés Gastesi,
3159:Farkas, Hershel M.;
3003:RiemannâRoch theorem
2865: â 1), by
2796:Euler characteristic
2794:, which relates the
2750:
2694:
2637:
2610:{\displaystyle 6g-6}
2592:
2566:
2543:
2512:
2484:
2439:
2419:
2371:
2343:
2294:
2259:
2234:
2209:
2189:
2141:
2086:
2056:
2034:
1994:
1918:
1851:
1823:
1779:
1727:
1715:) states that every
1684:
1673:{\displaystyle -1,0}
1655:
1627:
1456:
1407:
1397:meromorphic function
1352:
1347:Weierstrass function
1251:
500:
416:
200:RiemannâRoch theorem
4321:Riemannian geometry
4231:Riemann Xi function
4216:Local zeta function
3941:Structure of curves
3833:Quartic plane curve
3755:Hyperelliptic curve
3735:De Franchis theorem
3679:NagellâLutz theorem
3413:Siegel, Carl Ludwig
3127:Principal Functions
2971:Mapping class group
2627:Liouville's theorem
2466:mapping class group
2082:The Riemann sphere
1723:The Riemann sphere
1647:2-dimensional real
1621:sectional curvature
1610:and hence a torus.
1422:
1299:equations inside a
1007:if for every chart
304:conformal structure
36:Riemannian manifold
4241:Riemann hypothesis
3948:Divisors on curves
3740:Faltings's theorem
3689:Schoof's algorithm
3669:Modularity theorem
3454:Dover Publications
3234:, esp. chapter IV.
3200:Algebraic Geometry
2819:automorphism group
2792:algebraic topology
2772:
2742:genus, but not to
2711:
2671:
2607:
2578:
2549:
2518:
2490:
2445:
2425:
2397:
2357:
2329:
2277:
2242:
2217:
2195:
2167:
2112:
2064:
2042:
2020:
1972:
1901:
1831:
1819:The complex plane
1805:
1764:
1690:
1670:
1633:
1561:
1432:
1410:
1374:
1317:algebraic geometry
1293:projective variety
1269:
793:)) where â is the
636:
583:
480:
390:
328:Riemannian metrics
302:â together with a
263:: for every point
224:A Riemann surface
216:Conformal geometry
92:, particularly in
86:
32:Riemannian surface
4404:
4403:
4311:Riemannian circle
4251:Riemann invariant
4146:
4145:
4142:
4141:
4042:HasseâWitt matrix
3985:Weierstrass point
3932:Smooth completion
3901:TeichmĂŒller space
3803:Cubic plane curve
3723:
3722:
3637:Arithmetic theory
3618:Elliptic integral
3613:Elliptic function
3487:"Riemann surface"
3463:978-0-486-47004-7
3387:978-3-03719-103-3
3345:978-3-03719-055-5
3293:978-3-03719-029-6
3253:978-3-540-33065-3
3214:978-0-387-90244-9
3195:Hartshorne, Robin
3178:978-0-387-90465-8
2988:Branching theorem
2871:Hurwitz surfaces.
2708:
2686:Punctured spheres
2665:
2552:{\displaystyle X}
2521:{\displaystyle X}
2493:{\displaystyle X}
2457:TeichmĂŒller space
2198:{\displaystyle X}
1990:is isomorphic to
1741:
1693:{\displaystyle 1}
1636:{\displaystyle X}
1190:maximum principle
810:compactifications
566:
560:
554:
551:
548:
544:
541:
538:
523:
517:
471:
466:
455:
442:
437:
426:
324:equivalence class
296:oriented manifold
237:complex dimension
168:complex structure
16:(Redirected from
4429:
4422:Bernhard Riemann
4417:Riemann surfaces
4392:
4391:
4246:Riemann integral
4226:Riemann (crater)
4180:Bernhard Riemann
4173:
4166:
4159:
4150:
3975:Jacobian variety
3945:
3848:Riemann surfaces
3838:Real plane curve
3798:Cramer's paradox
3778:BĂ©zout's theorem
3603:
3552:algebraic curves
3544:
3537:
3530:
3521:
3516:
3510:
3500:
3474:
3439:
3408:
3390:
3364:
3329:
3312:
3277:
3265:
3256:
3233:
3181:
3165:Riemann Surfaces
3145:
3144:
3121:
3115:
3114:
3107:Riemann Surfaces
3099:
3093:
3092:
3070:
3064:
3063:
3061:
3052:
3046:
3039:
3033:
3024:
2913:Macbeath surface
2882:, with order 48.
2845:upper half-plane
2781:
2779:
2778:
2773:
2768:
2767:
2720:
2718:
2717:
2712:
2710:
2709:
2704:
2699:
2680:
2678:
2677:
2672:
2667:
2666:
2661:
2656:
2650:
2616:
2614:
2613:
2608:
2587:
2585:
2584:
2579:
2558:
2556:
2555:
2550:
2527:
2525:
2524:
2519:
2499:
2497:
2496:
2491:
2454:
2452:
2451:
2446:
2434:
2432:
2431:
2426:
2406:
2404:
2403:
2398:
2381:
2366:
2364:
2363:
2358:
2356:
2338:
2336:
2335:
2330:
2322:
2314:
2306:
2301:
2286:
2284:
2283:
2278:
2276:
2271:
2266:
2251:
2249:
2248:
2243:
2241:
2226:
2224:
2223:
2218:
2216:
2204:
2202:
2201:
2196:
2176:
2174:
2173:
2168:
2163:
2155:
2154:
2149:
2121:
2119:
2118:
2113:
2108:
2100:
2099:
2094:
2073:
2071:
2070:
2065:
2063:
2051:
2049:
2048:
2043:
2041:
2029:
2027:
2026:
2021:
2016:
2008:
2007:
2002:
1981:
1979:
1978:
1973:
1953:
1942:
1925:
1913:upper half-plane
1910:
1908:
1907:
1902:
1891:
1883:
1875:
1858:
1840:
1838:
1837:
1832:
1830:
1814:
1812:
1811:
1806:
1801:
1793:
1792:
1787:
1773:
1771:
1770:
1765:
1751:
1743:
1742:
1737:
1732:
1717:simply connected
1699:
1697:
1696:
1691:
1679:
1677:
1676:
1671:
1643:admits a unique
1642:
1640:
1639:
1634:
1570:
1568:
1567:
1562:
1557:
1556:
1532:
1531:
1519:
1518:
1488:
1487:
1469:
1441:
1439:
1438:
1433:
1418:
1383:
1381:
1380:
1375:
1364:
1363:
1301:projective space
1278:
1276:
1275:
1270:
1268:
1267:
1262:
1245:algebraic curves
1235:of the surface.
1023:in the atlas of
1019:and every chart
955:
933:
911:
889:
867:
720:âP/âx, âP/ây = 0
683:Algebraic curves
592:
590:
589:
584:
579:
578:
564:
558:
552:
549:
546:
542:
539:
536:
535:
534:
529:
521:
515:
514:
513:
508:
489:
487:
486:
481:
476:
469:
468:
467:
465:
463:
458:
453:
452:
451:
440:
439:
438:
436:
434:
429:
424:
423:
344:Euclidean metric
233:complex manifold
212:Complex manifold
106:Bernhard Riemann
102:complex manifold
94:complex analysis
21:
4437:
4436:
4432:
4431:
4430:
4428:
4427:
4426:
4407:
4406:
4405:
4400:
4380:
4291:Riemann surface
4266:Riemann problem
4182:
4177:
4147:
4138:
4110:
4101:Delta invariant
4079:
4061:
4030:
3994:
3955:AbelâJacobi map
3936:
3910:
3906:Torelli theorem
3876:Dessin d'enfant
3856:Belyi's theorem
3842:
3828:PlĂŒcker formula
3759:
3750:Hurwitz surface
3719:
3698:
3632:
3606:Analytic theory
3598:Elliptic curves
3592:
3573:Projective line
3560:Rational curves
3554:
3548:
3508:
3503:
3485:
3482:
3477:
3464:
3442:
3411:
3394:
3388:
3367:
3346:
3315:
3294:
3275:
3268:
3259:
3254:
3244:Springer-Verlag
3237:
3215:
3205:Springer-Verlag
3193:
3179:
3169:Springer-Verlag
3158:
3154:
3149:
3148:
3142:
3134:, p. 199,
3123:
3122:
3118:
3101:
3100:
3096:
3089:
3072:
3071:
3067:
3059:
3055:Nollet, Scott.
3054:
3053:
3049:
3041:See (Jost
3040:
3036:
3025:
3021:
3016:
2984:
2966:Lorentz surface
2961:KĂ€hler manifold
2956:Dessin d'enfant
2952:
2924:
2906:Bring's surface
2852:Abelian variety
2811:
2759:
2748:
2747:
2736:
2692:
2691:
2688:
2635:
2634:
2623:
2590:
2589:
2564:
2563:
2541:
2540:
2510:
2509:
2482:
2481:
2478:
2437:
2436:
2417:
2416:
2369:
2368:
2341:
2340:
2292:
2291:
2257:
2256:
2232:
2231:
2207:
2206:
2187:
2186:
2183:
2144:
2139:
2138:
2089:
2084:
2083:
2080:
2054:
2053:
2032:
2031:
1997:
1992:
1991:
1988:universal cover
1916:
1915:
1849:
1848:
1821:
1820:
1782:
1777:
1776:
1725:
1724:
1682:
1681:
1653:
1652:
1625:
1624:
1616:
1594:
1587:
1580:
1548:
1523:
1510:
1479:
1462:
1454:
1453:
1405:
1404:
1355:
1350:
1349:
1289:
1254:
1249:
1248:
1241:
1233:AbelâJacobi map
1229:theta functions
1167:
1084:
976:
969:
956:
947:
934:
925:
912:
903:
890:
881:
868:
859:
846:singular points
724:algebraic curve
685:
680:
668:elliptic curves
614:projective line
567:
524:
503:
498:
497:
443:
414:
413:
370:Every nonempty
356:
285:transition maps
245:Hausdorff space
243:is a connected
218:
208:
98:Riemann surface
46:
39:
28:
23:
22:
15:
12:
11:
5:
4435:
4433:
4425:
4424:
4419:
4409:
4408:
4402:
4401:
4399:
4398:
4385:
4382:
4381:
4379:
4378:
4373:
4368:
4363:
4358:
4353:
4348:
4343:
4338:
4333:
4328:
4323:
4318:
4313:
4308:
4303:
4298:
4293:
4288:
4283:
4281:Riemann sphere
4278:
4276:Riemann solver
4273:
4268:
4263:
4258:
4253:
4248:
4243:
4238:
4233:
4228:
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4187:
4184:
4183:
4178:
4176:
4175:
4168:
4161:
4153:
4144:
4143:
4140:
4139:
4137:
4136:
4131:
4126:
4120:
4118:
4116:Vector bundles
4112:
4111:
4109:
4108:
4103:
4098:
4093:
4088:
4083:
4077:
4071:
4069:
4063:
4062:
4060:
4059:
4054:
4049:
4044:
4038:
4036:
4032:
4031:
4029:
4028:
4023:
4018:
4013:
4008:
4002:
4000:
3996:
3995:
3993:
3992:
3987:
3982:
3977:
3972:
3967:
3962:
3957:
3951:
3949:
3942:
3938:
3937:
3935:
3934:
3929:
3924:
3918:
3916:
3912:
3911:
3909:
3908:
3903:
3898:
3893:
3888:
3883:
3878:
3873:
3868:
3863:
3858:
3852:
3850:
3844:
3843:
3841:
3840:
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3769:
3767:
3761:
3760:
3758:
3757:
3752:
3747:
3742:
3737:
3731:
3729:
3725:
3724:
3721:
3720:
3718:
3717:
3712:
3706:
3704:
3700:
3699:
3697:
3696:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3656:
3651:
3646:
3640:
3638:
3634:
3633:
3631:
3630:
3625:
3620:
3615:
3609:
3607:
3600:
3594:
3593:
3591:
3590:
3585:
3583:Riemann sphere
3580:
3575:
3570:
3564:
3562:
3556:
3555:
3549:
3547:
3546:
3539:
3532:
3524:
3518:
3517:
3501:
3481:
3480:External links
3478:
3476:
3475:
3462:
3440:
3409:
3391:
3386:
3365:
3344:
3313:
3292:
3266:
3257:
3252:
3235:
3213:
3191:
3182:
3177:
3155:
3153:
3150:
3147:
3146:
3140:
3116:
3094:
3087:
3065:
3047:
3034:
3018:
3017:
3015:
3012:
3011:
3010:
3005:
3000:
2995:
2990:
2983:
2980:
2979:
2978:
2973:
2968:
2963:
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2948:
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2919:
2918:
2917:
2916:
2909:
2902:
2883:
2873:
2855:
2848:
2841:
2830:
2815:isometry group
2810:
2807:
2771:
2766:
2762:
2758:
2755:
2735:
2732:
2707:
2703:
2687:
2684:
2670:
2664:
2660:
2653:
2649:
2645:
2642:
2622:
2619:
2606:
2603:
2600:
2597:
2577:
2574:
2571:
2548:
2517:
2506:Fuchsian model
2502:Fuchsian group
2489:
2477:
2474:
2444:
2424:
2409:
2408:
2396:
2393:
2390:
2387:
2384:
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2153:
2148:
2111:
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2103:
2098:
2093:
2079:
2076:
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2040:
2019:
2015:
2011:
2006:
2001:
1984:
1983:
1971:
1968:
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1962:
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1754:
1750:
1746:
1740:
1736:
1689:
1669:
1666:
1663:
1660:
1649:Riemann metric
1632:
1615:
1612:
1592:
1585:
1578:
1572:
1571:
1560:
1555:
1551:
1547:
1544:
1541:
1538:
1535:
1530:
1526:
1522:
1517:
1513:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1486:
1482:
1478:
1475:
1472:
1468:
1465:
1461:
1431:
1428:
1425:
1421:
1417:
1413:
1373:
1370:
1367:
1362:
1358:
1330: :=
1321:Chow's theorem
1288:
1285:
1266:
1261:
1258:
1240:
1237:
1205:function field
1198:Riemann sphere
1175:Stein manifold
1166:
1163:
1136:complex number
1083:
1080:
975:
972:
971:
970:
957:
950:
948:
935:
928:
926:
913:
906:
904:
891:
884:
882:
869:
862:
858:
857:
831:
830:
798:
781:) is sent to (
748:
747:
727:
717:
716:
686:
684:
681:
679:
678:
671:
637:
598:transition map
594:
593:
582:
577:
574:
570:
563:
557:
533:
528:
520:
512:
507:
491:
490:
479:
475:
462:
450:
446:
433:
422:
403:Riemann sphere
391:
368:
357:
355:
352:
340:
339:
292:
257:open unit disk
207:
204:
196:Chow's theorem
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4434:
4423:
4420:
4418:
4415:
4414:
4412:
4397:
4396:
4387:
4386:
4383:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4188:
4185:
4181:
4174:
4169:
4167:
4162:
4160:
4155:
4154:
4151:
4135:
4132:
4130:
4127:
4125:
4122:
4121:
4119:
4117:
4113:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4080:
4073:
4072:
4070:
4068:
4067:Singularities
4064:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4039:
4037:
4033:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4003:
4001:
3997:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3952:
3950:
3946:
3943:
3939:
3933:
3930:
3928:
3925:
3923:
3920:
3919:
3917:
3915:Constructions
3913:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3886:Klein quartic
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3866:Bolza surface
3864:
3862:
3861:Bring's curve
3859:
3857:
3854:
3853:
3851:
3849:
3845:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3793:Conic section
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3773:AF+BG theorem
3771:
3770:
3768:
3766:
3762:
3756:
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3732:
3730:
3726:
3716:
3713:
3711:
3708:
3707:
3705:
3701:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3641:
3639:
3635:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3610:
3608:
3604:
3601:
3599:
3595:
3589:
3588:Twisted cubic
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3565:
3563:
3561:
3557:
3553:
3545:
3540:
3538:
3533:
3531:
3526:
3525:
3522:
3514:
3507:
3504:McMullen, C.
3502:
3498:
3494:
3493:
3488:
3484:
3483:
3479:
3473:
3469:
3465:
3459:
3455:
3451:
3450:
3445:
3444:Weyl, Hermann
3441:
3438:
3434:
3430:
3426:
3422:
3418:
3414:
3410:
3406:
3402:
3398:
3392:
3389:
3383:
3379:
3375:
3371:
3366:
3363:
3359:
3355:
3351:
3347:
3341:
3337:
3333:
3328:
3323:
3319:
3314:
3311:
3307:
3303:
3299:
3295:
3289:
3285:
3281:
3274:
3273:
3267:
3263:
3258:
3255:
3249:
3245:
3241:
3236:
3232:
3228:
3224:
3220:
3216:
3210:
3206:
3202:
3201:
3196:
3192:
3189:
3188:
3183:
3180:
3174:
3170:
3166:
3162:
3157:
3156:
3151:
3143:
3141:9781468480382
3137:
3133:
3129:
3128:
3120:
3117:
3113:, p. 204
3112:
3108:
3104:
3103:Ahlfors, Lars
3098:
3095:
3090:
3084:
3080:
3076:
3069:
3066:
3058:
3051:
3048:
3044:
3038:
3035:
3032:
3028:
3023:
3020:
3013:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2985:
2981:
2977:
2976:Serre duality
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2953:
2949:
2947:
2945:
2941:
2936:
2934:
2930:
2921:
2914:
2910:
2907:
2904:For genus 4,
2903:
2900:
2896:
2892:
2888:
2887:Klein quartic
2884:
2881:
2880:Bolza surface
2877:
2876:
2874:
2872:
2868:
2864:
2860:
2856:
2853:
2849:
2846:
2842:
2839:
2835:
2831:
2828:
2824:
2823:
2822:
2820:
2816:
2808:
2806:
2804:
2799:
2797:
2793:
2789:
2785:
2769:
2764:
2760:
2753:
2745:
2741:
2733:
2731:
2729:
2725:
2724:pair of pants
2705:
2685:
2683:
2668:
2662:
2651:
2643:
2632:
2628:
2620:
2618:
2604:
2601:
2598:
2595:
2575:
2572:
2569:
2562:
2546:
2537:
2535:
2531:
2515:
2507:
2503:
2487:
2475:
2473:
2471:
2470:modular curve
2467:
2463:
2458:
2442:
2422:
2414:
2394:
2391:
2385:
2349:
2346:
2323:
2315:
2303:
2289:
2268:
2255:The quotient
2254:
2230:
2229:
2228:
2192:
2180:
2178:
2151:
2136:
2132:
2128:
2125:
2096:
2077:
2075:
2004:
1989:
1966:
1963:
1957:
1943:
1935:
1932:
1926:
1914:
1895:
1892:
1884:
1876:
1868:
1865:
1859:
1847:
1843:
1818:
1815:
1789:
1752:
1744:
1738:
1722:
1721:
1720:
1718:
1714:
1710:
1705:
1703:
1687:
1667:
1664:
1661:
1658:
1650:
1646:
1630:
1622:
1613:
1611:
1609:
1605:
1601:
1598:
1591:
1584:
1577:
1558:
1553:
1549:
1545:
1539:
1528:
1524:
1520:
1515:
1504:
1492:
1489:
1484:
1473:
1466:
1452:
1451:
1450:
1448:
1444:
1426:
1419:
1415:
1402:
1398:
1394:
1391:
1388: +
1387:
1368:
1360:
1348:
1344:
1341:
1338: +
1337:
1333:
1329:
1324:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1286:
1284:
1282:
1264:
1246:
1238:
1236:
1234:
1230:
1226:
1225:Siegel (1955)
1222:
1218:
1214:
1210:
1206:
1202:
1199:
1195:
1191:
1187:
1183:
1178:
1176:
1172:
1164:
1162:
1160:
1156:
1152:
1148:
1144:
1140:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1082:Orientability
1081:
1079:
1076:
1072:
1068:
1064:
1060:
1059:
1058:biholomorphic
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1005:
1000:
996:
992:
988:
984:
981:
973:
968:
964:
960:
954:
949:
946:
942:
938:
932:
927:
924:
920:
916:
910:
905:
902:
898:
894:
888:
883:
880:
876:
872:
866:
861:
855:
851:
847:
843:
839:
836:is a complex
835:
828:
824:
820:
817:
816:
815:
811:
807:
805:
799:
796:
792:
788:
784:
780:
776:
773: +
772:
768:
764:
760:
757:depending on
756:
752:
746:
742:
738:
734:
731:
730:
728:
725:
721:
715:
711:
707:
703:
699:
698:
696:
692:
688:
687:
682:
676:
672:
669:
665:
661:
657:
654: +
653:
649:
645:
642:
638:
632:
628:
626:
622:
618:
615:
611:
607:
603:
599:
580:
575:
572:
568:
555:
531:
510:
496:
495:
494:
477:
448:
444:
412:
411:
409:
405:
404:
399:
396:
392:
386:
381:
377:
373:
369:
366:
363:
362:complex plane
359:
358:
353:
351:
349:
345:
337:
333:
329:
326:of so-called
325:
321:
317:
313:
309:
305:
301:
297:
293:
290:
286:
282:
278:
274:
273:neighbourhood
270:
266:
262:
261:complex plane
258:
254:
250:
246:
242:
238:
234:
231:
227:
223:
222:
221:
217:
213:
205:
203:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
156:
154:
150:
146:
142:
138:
134:
130:
125:
123:
119:
115:
111:
110:complex plane
107:
103:
99:
95:
91:
83:
79:
75:
71:
67:
63:
59:
55:
50:
44:
37:
33:
19:
4393:
4290:
4261:Riemann form
4075:
4052:Prym variety
4026:Stable curve
4016:Hodge bundle
4006:ELSV formula
3847:
3808:Fermat curve
3765:Plane curves
3728:Higher genus
3703:Applications
3628:Modular form
3513:Harvard Math
3512:
3490:
3448:
3420:
3416:
3396:
3369:
3327:math/0511271
3317:
3271:
3261:
3239:
3198:
3185:
3164:
3126:
3119:
3106:
3097:
3078:
3068:
3050:
3037:
3031:Miranda 1995
3022:
2943:
2939:
2937:
2932:
2928:
2925:
2891:simple group
2870:
2862:
2858:
2837:
2827:Möbius group
2812:
2800:
2743:
2739:
2737:
2689:
2624:
2538:
2479:
2462:moduli space
2410:
2184:
2081:
1985:
1706:
1617:
1607:
1603:
1599:
1589:
1582:
1575:
1573:
1446:
1400:
1392:
1389:
1385:
1342:
1339:
1335:
1331:
1327:
1325:
1290:
1242:
1239:Algebraicity
1220:
1216:
1211:is a finite
1208:
1200:
1185:
1181:
1179:
1170:
1168:
1158:
1154:
1150:
1142:
1138:
1131:
1123:
1119:
1115:
1111:
1107:
1103:
1099:
1095:
1091:
1085:
1074:
1070:
1062:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1008:
1002:
998:
994:
990:
986:
982:
977:
966:
962:
958:
944:
940:
936:
922:
918:
914:
900:
896:
892:
878:
874:
870:
853:
849:
841:
833:
826:
822:
818:
803:
790:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
744:
740:
736:
732:
719:
713:
709:
705:
701:
694:
690:
667:
663:
659:
655:
651:
647:
643:
624:
620:
616:
609:
605:
601:
595:
492:
401:
397:
379:
375:
364:
347:
341:
335:
315:
311:
307:
281:homeomorphic
276:
268:
264:
240:
225:
219:
184:Klein bottle
180:Möbius strip
157:
152:
148:
144:
140:
136:
126:
97:
87:
81:
77:
73:
69:
65:
61:
57:
53:
4286:Riemann sum
4081:singularity
3927:Polar curve
3378:10.4171/103
3336:10.4171/055
3284:10.4171/029
2290:A quotient
1597:j-invariant
1147:determinant
1130:in a point
1055:are called
1004:holomorphic
877:) = arcsin
840:of degree 2
604:. As sets,
372:open subset
289:holomorphic
271:there is a
206:Definitions
90:mathematics
4411:Categories
3922:Dual curve
3550:Topics in
3405:1044.01520
3161:Kra, Irwin
3152:References
3088:0691081387
2933:hyperbolic
2857:For genus
2455:gives the
1297:polynomial
1088:orientable
1027:, the map
1001:is called
838:polynomial
800:Likewise,
761:. A point
332:equivalent
176:metrizable
172:orientable
153:w = log(z)
4035:Morphisms
3783:Bitangent
3497:EMS Press
3446:(2009) ,
3429:0065-5295
3423:: 71â77,
3310:119593165
2929:parabolic
2757:↦
2706:^
2663:^
2652:⊂
2644:⊂
2641:Δ
2602:−
2573:≥
2443:τ
2423:τ
2386:τ
2350:∈
2347:τ
2324:τ
1936:∈
1869:∈
1846:open disk
1759:∞
1753:∪
1739:^
1659:−
1546:−
1534:℘
1521:−
1499:℘
1464:℘
1416:τ
1412:℘
1361:τ
1357:℘
1213:extension
1165:Functions
1067:bijective
712:) = 0} â
704:) :
573:−
562:↦
532:×
519:→
511:×
461:↩
432:↪
230:connected
4395:Category
3362:16687772
3231:13348052
3197:(1977),
3163:(1980),
2950:See also
2899:PSL(3,2)
2895:PSL(2,7)
2834:subgroup
2629:and the
1645:complete
1467:′
1443:generate
1420:′
1313:analytic
1305:embedded
1231:and the
1153:equals |
1128:Jacobian
980:function
899:) = log
806:surfaces
662:) where
634:A torus.
395:2-sphere
354:Examples
279:that is
198:and the
164:manifold
114:topology
60:) = log(
4106:Tacnode
4091:Crunode
3499:, 2001
3472:0069903
3437:0074061
3354:2524085
3302:2284826
3223:0463157
2927:called
2252:itself;
1345:). The
1011:in the
848:. When
785:) = (â(
641:2-torus
623:- {0})/
300:surface
259:of the
255:to the
160:surface
4086:Acnode
3999:Moduli
3470:
3460:
3435:
3427:
3403:
3384:
3360:
3352:
3342:
3308:
3300:
3290:
3250:
3229:
3221:
3211:
3175:
3138:
3085:
2744:higher
2534:sphere
2339:where
2131:freely
2127:acting
1126:whose
832:where
802:genus
777:
658:
565:
559:
553:
550:
547:
543:
540:
537:
522:
516:
470:
454:
441:
425:
253:charts
141:log(z)
129:graphs
118:sphere
3509:(PDF)
3358:S2CID
3322:arXiv
3306:S2CID
3276:(PDF)
3060:(PDF)
3014:Notes
2740:lower
2561:genus
2530:torus
2413:torus
2367:with
2124:group
1395:is a
1307:into
1013:atlas
789:),â'(
596:This
320:angle
249:atlas
228:is a
151:with
135:like
122:torus
120:or a
4096:Cusp
3458:ISBN
3425:ISSN
3421:1955
3382:ISBN
3340:ISBN
3288:ISBN
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