296:
1070:
939:
531:
445:
2066:
161:
3000:-expansions of their Hauptmoduln were computed already in the 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster.
2238:
1847:
593:
1455:
1968:
1884:
958:
2024:
2930:
2672:
2469:
1322:
791:– the cusps are the points lying over ∞ (red dots), while the vertices and centers of the edges (black and white dots) are the points lying over 0 and 1. The Galois group of the covering
2782:
2727:
2524:
783:
of genus 3 with 24 cusps. It can be interpreted as a surface with three handles tiled by 24 heptagons, with a cusp at the center of each face. These tilings can be understood via
2062:
1672:
2872:
2614:
2569:
2411:
96:, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to
2975:
2827:
1720:
1694:
2107:
2368:
1904:
854:
2261:
2347:
467:
381:
3305:
1598:-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined over
291:{\displaystyle \Gamma (N)=\left\{{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}:\ a\equiv d\equiv 1\mod N{\text{ and }}b,c\equiv 0\mod N\right\}.}
3087:, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.
2116:
1725:
341:(Γ) is obtained by adding finitely many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on the
3058:
in the 1970s, and the subsequent observation relating it to the monster group is due to Ogg, who wrote up a paper offering a bottle of
542:
1484:), and the fact that PSL(2, 2) has order 6, rather than 3. There is a more complicated formula for the genus of the modular curve
1513:
3240:
3034:
is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are precisely the prime factors of the order of the
1085:
1174:
1377:
1065:{\displaystyle \left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\ a\equiv d\equiv 1\mod N,c\equiv 0\mod N\right\}.}
1913:
1852:
3292:
3156:
3070:
1973:
137:
2890:
2632:
2429:
1258:
3101:
3062:
whiskey to anyone who could explain this fact, which was a starting point for the theory of monstrous moonshine.
2745:
2690:
2487:
3267:
77:
1906:
is minimal among all integral
Weierstrass models for the same curve. The following table contains the unique
145:
2992:
Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the
829:) can be restated as follows: it is the subgroup of the modular group which is the kernel of the reduction
661:
2029:
1639:
1229:
2845:
2587:
2542:
2384:
140:. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2,
3232:
3096:
2948:
2800:
1615:
54:
43:
3347:
3337:
1476:= 2 or 3, one must additionally take into account the ramification, that is, the presence of order
1233:
1703:
1677:
1549:
2026:. The last column of this table refers to the home page of the respective elliptic modular curve
1529:
818:
3151:. Lecture Notes in Mathematics. Vol. 476. Berlin, Heidelberg: Springer-Verlag. p. 79.
934:{\displaystyle \left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\ c\equiv 0\mod N\right\},}
3342:
2076:
2353:
1889:
749:(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular
3313:
Seminaire
Delange-Pisot-Poitou. Theorie des nombres, tome 16, no. 1 (1974–1975), exp. no. 7
3320:
3250:
3016:
1332:
784:
762:
642:
124:). The latter fact and its generalizations are of fundamental importance in number theory.
8:
2993:
1525:
1521:
1090:
101:
58:
3173:
3106:
2246:
1193:
830:
24:
3215:
2267:
1143:. These curves have been studied in great detail, and in particular, it is known that
3288:
3261:
3236:
3152:
3066:
3047:
1364:
1166:
1110:
526:{\displaystyle \{\infty \}\cup \{\tau \in \mathbf {H} \mid {\text{Im}}(\tau )>r\}}
440:{\displaystyle \{\infty \}\cup \{\tau \in \mathbf {H} \mid {\text{Im}}(\tau )>r\}}
3074:
3059:
3055:
1162:
450:
310:
112:
47:
3316:
3246:
3111:
105:
39:
35:
3136:, Le Mathématicien, vol. 2 (2nd ed.), Presses Universitaires de France
3280:
1697:
1189:
1170:
1080:
788:
624:
97:
93:
89:
2979:
2934:
2876:
2831:
2786:
2731:
2676:
2618:
2573:
2528:
2473:
2415:
1192:Γ other than subgroups of the modular group; a class of them constructed from
765:
on the
Riemann sphere. This group is a simple group of order 60 isomorphic to
3331:
3035:
780:
677:
62:
20:
3276:
3224:
3081:
1517:
1245:
1076:
1533:
1165:. The "best models" can be very different from those taken directly from
750:
3301:
3051:
3004:
1548:*. The traditional name for such a generator, which is unique up to a
844:) is the larger subgroup of matrices which are upper triangular modulo
318:
1161:
The equations defining modular curves are the best-known examples of
3080:
that are meromorphic and can have poles at the cusps, as opposed to
1135:), the level structure is, respectively, a cyclic subgroup of order
3231:, Publications of the Mathematical Society of Japan, vol. 11,
800:
3003:
Another connection is that the modular curve corresponding to the
2233:{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}
1842:{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}
1105:) is the moduli space for elliptic curves with a basis for the
668:. Once again, a complex structure can be put on the quotient Γ\
588:{\displaystyle {\begin{pmatrix}a&-m\\c&n\end{pmatrix}}}
3229:
Introduction to the arithmetic theory of automorphic functions
1696:
equals one of the 12 values listed in the following table. As
69:). The term modular curve can also be used to refer to the
3114:, a generalization of modular curves to higher dimensions
2952:
2894:
2849:
2804:
2749:
2694:
2636:
2591:
2546:
2491:
2433:
2388:
2033:
1977:
1917:
1856:
1681:
1643:
1388:
972:
868:
551:
190:
2951:
2893:
2848:
2803:
2748:
2693:
2635:
2590:
2545:
2490:
2432:
2387:
2356:
2270:
2249:
2119:
2079:
2032:
1976:
1916:
1892:
1855:
1728:
1706:
1680:
1642:
1610:. The converse statement, that only these values of
1380:
1261:
961:
857:
545:
470:
384:
164:
3065:
The relation runs very deep and, as demonstrated by
1910:, minimal, integral Weierstrass models, which means
1516:
of a modular curve (or, occasionally, of some other
623:* into a topological space which is a subset of the
80:
obtained by adding finitely many points (called the
1089:and for this reason they play an important role in
2987:
2969:
2924:
2866:
2821:
2776:
2721:
2666:
2608:
2563:
2518:
2463:
2405:
2362:
2341:
2255:
2232:
2101:
2067:The L-functions and modular forms database (LMFDB)
2056:
2018:
1962:
1898:
1878:
1841:
1714:
1688:
1666:
1590:= 12. Since each of these curves is defined over
1450:{\displaystyle g={\tfrac {1}{24}}(p+2)(p-3)(p-5).}
1449:
1316:
1064:
933:
587:
525:
439:
290:
3073:. Work in this area underlined the importance of
1963:{\displaystyle \textstyle a_{1},a_{3}\in \{0,1\}}
1722:, they have minimal, integral Weierstrass models
799:(1) is a simple group of order 168 isomorphic to
3329:
1879:{\displaystyle \textstyle a_{j}\in \mathbb {Z} }
100:, and, moreover, prove that modular curves are
3274:
2019:{\displaystyle \textstyle a_{2}\in \{-1,0,1\}}
1528:zero means such a function field has a single
332:
3181:Bulletin de la Société Mathématique de France
1075:These curves have a direct interpretation as
2996:conjectures. First several coefficients of
2925:{\displaystyle \textstyle -2^{4}\cdot 3^{3}}
2667:{\displaystyle \textstyle -2^{8}\cdot 5^{2}}
2464:{\displaystyle \textstyle -2^{6}\cdot 7^{3}}
2012:
1991:
1956:
1944:
1317:{\displaystyle -\pi \chi (X(p))=|G|\cdot D,}
672:* turning it into a Riemann surface denoted
520:
483:
477:
471:
434:
397:
391:
385:
2777:{\displaystyle \textstyle 2^{8}\cdot 3^{2}}
2722:{\displaystyle \textstyle 3^{4}\cdot 7^{2}}
2519:{\displaystyle \textstyle 3^{4}\cdot 5^{4}}
1886:and the absolute value of the discriminant
1552:and can be appropriately normalized, is a
3147:Birch, Bryan; Kuyk, Willem, eds. (1975).
3146:
1871:
1708:
1220:)/{1, −1}, which is equal to PSL(2,
1050:
1049:
1029:
1028:
919:
918:
806:There is an explicit classical model for
276:
275:
247:
246:
84:) to this quotient (via an action on the
952:) is the intermediate group defined by:
692:The most common examples are the curves
3223:
3171:
1351:−1)/2 is the order of the group PSL(2,
1216:(1) is Galois, with Galois group SL(2,
146:principal congruence subgroup of level
3330:
3306:"Automorphismes de courbes modulaires"
1371:) triangle. This results in a formula
1196:is also of interest in number theory.
680:. This space is a compactification of
127:
3131:
1621:
761:(1) is realized by the action of the
641:}, breaking it up into finitely many
3149:Modular functions of one variable IV
1177:connecting pairs of modular curves.
65:of integral 2×2 matrices SL(2,
3300:
3204:
2057:{\displaystyle \textstyle X_{0}(N)}
1667:{\displaystyle \textstyle X_{0}(N)}
825:modular curve. The definition of Γ(
13:
2867:{\displaystyle \textstyle -2^{12}}
2609:{\displaystyle \textstyle -19^{3}}
2564:{\displaystyle \textstyle -17^{4}}
2406:{\displaystyle \textstyle -11^{5}}
2357:
1893:
722:) associated with the subgroups Γ(
634:). The group Γ acts on the subset
474:
388:
165:
144:), i.e. a subgroup containing the
136:) acts on the upper half-plane by
14:
3359:
2970:{\displaystyle \textstyle -7^{3}}
2822:{\displaystyle \textstyle -3^{9}}
1236:, one can calculate the genus of
1173:may be studied geometrically, as
343:extended complex upper-half plane
138:fractional linear transformations
88:). The points of a modular curve
86:extended complex upper-half plane
3030:) has genus zero if and only if
1674:are of genus one if and only if
1536:generates the function field of
493:
407:
3174:"Courbes modulaires de genre 1"
3102:Moduli stack of elliptic curves
2988:Relation with the Monster group
1045:
1024:
914:
271:
242:
3198:
3165:
3140:
3125:
3071:generalized Kac–Moody algebras
2336:
2271:
2096:
2090:
2050:
2044:
1660:
1654:
1532:as generator: for example the
1441:
1429:
1426:
1414:
1411:
1399:
1301:
1293:
1286:
1283:
1277:
1271:
511:
505:
425:
419:
359:}. We introduce a topology on
174:
168:
1:
3118:
1606:-torsion for these values of
1503:
337:A common compactification of
313:can be put on the quotient Γ\
132:The modular group SL(2,
3285:Encyclopaedia of Mathematics
3216:Equations For Modular Curves
1715:{\displaystyle \mathbb {Q} }
1689:{\displaystyle \textstyle N}
1586: = 1, ..., 10 and
662:Alexandroff compactification
649:. If Γ acts transitively on
7:
3132:Serre, Jean-Pierre (1977),
3090:
821:; this is sometimes called
687:
333:Compactified modular curves
71:compactified modular curves
10:
3364:
3266:: CS1 maint: postscript (
3253:, Kanô Memorial Lectures,
3233:Princeton University Press
1562:principal modular function
1496:that involves divisors of
151:for some positive integer
2113:
321:Riemann surface called a
155:, which is defined to be
104:either over the field of
3172:Ligozat, Gerard (1975).
2102:{\displaystyle X_{0}(N)}
1520:that turns out to be an
1199:
363:* by taking as a basis:
2363:{\displaystyle \Delta }
1899:{\displaystyle \Delta }
1616:Mazur's torsion theorem
1530:transcendental function
1472:(11) has genus 26. For
1230:Riemann–Hurwitz formula
1228:is prime. Applying the
819:classical modular curve
607:are integers such that
325:, and commonly denoted
92:isomorphism classes of
38:, or the corresponding
3214:Steven D. Galbraith -
3097:Manin–Drinfeld theorem
2971:
2926:
2868:
2823:
2778:
2723:
2668:
2610:
2565:
2520:
2465:
2407:
2364:
2343:
2257:
2234:
2103:
2058:
2020:
1964:
1900:
1880:
1843:
1716:
1690:
1668:
1582:) have genus zero for
1510:modular function field
1451:
1367:of the spherical (2,3,
1318:
1154:) can be defined over
1066:
935:
589:
527:
441:
292:
2972:
2927:
2869:
2824:
2779:
2724:
2669:
2611:
2566:
2521:
2466:
2408:
2365:
2344:
2258:
2235:
2104:
2059:
2021:
1965:
1901:
1881:
1844:
1717:
1691:
1669:
1550:Möbius transformation
1468:(7) has genus 3, and
1452:
1319:
1188:compact do occur for
1139:and a point of order
1067:
936:
590:
528:
464:> 0, the image of
442:
293:
3134:Cours d'arithmétique
3038:. The result about Γ
2949:
2891:
2846:
2801:
2746:
2691:
2633:
2588:
2543:
2488:
2430:
2385:
2354:
2268:
2247:
2117:
2077:
2030:
1974:
1914:
1890:
1853:
1726:
1704:
1678:
1640:
1378:
1359:= π − π/2 − π/3 − π/
1333:Euler characteristic
1259:
1234:Gauss–Bonnet theorem
959:
855:
772:and PSL(2, 5).
543:
468:
382:
162:
3275:Panchishkin, A.A.;
3069:, it also involves
2994:monstrous moonshine
2110:
1636:The modular curves
1522:irreducible variety
1480:elements in PSL(2,
1194:quaternion algebras
1091:arithmetic geometry
533:under the action of
367:any open subset of
128:Analytic definition
59:congruence subgroup
42:, constructed as a
3107:Modularity theorem
2967:
2966:
2922:
2921:
2864:
2863:
2819:
2818:
2774:
2773:
2719:
2718:
2664:
2663:
2606:
2605:
2561:
2560:
2516:
2515:
2461:
2460:
2403:
2402:
2360:
2339:
2253:
2230:
2099:
2073:
2054:
2053:
2016:
2015:
1960:
1959:
1896:
1876:
1875:
1839:
1712:
1686:
1685:
1664:
1663:
1447:
1397:
1314:
1062:
997:
931:
893:
775:The modular curve
745:The modular curve
585:
579:
523:
437:
288:
215:
25:algebraic geometry
3242:978-0-691-08092-5
3067:Richard Borcherds
3048:Jean-Pierre Serre
2985:
2984:
2256:{\displaystyle N}
1464:(5) has genus 0,
1396:
1167:elliptic function
1163:modular equations
1008:
904:
785:dessins d'enfants
763:icosahedral group
676:(Γ) which is now
503:
417:
311:complex structure
301:The minimal such
254:
226:
78:compactifications
16:Algebraic variety
3355:
3348:Riemann surfaces
3338:Algebraic curves
3323:
3310:
3297:
3271:
3265:
3257:
3207:
3202:
3196:
3195:
3193:
3192:
3178:
3169:
3163:
3162:
3144:
3138:
3137:
3129:
3056:John G. Thompson
2976:
2974:
2973:
2968:
2965:
2964:
2931:
2929:
2928:
2923:
2920:
2919:
2907:
2906:
2873:
2871:
2870:
2865:
2862:
2861:
2828:
2826:
2825:
2820:
2817:
2816:
2783:
2781:
2780:
2775:
2772:
2771:
2759:
2758:
2728:
2726:
2725:
2720:
2717:
2716:
2704:
2703:
2673:
2671:
2670:
2665:
2662:
2661:
2649:
2648:
2615:
2613:
2612:
2607:
2604:
2603:
2570:
2568:
2567:
2562:
2559:
2558:
2525:
2523:
2522:
2517:
2514:
2513:
2501:
2500:
2470:
2468:
2467:
2462:
2459:
2458:
2446:
2445:
2412:
2410:
2409:
2404:
2401:
2400:
2369:
2367:
2366:
2361:
2348:
2346:
2345:
2342:{\displaystyle }
2340:
2335:
2334:
2322:
2321:
2309:
2308:
2296:
2295:
2283:
2282:
2262:
2260:
2259:
2254:
2239:
2237:
2236:
2231:
2229:
2228:
2213:
2212:
2200:
2199:
2190:
2189:
2177:
2176:
2161:
2160:
2142:
2141:
2129:
2128:
2111:
2108:
2106:
2105:
2100:
2089:
2088:
2072:
2063:
2061:
2060:
2055:
2043:
2042:
2025:
2023:
2022:
2017:
1987:
1986:
1969:
1967:
1966:
1961:
1940:
1939:
1927:
1926:
1905:
1903:
1902:
1897:
1885:
1883:
1882:
1877:
1874:
1866:
1865:
1848:
1846:
1845:
1840:
1838:
1837:
1822:
1821:
1809:
1808:
1799:
1798:
1786:
1785:
1770:
1769:
1751:
1750:
1738:
1737:
1721:
1719:
1718:
1713:
1711:
1695:
1693:
1692:
1687:
1673:
1671:
1670:
1665:
1653:
1652:
1456:
1454:
1453:
1448:
1398:
1389:
1323:
1321:
1320:
1315:
1304:
1296:
1071:
1069:
1068:
1063:
1058:
1054:
1006:
1002:
1001:
940:
938:
937:
932:
927:
923:
902:
898:
897:
795:(7) →
655:
640:
594:
592:
591:
586:
584:
583:
532:
530:
529:
524:
504:
501:
496:
451:coprime integers
446:
444:
443:
438:
418:
415:
410:
378:> 0, the set
358:
297:
295:
294:
289:
284:
280:
255:
252:
224:
220:
219:
113:cyclotomic field
106:rational numbers
48:upper half-plane
3363:
3362:
3358:
3357:
3356:
3354:
3353:
3352:
3328:
3327:
3326:
3308:
3295:
3281:"Modular curve"
3259:
3258:
3243:
3211:
3210:
3203:
3199:
3190:
3188:
3176:
3170:
3166:
3159:
3145:
3141:
3130:
3126:
3121:
3112:Shimura variety
3093:
3041:
3020:
3010:
2990:
2960:
2956:
2950:
2947:
2946:
2915:
2911:
2902:
2898:
2892:
2889:
2888:
2857:
2853:
2847:
2844:
2843:
2812:
2808:
2802:
2799:
2798:
2767:
2763:
2754:
2750:
2747:
2744:
2743:
2712:
2708:
2699:
2695:
2692:
2689:
2688:
2657:
2653:
2644:
2640:
2634:
2631:
2630:
2599:
2595:
2589:
2586:
2585:
2554:
2550:
2544:
2541:
2540:
2509:
2505:
2496:
2492:
2489:
2486:
2485:
2454:
2450:
2441:
2437:
2431:
2428:
2427:
2396:
2392:
2386:
2383:
2382:
2355:
2352:
2351:
2330:
2326:
2317:
2313:
2304:
2300:
2291:
2287:
2278:
2274:
2269:
2266:
2265:
2248:
2245:
2244:
2224:
2220:
2208:
2204:
2195:
2191:
2185:
2181:
2172:
2168:
2156:
2152:
2137:
2133:
2124:
2120:
2118:
2115:
2114:
2084:
2080:
2078:
2075:
2074:
2038:
2034:
2031:
2028:
2027:
1982:
1978:
1975:
1972:
1971:
1935:
1931:
1922:
1918:
1915:
1912:
1911:
1891:
1888:
1887:
1870:
1861:
1857:
1854:
1851:
1850:
1833:
1829:
1817:
1813:
1804:
1800:
1794:
1790:
1781:
1777:
1765:
1761:
1746:
1742:
1733:
1729:
1727:
1724:
1723:
1707:
1705:
1702:
1701:
1698:elliptic curves
1679:
1676:
1675:
1648:
1644:
1641:
1638:
1637:
1634:
1627:
1577:
1506:
1492:) of any level
1387:
1379:
1376:
1375:
1327:where χ = 2 − 2
1300:
1292:
1260:
1257:
1256:
1202:
1190:Fuchsian groups
1175:correspondences
1171:Hecke operators
1149:
1130:
1119:
1086:level structure
1081:elliptic curves
996:
995:
990:
984:
983:
978:
968:
967:
966:
962:
960:
957:
956:
947:
892:
891:
886:
880:
879:
874:
864:
863:
862:
858:
856:
853:
852:
839:
812:
789:Belyi functions
771:
753:. The covering
737:
729:
717:
706:
690:
656:}, the space Γ\
650:
635:
578:
577:
572:
566:
565:
557:
547:
546:
544:
541:
540:
500:
492:
469:
466:
465:
414:
406:
383:
380:
379:
349:
335:
253: and
251:
214:
213:
208:
202:
201:
196:
186:
185:
184:
180:
163:
160:
159:
130:
123:
98:complex numbers
94:elliptic curves
46:of the complex
40:algebraic curve
36:Riemann surface
17:
12:
11:
5:
3361:
3351:
3350:
3345:
3340:
3325:
3324:
3302:Ogg, Andrew P.
3298:
3293:
3272:
3241:
3220:
3219:
3218:
3209:
3208:
3197:
3164:
3157:
3139:
3123:
3122:
3120:
3117:
3116:
3115:
3109:
3104:
3099:
3092:
3089:
3039:
3018:
3008:
2989:
2986:
2983:
2982:
2977:
2963:
2959:
2955:
2944:
2942:
2938:
2937:
2932:
2918:
2914:
2910:
2905:
2901:
2897:
2886:
2884:
2880:
2879:
2874:
2860:
2856:
2852:
2841:
2839:
2835:
2834:
2829:
2815:
2811:
2807:
2796:
2794:
2790:
2789:
2784:
2770:
2766:
2762:
2757:
2753:
2741:
2739:
2735:
2734:
2729:
2715:
2711:
2707:
2702:
2698:
2686:
2684:
2680:
2679:
2674:
2660:
2656:
2652:
2647:
2643:
2639:
2628:
2626:
2622:
2621:
2616:
2602:
2598:
2594:
2583:
2581:
2577:
2576:
2571:
2557:
2553:
2549:
2538:
2536:
2532:
2531:
2526:
2512:
2508:
2504:
2499:
2495:
2483:
2481:
2477:
2476:
2471:
2457:
2453:
2449:
2444:
2440:
2436:
2425:
2423:
2419:
2418:
2413:
2399:
2395:
2391:
2380:
2378:
2374:
2373:
2370:
2359:
2349:
2338:
2333:
2329:
2325:
2320:
2316:
2312:
2307:
2303:
2299:
2294:
2290:
2286:
2281:
2277:
2273:
2263:
2252:
2241:
2240:
2227:
2223:
2219:
2216:
2211:
2207:
2203:
2198:
2194:
2188:
2184:
2180:
2175:
2171:
2167:
2164:
2159:
2155:
2151:
2148:
2145:
2140:
2136:
2132:
2127:
2123:
2098:
2095:
2092:
2087:
2083:
2052:
2049:
2046:
2041:
2037:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1985:
1981:
1958:
1955:
1952:
1949:
1946:
1943:
1938:
1934:
1930:
1925:
1921:
1895:
1873:
1869:
1864:
1860:
1836:
1832:
1828:
1825:
1820:
1816:
1812:
1807:
1803:
1797:
1793:
1789:
1784:
1780:
1776:
1773:
1768:
1764:
1760:
1757:
1754:
1749:
1745:
1741:
1736:
1732:
1710:
1684:
1662:
1659:
1656:
1651:
1647:
1633:
1632:) of genus one
1625:
1620:
1614:can occur, is
1575:
1514:function field
1505:
1502:
1458:
1457:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1395:
1392:
1386:
1383:
1365:angular defect
1325:
1324:
1313:
1310:
1307:
1303:
1299:
1295:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1201:
1198:
1147:
1128:
1117:
1097:modular curve
1073:
1072:
1061:
1057:
1053:
1048:
1044:
1041:
1038:
1035:
1032:
1027:
1023:
1020:
1017:
1014:
1011:
1005:
1000:
994:
991:
989:
986:
985:
982:
979:
977:
974:
973:
971:
965:
945:
942:
941:
930:
926:
922:
917:
913:
910:
907:
901:
896:
890:
887:
885:
882:
881:
878:
875:
873:
870:
869:
867:
861:
837:
810:
801:PSL(2, 7)
769:
735:
727:
715:
704:
689:
686:
660:* becomes the
625:Riemann sphere
617:
616:
597:
596:
595:
582:
576:
573:
571:
568:
567:
564:
561:
558:
556:
553:
552:
550:
535:
534:
522:
519:
516:
513:
510:
507:
499:
495:
491:
488:
485:
482:
479:
476:
473:
447:
436:
433:
430:
427:
424:
421:
413:
409:
405:
402:
399:
396:
393:
390:
387:
372:
348:* =
334:
331:
305:is called the
299:
298:
287:
283:
279:
274:
270:
267:
264:
261:
258:
250:
245:
241:
238:
235:
232:
229:
223:
218:
212:
209:
207:
204:
203:
200:
197:
195:
192:
191:
189:
183:
179:
176:
173:
170:
167:
129:
126:
119:
76:(Γ) which are
15:
9:
6:
4:
3:
2:
3360:
3349:
3346:
3344:
3343:Modular forms
3341:
3339:
3336:
3335:
3333:
3322:
3318:
3315:(in French),
3314:
3307:
3303:
3299:
3296:
3294:1-4020-0609-8
3290:
3286:
3282:
3278:
3277:Parshin, A.N.
3273:
3269:
3263:
3256:
3252:
3248:
3244:
3238:
3234:
3230:
3226:
3225:Shimura, Goro
3222:
3221:
3217:
3213:
3212:
3206:
3201:
3186:
3182:
3175:
3168:
3160:
3158:3-540-07392-2
3154:
3150:
3143:
3135:
3128:
3124:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3094:
3088:
3086:
3085:
3079:
3078:
3072:
3068:
3063:
3061:
3060:Jack Daniel's
3057:
3053:
3049:
3045:
3037:
3036:monster group
3033:
3029:
3025:
3021:
3014:
3006:
3001:
2999:
2995:
2981:
2978:
2961:
2957:
2953:
2945:
2943:
2940:
2939:
2936:
2933:
2916:
2912:
2908:
2903:
2899:
2895:
2887:
2885:
2882:
2881:
2878:
2875:
2858:
2854:
2850:
2842:
2840:
2837:
2836:
2833:
2830:
2813:
2809:
2805:
2797:
2795:
2792:
2791:
2788:
2785:
2768:
2764:
2760:
2755:
2751:
2742:
2740:
2737:
2736:
2733:
2730:
2713:
2709:
2705:
2700:
2696:
2687:
2685:
2682:
2681:
2678:
2675:
2658:
2654:
2650:
2645:
2641:
2637:
2629:
2627:
2624:
2623:
2620:
2617:
2600:
2596:
2592:
2584:
2582:
2579:
2578:
2575:
2572:
2555:
2551:
2547:
2539:
2537:
2534:
2533:
2530:
2527:
2510:
2506:
2502:
2497:
2493:
2484:
2482:
2479:
2478:
2475:
2472:
2455:
2451:
2447:
2442:
2438:
2434:
2426:
2424:
2421:
2420:
2417:
2414:
2397:
2393:
2389:
2381:
2379:
2376:
2375:
2371:
2350:
2331:
2327:
2323:
2318:
2314:
2310:
2305:
2301:
2297:
2292:
2288:
2284:
2279:
2275:
2264:
2250:
2243:
2242:
2225:
2221:
2217:
2214:
2209:
2205:
2201:
2196:
2192:
2186:
2182:
2178:
2173:
2169:
2165:
2162:
2157:
2153:
2149:
2146:
2143:
2138:
2134:
2130:
2125:
2121:
2112:
2093:
2085:
2081:
2071:
2069:
2068:
2047:
2039:
2035:
2009:
2006:
2003:
2000:
1997:
1994:
1988:
1983:
1979:
1953:
1950:
1947:
1941:
1936:
1932:
1928:
1923:
1919:
1909:
1867:
1862:
1858:
1834:
1830:
1826:
1823:
1818:
1814:
1810:
1805:
1801:
1795:
1791:
1787:
1782:
1778:
1774:
1771:
1766:
1762:
1758:
1755:
1752:
1747:
1743:
1739:
1734:
1730:
1699:
1682:
1657:
1649:
1645:
1631:
1624:
1619:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1574:
1569:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1540:(1) = PSL(2,
1539:
1535:
1531:
1527:
1523:
1519:
1515:
1511:
1508:In general a
1501:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1471:
1467:
1463:
1444:
1438:
1435:
1432:
1423:
1420:
1417:
1408:
1405:
1402:
1393:
1390:
1384:
1381:
1374:
1373:
1372:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1311:
1308:
1305:
1297:
1289:
1280:
1274:
1268:
1265:
1262:
1255:
1254:
1253:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1204:The covering
1197:
1195:
1191:
1187:
1183:
1180:Quotients of
1178:
1176:
1172:
1168:
1164:
1159:
1157:
1153:
1146:
1142:
1138:
1134:
1127:
1123:
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1087:
1082:
1078:
1077:moduli spaces
1059:
1055:
1051:
1046:
1042:
1039:
1036:
1033:
1030:
1025:
1021:
1018:
1015:
1012:
1009:
1003:
998:
992:
987:
980:
975:
969:
963:
955:
954:
953:
951:
928:
924:
920:
915:
911:
908:
905:
899:
894:
888:
883:
876:
871:
865:
859:
851:
850:
849:
847:
843:
835:
832:
828:
824:
820:
816:
809:
804:
802:
798:
794:
790:
786:
782:
781:Klein quartic
778:
773:
768:
764:
760:
756:
752:
748:
743:
741:
733:
725:
721:
714:
710:
703:
699:
695:
685:
683:
679:
675:
671:
667:
663:
659:
653:
648:
644:
638:
633:
629:
626:
622:
614:
610:
606:
602:
598:
580:
574:
569:
562:
559:
554:
548:
539:
538:
537:
536:
517:
514:
508:
497:
489:
486:
480:
463:
459:
455:
452:
448:
431:
428:
422:
411:
403:
400:
394:
377:
373:
370:
366:
365:
364:
362:
356:
352:
347:
344:
340:
330:
328:
324:
323:modular curve
320:
316:
312:
308:
304:
285:
281:
277:
272:
268:
265:
262:
259:
256:
248:
243:
239:
236:
233:
230:
227:
221:
216:
210:
205:
198:
193:
187:
181:
177:
171:
158:
157:
156:
154:
150:
149:
143:
139:
135:
125:
122:
117:
114:
110:
107:
103:
99:
95:
91:
87:
83:
79:
75:
72:
68:
64:
63:modular group
60:
56:
52:
49:
45:
41:
37:
33:
30:
29:modular curve
26:
22:
21:number theory
3312:
3284:
3254:
3228:
3200:
3189:. Retrieved
3184:
3180:
3167:
3148:
3142:
3133:
3127:
3083:
3076:
3064:
3046:) is due to
3043:
3031:
3027:
3023:
3012:
3002:
2997:
2991:
2065:
1907:
1635:
1629:
1622:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1572:
1570:
1565:
1561:
1557:
1553:
1545:
1541:
1537:
1518:moduli space
1509:
1507:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1459:
1368:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1328:
1326:
1249:
1241:
1237:
1225:
1221:
1217:
1213:
1209:
1205:
1203:
1185:
1181:
1179:
1160:
1155:
1151:
1144:
1140:
1136:
1132:
1125:
1121:
1114:
1106:
1102:
1098:
1094:
1093:. The level
1084:
1074:
949:
943:
845:
841:
833:
826:
822:
814:
807:
805:
796:
792:
776:
774:
766:
758:
754:
746:
744:
739:
731:
723:
719:
712:
708:
701:
697:
693:
691:
681:
673:
669:
665:
657:
651:
646:
636:
631:
627:
620:
618:
612:
608:
604:
600:
461:
457:
453:
375:
368:
360:
354:
350:
345:
342:
338:
336:
326:
322:
317:to obtain a
314:
306:
302:
300:
152:
147:
141:
133:
131:
120:
115:
108:
85:
81:
73:
70:
66:
50:
31:
28:
18:
3026:) in SL(2,
2109:of genus 1
1849:. This is,
1571:The spaces
1566:Hauptmoduln
779:(7) is the
751:icosahedron
645:called the
619:This turns
90:parametrize
3332:Categories
3205:Ogg (1974)
3191:2022-11-06
3119:References
3052:Andrew Ogg
3005:normalizer
1594:and has a
1554:Hauptmodul
1534:j-function
1504:Genus zero
647:cusps of Γ
319:noncompact
307:level of Γ
82:cusps of Γ
3227:(1994) ,
3077:functions
2954:−
2909:⋅
2896:−
2851:−
2806:−
2761:⋅
2706:⋅
2651:⋅
2638:−
2593:−
2548:−
2503:⋅
2448:⋅
2435:−
2390:−
2358:Δ
1995:−
1989:∈
1942:∈
1894:Δ
1868:∈
1564:, plural
1436:−
1421:−
1306:⋅
1269:χ
1266:π
1263:−
1244:). For a
1040:≡
1019:≡
1013:≡
909:≡
560:−
509:τ
498:∣
490:∈
487:τ
481:∪
475:∞
423:τ
412:∣
404:∈
401:τ
395:∪
389:∞
266:≡
237:≡
231:≡
166:Γ
61:Γ of the
34:(Γ) is a
3304:(1974),
3262:citation
3091:See also
3082:modular
3075:modular
1169:theory.
836:. Then Γ
734:), and Γ
688:Examples
460:and all
449:for all
374:for all
44:quotient
3321:0417184
3251:1291394
3187:: 44–45
1908:reduced
1363:is the
1355:), and
1331:is the
1111:torsion
817:), the
711:), and
678:compact
102:defined
53:by the
3319:
3291:
3249:
3239:
3155:
2372:LMFDB
1248:level
1124:) and
1113:. For
1007:
903:
831:modulo
757:(5) →
643:orbits
599:where
225:
55:action
3309:(PDF)
3177:(PDF)
3084:forms
3015:) of
1700:over
1602:with
1526:Genus
1512:is a
1460:Thus
1339:| = (
1252:≥ 5,
1246:prime
1224:) if
1200:Genus
1184:that
1083:with
944:and Γ
684:(Γ).
664:of Γ\
329:(Γ).
111:or a
57:of a
3289:ISBN
3268:link
3237:ISBN
3153:ISBN
3054:and
2980:link
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