424:
412:
1256:
2400:
2411:
2718:
1930:
2918:
will yield the conjectured result in the introduction. This conjecture is based on extensive numerical computations of eigenvalues of the surface and other surfaces of genus 2. In particular, the spectrum of the Bolza surface is known to a very high accuracy
1079:
2256:
1572:
1030:
2139:
1475:
1966:
of the surface. A set of
Fenchel-Nielsen coordinates for a surface of genus 2 consists of three pairs, each pair being a length and twist. Perhaps the simplest such set of coordinates for the Bolza surface is
1715:
2453:
534:
2053:
2747:
to retrieve information about the representation theory of the group. In particular, there are four 1-dimensional, two 2-dimensional, four 3-dimensional, and three 4-dimensional irreducible representations, and
3432:
3494:
1826:
2221:
3636:
2857:
677:
229:
1299:
1811:
3544:
4727:
1251:{\displaystyle g_{k}={\begin{pmatrix}1+{\sqrt {2}}&(2+{\sqrt {2}})\alpha e^{\tfrac {ik\pi }{4}}\\(2+{\sqrt {2}})\alpha e^{-{\tfrac {ik\pi }{4}}}&1+{\sqrt {2}}\end{pmatrix}},}
2874:
Plots of the three eigenfunctions corresponding to the first positive eigenvalue of the Bolza surface. Functions are zero on the light blue lines. These plots were produced using
1820:
exist to calculate the side lengths of a (2,3,8) triangles explicitly. The systole is equal to four times the length of the side of medial length in a (2,3,8) triangle, that is,
921:
591:
727:
310:
172:
4758:
2902:. The first eigenspace (that is, the eigenspace corresponding to the first positive eigenvalue) of the Bolza surface is three-dimensional, and the second, four-dimensional (
2395:{\displaystyle t={\frac {\operatorname {\rm {arcosh}} \left({\sqrt {{\tfrac {2}{7}}(3+{\sqrt {2}})}}\right)}{\operatorname {\rm {arcosh}} (1+{\sqrt {2}})}}\approx 0.321281.}
3327:
1602:
1337:
1071:
278:
4663:
3385:
3291:
3255:
3219:
3183:
3147:
3111:
3075:
3039:
3003:
2967:
4045:
2248:
1960:
1629:
140:
110:
2900:
888:
865:
827:
789:
747:
467:
3499:
respectively, where all decimal places are believed to be correct. It is conjectured that the spectral determinant is maximized in genus 2 for the Bolza surface.
540:-two subgroup of the group of reflections, which consists of products of an even number of reflections, which has an abstract presentation in terms of generators
2741:
1766:
1738:
397:
334:
64:
2418:
The fundamental domain of the Bolza surface is a regular octagon in the
Poincaré disk; the four symmetric actions that generate the (full) symmetry group are:
371:
acting on functions on the Bolza surface is of interest to both mathematicians and physicists, since the surface is conjectured to maximize the first positive
2871:
3981:
Strohmaier, Alexander (2017). Girouard, Alexandre (ed.). "Compuration of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces".
1494:
929:
2058:
3847:
Strohmaier, A.; Uski, V. (2013). "An
Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces".
4468:
1345:
2713:{\displaystyle \langle R,\,S,\,T,\,U\mid R^{8}=S^{2}=T^{2}=U^{3}=RSRS=STST=RTR^{3}T=e,\,UR=R^{7}U^{2},\,U^{2}R=STU,\,US=SU^{2},\,UT=RSU\rangle ,}
4197:
4656:
1637:
480:
4560:
4157:
4038:
1970:
348:
Riemann surface, it arises as the ramified double cover of the
Riemann sphere, with ramification locus at the six vertices of a regular
3393:
1925:{\displaystyle \ell _{1}=4\operatorname {\rm {arcosh}} \left({\tfrac {\csc \left({\tfrac {\pi }{8}}\right)}{2}}\right)\approx 3.05714.}
4626:
4248:
4147:
4616:
3443:
4649:
4012:
3903:
3757:
416:
4326:
4031:
477:
defining the Bolza surface is a subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles
1963:
2147:
4473:
4384:
3717:
4394:
4321:
3556:
2754:
368:
360:
596:
4686:
4071:
2744:
4291:
4187:
4550:
4514:
1073:. Opposite sides of the octagon are identified under the action of the Fuchsian group. Its generators are the matrices
180:
4213:
4126:
1264:
4794:
4524:
4162:
4789:
4570:
1774:
4483:
4463:
4399:
4316:
4177:
4218:
3514:
4182:
4696:
4374:
2870:
4167:
3341:
1484:, which gives all of the possible lengths of geodesic loops. The shortest such length is called the
355:
The Bolza surface has attracted the attention of physicists, as it provides a relatively simple model for
4545:
4281:
4081:
3674:
897:
423:
4243:
4192:
2447:
These are shown by the bold lines in the adjacent figure. They satisfy the following set of relations:
543:
682:
286:
148:
4706:
4621:
4493:
4404:
4152:
3939:
3866:
3799:
2915:
1481:
113:
4458:
1816:
It is possible to obtain an equivalent closed form of the systole directly from the triangle group.
4336:
4301:
4258:
4238:
3654:
3642:
3305:
2911:
2410:
1817:
1580:
1304:
1038:
537:
349:
345:
237:
175:
44:
3350:
3269:
3233:
3197:
3161:
3125:
3089:
3053:
3017:
2981:
2945:
4732:
4711:
4588:
4172:
3990:
3963:
3882:
3856:
3774:
3743:
3725:
3715:
Katz, M.; Sabourau, S. (2006). "An optimal systolic inequality for CAT(0) metrics in genus two".
3704:
3508:
70:
4691:
4379:
4359:
4331:
427:
The fundamental domain of the Bolza surface in the
Poincaré disk; opposite sides are identified.
231:
of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation
3927:
2226:
1938:
1607:
119:
76:
4768:
4672:
4488:
4435:
4306:
4121:
4116:
4008:
3955:
3899:
3835:
1741:
337:
313:
2885:
873:
832:
794:
756:
732:
434:
4763:
4742:
4478:
4364:
4341:
4000:
3947:
3874:
3807:
3766:
3735:
3696:
3669:
3664:
470:
376:
4593:
4409:
4351:
4253:
4076:
4055:
364:
40:
4276:
3943:
3870:
3803:
2437:– reflection in the side of one of the 16 (4,4,4) triangles that tessellate the octagon;
4737:
4101:
4086:
4063:
3989:. Montréal: Centre de Recherches Mathématiques and American Mathematical Society: 194.
3345:
2726:
1751:
1723:
1567:{\displaystyle \ell _{1}=2\operatorname {\rm {arcosh}} (1+{\sqrt {2}})\approx 3.05714.}
1025:{\displaystyle p_{k}=2^{-1/4}e^{i\left({\tfrac {\pi }{8}}+{\tfrac {k\pi }{4}}\right)},}
750:
474:
382:
319:
49:
3832:Über das Spektrum des Laplace-Operators auf einer Schar kompakter Riemannscher Flächen
3687:
Bolza, Oskar (1887), "On Binary
Sextics with Linear Transformations into Themselves",
2923:). The following table gives the first ten positive eigenvalues of the Bolza surface.
2134:{\displaystyle \ell _{2}=2\operatorname {\rm {arcosh}} (3+2{\sqrt {2}})\approx 4.8969}
4783:
4608:
4389:
4296:
4091:
3811:
3778:
3659:
891:
356:
67:
3967:
3886:
3747:
4555:
4529:
4519:
4509:
4311:
4131:
411:
143:
2927:
Numerical computations of the first ten positive eigenvalues of the Bolza surface
3755:
Schmutz, P. (1993). "Riemann surfaces with shortest geodesic of maximal length".
2875:
4430:
4268:
32:
20:
3951:
1470:{\displaystyle g_{0}g_{1}^{-1}g_{2}g_{3}^{-1}g_{0}^{-1}g_{1}g_{2}^{-1}g_{3}=1.}
894:, the fundamental domain of the Bolza surface is a regular octagon with angles
4425:
3878:
372:
4023:
4286:
3839:
3739:
829:
group does not have a realization in terms of a quaternion algebra, but the
400:
3959:
3787:
415:
The tiling of the Bolza surface by reflection domains is a quotient of the
1710:{\displaystyle \ell _{n}=2\operatorname {\rm {arcosh}} (m+n{\sqrt {2}}),}
352:
inscribed in the sphere, as can be readily seen from the equation above.
3822:
Properties of
Eigenvalues on Riemann Surfaces with Large Symmetry Groups
3820:
3730:
529:{\displaystyle {\tfrac {\pi }{2}},{\tfrac {\pi }{3}},{\tfrac {\pi }{8}}}
4598:
4583:
4004:
3770:
3708:
2743:
is the trivial (identity) action. One may use this set of relations in
2048:{\displaystyle (\ell _{2},{\tfrac {1}{2}};\;\ell _{1},0;\;\ell _{1},0)}
536:. The group of orientation preserving isometries is a subgroup of the
4578:
4641:
3700:
3995:
336:
hyperbolic surfaces, the Bolza surface maximizes the length of the
3861:
2869:
2409:
422:
410:
3427:{\displaystyle \det {}_{\zeta }(\Delta )\approx 4.72273280444557}
1339:, along with their inverses. The generators satisfy the relation
4645:
4027:
3489:{\displaystyle \zeta _{\Delta }(-1/2)\approx -0.65000636917383}
2882:
Here, spectral theory refers to the spectrum of the
Laplacian,
2414:
The four generators of the symmetry group of the Bolza surface
1744:(but omitting 4, 24, 48, 72, 140, and various higher values) (
174:). The full automorphism group (including reflections) is the
3898:. Graduate Texts in Math. Vol. 219. New York: Springer.
2443:– rotation of order 3 about the centre of a (4,4,4) triangle.
749:
defining the Bolza surface is also a subgroup of the (3,3,4)
2914:
of the nodal lines of functions in the first eigenspace in
2216:{\displaystyle (\ell _{1},t;\;\ell _{1},t;\;\ell _{1},t)}
1631:
of the length spectrum for the Bolza surface is given by
3631:{\displaystyle i^{2}=-3,\;j^{2}={\sqrt {2}},\;ij=-ji,}
2852:{\displaystyle 4(1^{2})+2(2^{2})+4(3^{2})+3(4^{2})=96}
2425:– rotation of order 8 about the centre of the octagon;
2301:
1991:
1890:
1876:
1745:
1201:
1146:
1101:
996:
981:
902:
515:
500:
485:
3926:
Aurich, R.; Sieber, M.; Steiner, F. (1 August 1988).
3559:
3517:
3446:
3396:
3353:
3308:
3272:
3259:
23.0785584813816351550752062995745529967807846993874
3236:
3223:
20.5198597341420020011497712606420998241440266544635
3200:
3164:
3128:
3092:
3056:
3020:
2984:
2948:
2888:
2757:
2729:
2456:
2259:
2229:
2150:
2061:
1973:
1941:
1829:
1777:
1754:
1726:
1640:
1610:
1583:
1497:
1488:
of the surface. The systole of the Bolza surface is
1348:
1307:
1267:
1082:
1041:
932:
900:
876:
835:
797:
759:
735:
685:
672:{\displaystyle s_{2}{}^{2}=s_{3}{}^{3}=s_{8}{}^{8}=1}
599:
546:
483:
437:
385:
322:
289:
240:
183:
151:
122:
79:
52:
4728:
Gromov's systolic inequality for essential manifolds
3331:
30.833042737932549674243957560470189329562655076386
3295:
28.079605737677729081562207945001124964945310994142
359:; in this context, it is usually referred to as the
4751:
4720:
4679:
4607:
4569:
4538:
4502:
4451:
4444:
4418:
4350:
4267:
4231:
4206:
4140:
4109:
4100:
4062:
3825:(PhD thesis, unpublished). Loughborough University.
3788:"Periodic orbits on the regular hyperbolic octagon"
3187:18.65881962726019380629623466134099363131475471461
3151:15.04891613326704874618158434025881127570452711372
3115:14.72621678778883204128931844218483598373384446932
3630:
3546:generated as an associative algebra by generators
3538:
3488:
3426:
3379:
3321:
3285:
3249:
3213:
3177:
3141:
3105:
3069:
3033:
2997:
2961:
2894:
2851:
2735:
2712:
2394:
2242:
2215:
2133:
2047:
1954:
1924:
1805:
1760:
1732:
1709:
1623:
1596:
1566:
1469:
1331:
1293:
1250:
1065:
1024:
915:
882:
859:
821:
783:
741:
721:
671:
585:
528:
461:
391:
328:
304:
272:
223:
166:
134:
104:
58:
3786:Aurich, R.; Bogomolny, E.B.; Steiner, F. (1991).
3007:3.8388872588421995185866224504354645970819150157
2223:, where all three of the lengths are the systole
431:The Bolza surface is conformally equivalent to a
224:{\displaystyle GL_{2}(3)\rtimes \mathbb {Z} _{2}}
4759:Gromov's inequality for complex projective space
3928:"Quantum Chaos of the Hadamard–Gutzwiller Model"
3397:
3079:8.249554815200658121890106450682456568390578132
3043:5.353601341189050410918048311031446376357372198
2144:There is also a "symmetric" set of coordinates
1294:{\displaystyle \alpha ={\sqrt {{\sqrt {2}}-1}}}
2920:
4657:
4039:
8:
2704:
2457:
1797:
1778:
1806:{\displaystyle \vert m-n{\sqrt {2}}\vert .}
375:of the Laplacian among all compact, closed
4664:
4650:
4642:
4448:
4106:
4046:
4032:
4024:
3606:
3582:
2193:
2173:
2025:
2005:
3994:
3860:
3729:
3596:
3587:
3564:
3558:
3539:{\displaystyle \mathbb {Q} ({\sqrt {2}})}
3526:
3519:
3518:
3516:
3466:
3451:
3445:
3403:
3401:
3395:
3366:
3352:
3313:
3307:
3277:
3271:
3241:
3235:
3205:
3199:
3169:
3163:
3133:
3127:
3097:
3091:
3061:
3055:
3025:
3019:
2989:
2983:
2953:
2947:
2887:
2834:
2812:
2790:
2768:
2756:
2728:
2685:
2676:
2659:
2635:
2630:
2621:
2611:
2597:
2579:
2530:
2517:
2504:
2491:
2480:
2473:
2466:
2455:
2373:
2340:
2339:
2321:
2300:
2298:
2270:
2269:
2266:
2258:
2250:and all three of the twists are given by
2234:
2228:
2198:
2178:
2158:
2149:
2115:
2079:
2078:
2066:
2060:
2030:
2010:
1990:
1981:
1972:
1946:
1940:
1889:
1875:
1847:
1846:
1834:
1828:
1790:
1776:
1768:is the unique odd integer that minimizes
1753:
1725:
1694:
1658:
1657:
1645:
1639:
1615:
1609:
1588:
1582:
1548:
1515:
1514:
1502:
1496:
1455:
1442:
1437:
1427:
1414:
1409:
1396:
1391:
1381:
1368:
1363:
1353:
1347:
1306:
1276:
1274:
1266:
1230:
1200:
1196:
1179:
1145:
1128:
1110:
1096:
1087:
1081:
1040:
995:
980:
971:
957:
950:
937:
931:
901:
899:
875:
834:
796:
758:
734:
713:
700:
690:
684:
657:
655:
648:
635:
633:
626:
613:
611:
604:
598:
577:
564:
551:
545:
514:
499:
484:
482:
436:
384:
321:
296:
292:
291:
288:
258:
245:
239:
215:
211:
210:
191:
182:
158:
154:
153:
150:
121:
87:
78:
51:
3896:The Arithmetic of Hyperbolic 3-Manifolds
2925:
753:, which is a subgroup of index 2 in the
3918:
341:
66:with the highest possible order of the
4469:Clifford's theorem on special divisors
3849:Communications in Mathematical Physics
1480:These generators are connected to the
2907:
36:
7:
3511:can be taken to be the algebra over
2910:). It is thought that investigating
2903:
1746:Aurich, Bogomolny & Steiner 1991
3834:(PhD thesis). University of Basel.
3507:Following MacLachlan and Reid, the
316:of the affine curve. Of all genus
27:, alternatively, complex algebraic
4627:Vector bundles on algebraic curves
4561:Weber's theorem (Algebraic curves)
4158:Hasse's theorem on elliptic curves
4148:Counting points on elliptic curves
3452:
3412:
2889:
2356:
2353:
2350:
2347:
2344:
2341:
2286:
2283:
2280:
2277:
2274:
2271:
2095:
2092:
2089:
2086:
2083:
2080:
1863:
1860:
1857:
1854:
1851:
1848:
1674:
1671:
1668:
1665:
1662:
1659:
1531:
1528:
1525:
1522:
1519:
1516:
916:{\displaystyle {\tfrac {\pi }{4}}}
877:
736:
14:
3894:Maclachlan, C.; Reid, A. (2003).
3641:with an appropriate choice of an
586:{\displaystyle s_{2},s_{3},s_{8}}
417:order-3 bisected octagonal tiling
16:In mathematics, a Riemann surface
722:{\displaystyle s_{2}s_{3}=s_{8}}
305:{\displaystyle \mathbb {C} ^{2}}
167:{\displaystyle \mathbb {F} _{3}}
4249:Hurwitz's automorphisms theorem
3689:American Journal of Mathematics
4474:Gonality of an algebraic curve
4385:Differential of the first kind
3792:Physica D: Nonlinear Phenomena
3533:
3523:
3474:
3457:
3415:
3409:
3374:
3357:
2840:
2827:
2818:
2805:
2796:
2783:
2774:
2761:
2431:– reflection in the real line;
2380:
2364:
2328:
2312:
2210:
2151:
2122:
2103:
2042:
1974:
1701:
1682:
1555:
1539:
1186:
1170:
1135:
1119:
854:
836:
816:
798:
778:
760:
456:
438:
203:
197:
99:
93:
1:
4617:Birkhoff–Grothendieck theorem
4327:Nagata's conjecture on curves
4198:Schoof–Elkies–Atkin algorithm
4072:Five points determine a conic
3322:{\displaystyle \lambda _{10}}
1597:{\displaystyle n^{\text{th}}}
1332:{\displaystyle k=0,\ldots ,3}
1066:{\displaystyle k=0,\ldots ,7}
273:{\displaystyle y^{2}=x^{5}-x}
4188:Supersingular elliptic curve
3812:10.1016/0167-2789(91)90053-C
3380:{\displaystyle \zeta (-1/2)}
3286:{\displaystyle \lambda _{9}}
3250:{\displaystyle \lambda _{8}}
3214:{\displaystyle \lambda _{7}}
3178:{\displaystyle \lambda _{6}}
3142:{\displaystyle \lambda _{5}}
3106:{\displaystyle \lambda _{4}}
3070:{\displaystyle \lambda _{3}}
3034:{\displaystyle \lambda _{2}}
2998:{\displaystyle \lambda _{1}}
2962:{\displaystyle \lambda _{0}}
4395:Riemann's existence theorem
4322:Hilbert's sixteenth problem
4214:Elliptic curve cryptography
4127:Fundamental pair of periods
1964:Fenchel–Nielsen coordinates
312:. The Bolza surface is the
4811:
4687:Loewner's torus inequality
4525:Moduli of algebraic curves
3952:10.1103/PhysRevLett.61.483
2921:Strohmaier & Uski 2013
473:. More specifically, the
3879:10.1007/s00220-012-1557-1
3387:of the Bolza surface are
2406:Symmetries of the surface
2243:{\displaystyle \ell _{1}}
1955:{\displaystyle \ell _{n}}
1624:{\displaystyle \ell _{n}}
369:Laplace–Beltrami operator
361:Hadamard–Gutzwiller model
135:{\displaystyle 2\times 2}
105:{\displaystyle GL_{2}(3)}
4292:Cayley–Bacharach theorem
4219:Elliptic curve primality
3983:Contemporary Mathematics
4721:1-systoles of manifolds
4697:Filling area conjecture
4551:Riemann–Hurwitz formula
4515:Gromov–Witten invariant
4375:Compact Riemann surface
4163:Mazur's torsion theorem
3932:Physical Review Letters
3740:10.2140/pjm.2006.227.95
2895:{\displaystyle \Delta }
883:{\displaystyle \Gamma }
860:{\displaystyle (3,3,4)}
822:{\displaystyle (2,3,8)}
784:{\displaystyle (2,3,8)}
742:{\displaystyle \Gamma }
469:triangle surface – see
462:{\displaystyle (2,3,8)}
399:with constant negative
4680:1-systoles of surfaces
4168:Modular elliptic curve
3632:
3540:
3490:
3428:
3381:
3323:
3287:
3251:
3215:
3179:
3143:
3107:
3071:
3035:
2999:
2963:
2896:
2879:
2853:
2737:
2714:
2415:
2396:
2244:
2217:
2135:
2049:
1956:
1926:
1807:
1762:
1734:
1711:
1625:
1598:
1568:
1471:
1333:
1295:
1252:
1067:
1026:
917:
884:
861:
823:
785:
743:
729:. The Fuchsian group
723:
673:
587:
530:
463:
428:
420:
393:
330:
306:
274:
225:
168:
136:
106:
73:in this genus, namely
60:
4082:Rational normal curve
3675:First Hurwitz triplet
3633:
3541:
3491:
3429:
3382:
3324:
3288:
3252:
3216:
3180:
3144:
3108:
3072:
3036:
3000:
2964:
2897:
2873:
2854:
2738:
2715:
2413:
2397:
2245:
2218:
2136:
2050:
1957:
1935:The geodesic lengths
1927:
1808:
1763:
1735:
1712:
1626:
1599:
1569:
1472:
1334:
1296:
1253:
1068:
1027:
918:
885:
862:
824:
786:
744:
724:
674:
588:
531:
464:
426:
414:
394:
331:
307:
275:
226:
169:
137:
107:
61:
4707:Systoles of surfaces
4622:Stable vector bundle
4494:Weil reciprocity law
4484:Riemann–Roch theorem
4464:Brill–Noether theory
4400:Riemann–Roch theorem
4317:Genus–degree formula
4178:Mordell–Weil theorem
4153:Division polynomials
3557:
3515:
3444:
3394:
3351:
3342:spectral determinant
3306:
3270:
3234:
3198:
3162:
3126:
3090:
3054:
3018:
2982:
2946:
2886:
2755:
2727:
2454:
2257:
2227:
2148:
2059:
1971:
1939:
1827:
1775:
1752:
1724:
1638:
1608:
1581:
1495:
1346:
1305:
1265:
1080:
1039:
930:
898:
874:
870:Under the action of
833:
795:
791:triangle group. The
757:
733:
683:
597:
544:
481:
435:
383:
320:
287:
238:
181:
149:
120:
114:general linear group
77:
50:
4712:Eisenstein integers
4445:Structure of curves
4337:Quartic plane curve
4259:Hyperelliptic curve
4239:De Franchis theorem
4183:Nagell–Lutz theorem
3944:1988PhRvL..61..483A
3871:2013CMaPh.317..827S
3804:1991PhyD...48...91A
3655:Hyperelliptic curve
2928:
1962:also appear in the
1450:
1422:
1404:
1376:
176:semi-direct product
4733:Essential manifold
4452:Divisors on curves
4244:Faltings's theorem
4193:Schoof's algorithm
4173:Modularity theorem
3830:Jenni, F. (1981).
3771:10.1007/BF01896258
3628:
3536:
3509:quaternion algebra
3503:Quaternion algebra
3486:
3424:
3377:
3319:
3283:
3247:
3211:
3175:
3139:
3103:
3067:
3031:
2995:
2959:
2926:
2892:
2880:
2849:
2733:
2710:
2416:
2392:
2310:
2240:
2213:
2131:
2045:
2000:
1952:
1922:
1910:
1899:
1803:
1758:
1730:
1707:
1621:
1594:
1564:
1467:
1433:
1405:
1387:
1359:
1329:
1291:
1248:
1239:
1218:
1163:
1063:
1022:
1010:
990:
913:
911:
880:
857:
819:
781:
739:
719:
669:
583:
526:
524:
509:
494:
459:
429:
421:
389:
326:
302:
270:
221:
164:
142:matrices over the
132:
102:
71:automorphism group
56:
4795:Systolic geometry
4777:
4776:
4769:Systolic category
4673:Systolic geometry
4639:
4638:
4635:
4634:
4546:Hasse–Witt matrix
4489:Weierstrass point
4436:Smooth completion
4405:TeichmĂĽller space
4307:Cubic plane curve
4227:
4226:
4141:Arithmetic theory
4122:Elliptic integral
4117:Elliptic function
3819:Cook, J. (2018).
3601:
3531:
3338:
3337:
2916:TeichmĂĽller space
2736:{\displaystyle e}
2384:
2378:
2331:
2326:
2309:
2120:
1999:
1909:
1898:
1795:
1761:{\displaystyle m}
1742:positive integers
1740:runs through the
1733:{\displaystyle n}
1699:
1591:
1553:
1289:
1281:
1235:
1217:
1184:
1162:
1133:
1115:
1009:
989:
910:
523:
508:
493:
392:{\displaystyle 2}
329:{\displaystyle 2}
314:smooth completion
112:of order 48 (the
59:{\displaystyle 2}
39:)), is a compact
4802:
4790:Riemann surfaces
4764:Systolic freedom
4743:Hermite constant
4666:
4659:
4652:
4643:
4479:Jacobian variety
4449:
4352:Riemann surfaces
4342:Real plane curve
4302:Cramer's paradox
4282:BĂ©zout's theorem
4107:
4056:algebraic curves
4048:
4041:
4034:
4025:
4019:
4018:
4005:10.1090/conm/700
3998:
3978:
3972:
3971:
3923:
3909:
3890:
3864:
3843:
3826:
3815:
3782:
3751:
3733:
3718:Pacific J. Math.
3711:
3670:Macbeath surface
3637:
3635:
3634:
3629:
3602:
3597:
3592:
3591:
3569:
3568:
3545:
3543:
3542:
3537:
3532:
3527:
3522:
3495:
3493:
3492:
3487:
3484:0.65000636917383
3470:
3456:
3455:
3433:
3431:
3430:
3425:
3422:4.72273280444557
3408:
3407:
3402:
3386:
3384:
3383:
3378:
3370:
3328:
3326:
3325:
3320:
3318:
3317:
3292:
3290:
3289:
3284:
3282:
3281:
3256:
3254:
3253:
3248:
3246:
3245:
3220:
3218:
3217:
3212:
3210:
3209:
3184:
3182:
3181:
3176:
3174:
3173:
3148:
3146:
3145:
3140:
3138:
3137:
3112:
3110:
3109:
3104:
3102:
3101:
3076:
3074:
3073:
3068:
3066:
3065:
3040:
3038:
3037:
3032:
3030:
3029:
3004:
3002:
3001:
2996:
2994:
2993:
2968:
2966:
2965:
2960:
2958:
2957:
2935:Numerical value
2929:
2901:
2899:
2898:
2893:
2858:
2856:
2855:
2850:
2839:
2838:
2817:
2816:
2795:
2794:
2773:
2772:
2742:
2740:
2739:
2734:
2719:
2717:
2716:
2711:
2681:
2680:
2640:
2639:
2626:
2625:
2616:
2615:
2584:
2583:
2535:
2534:
2522:
2521:
2509:
2508:
2496:
2495:
2401:
2399:
2398:
2393:
2385:
2383:
2379:
2374:
2360:
2359:
2337:
2336:
2332:
2327:
2322:
2311:
2302:
2299:
2290:
2289:
2267:
2249:
2247:
2246:
2241:
2239:
2238:
2222:
2220:
2219:
2214:
2203:
2202:
2183:
2182:
2163:
2162:
2140:
2138:
2137:
2132:
2121:
2116:
2099:
2098:
2071:
2070:
2054:
2052:
2051:
2046:
2035:
2034:
2015:
2014:
2001:
1992:
1986:
1985:
1961:
1959:
1958:
1953:
1951:
1950:
1931:
1929:
1928:
1923:
1915:
1911:
1905:
1904:
1900:
1891:
1877:
1867:
1866:
1839:
1838:
1812:
1810:
1809:
1804:
1796:
1791:
1767:
1765:
1764:
1759:
1739:
1737:
1736:
1731:
1716:
1714:
1713:
1708:
1700:
1695:
1678:
1677:
1650:
1649:
1630:
1628:
1627:
1622:
1620:
1619:
1603:
1601:
1600:
1595:
1593:
1592:
1589:
1573:
1571:
1570:
1565:
1554:
1549:
1535:
1534:
1507:
1506:
1476:
1474:
1473:
1468:
1460:
1459:
1449:
1441:
1432:
1431:
1421:
1413:
1403:
1395:
1386:
1385:
1375:
1367:
1358:
1357:
1338:
1336:
1335:
1330:
1300:
1298:
1297:
1292:
1290:
1282:
1277:
1275:
1257:
1255:
1254:
1249:
1244:
1243:
1236:
1231:
1221:
1220:
1219:
1213:
1202:
1185:
1180:
1165:
1164:
1158:
1147:
1134:
1129:
1116:
1111:
1092:
1091:
1072:
1070:
1069:
1064:
1031:
1029:
1028:
1023:
1018:
1017:
1016:
1012:
1011:
1005:
997:
991:
982:
966:
965:
961:
942:
941:
922:
920:
919:
914:
912:
903:
889:
887:
886:
881:
866:
864:
863:
858:
828:
826:
825:
820:
790:
788:
787:
782:
748:
746:
745:
740:
728:
726:
725:
720:
718:
717:
705:
704:
695:
694:
678:
676:
675:
670:
662:
661:
656:
653:
652:
640:
639:
634:
631:
630:
618:
617:
612:
609:
608:
592:
590:
589:
584:
582:
581:
569:
568:
556:
555:
535:
533:
532:
527:
525:
516:
510:
501:
495:
486:
471:Schwarz triangle
468:
466:
465:
460:
407:Triangle surface
398:
396:
395:
390:
377:Riemann surfaces
335:
333:
332:
327:
311:
309:
308:
303:
301:
300:
295:
279:
277:
276:
271:
263:
262:
250:
249:
230:
228:
227:
222:
220:
219:
214:
196:
195:
173:
171:
170:
165:
163:
162:
157:
141:
139:
138:
133:
111:
109:
108:
103:
92:
91:
65:
63:
62:
57:
4810:
4809:
4805:
4804:
4803:
4801:
4800:
4799:
4780:
4779:
4778:
4773:
4752:Higher systoles
4747:
4716:
4692:Pu's inequality
4675:
4670:
4640:
4631:
4603:
4594:Delta invariant
4565:
4534:
4498:
4459:Abel–Jacobi map
4440:
4414:
4410:Torelli theorem
4380:Dessin d'enfant
4360:Belyi's theorem
4346:
4332:PlĂĽcker formula
4263:
4254:Hurwitz surface
4223:
4202:
4136:
4110:Analytic theory
4102:Elliptic curves
4096:
4077:Projective line
4064:Rational curves
4058:
4052:
4022:
4015:
3980:
3979:
3975:
3925:
3924:
3920:
3906:
3893:
3846:
3829:
3818:
3785:
3754:
3731:math.DG/0501017
3714:
3701:10.2307/2369402
3686:
3683:
3651:
3583:
3560:
3555:
3554:
3550:and relations
3513:
3512:
3505:
3447:
3442:
3441:
3400:
3392:
3391:
3349:
3348:
3309:
3304:
3303:
3273:
3268:
3267:
3237:
3232:
3231:
3201:
3196:
3195:
3165:
3160:
3159:
3129:
3124:
3123:
3093:
3088:
3087:
3057:
3052:
3051:
3021:
3016:
3015:
2985:
2980:
2979:
2949:
2944:
2943:
2884:
2883:
2868:
2866:Spectral theory
2830:
2808:
2786:
2764:
2753:
2752:
2725:
2724:
2672:
2631:
2617:
2607:
2575:
2526:
2513:
2500:
2487:
2452:
2451:
2408:
2338:
2294:
2268:
2255:
2254:
2230:
2225:
2224:
2194:
2174:
2154:
2146:
2145:
2062:
2057:
2056:
2026:
2006:
1977:
1969:
1968:
1942:
1937:
1936:
1885:
1878:
1871:
1830:
1825:
1824:
1773:
1772:
1750:
1749:
1722:
1721:
1641:
1636:
1635:
1611:
1606:
1605:
1584:
1579:
1578:
1498:
1493:
1492:
1482:length spectrum
1451:
1423:
1377:
1349:
1344:
1343:
1303:
1302:
1263:
1262:
1238:
1237:
1222:
1203:
1192:
1167:
1166:
1148:
1141:
1117:
1097:
1083:
1078:
1077:
1037:
1036:
998:
979:
975:
967:
946:
933:
928:
927:
923:and corners at
896:
895:
872:
871:
831:
830:
793:
792:
755:
754:
731:
730:
709:
696:
686:
681:
680:
654:
644:
632:
622:
610:
600:
595:
594:
573:
560:
547:
542:
541:
479:
478:
433:
432:
409:
381:
380:
365:spectral theory
318:
317:
290:
285:
284:
254:
241:
236:
235:
209:
187:
179:
178:
152:
147:
146:
118:
117:
83:
75:
74:
48:
47:
41:Riemann surface
33:Oskar Bolza
31:(introduced by
17:
12:
11:
5:
4808:
4806:
4798:
4797:
4792:
4782:
4781:
4775:
4774:
4772:
4771:
4766:
4761:
4755:
4753:
4749:
4748:
4746:
4745:
4740:
4738:Filling radius
4735:
4730:
4724:
4722:
4718:
4717:
4715:
4714:
4709:
4704:
4699:
4694:
4689:
4683:
4681:
4677:
4676:
4671:
4669:
4668:
4661:
4654:
4646:
4637:
4636:
4633:
4632:
4630:
4629:
4624:
4619:
4613:
4611:
4609:Vector bundles
4605:
4604:
4602:
4601:
4596:
4591:
4586:
4581:
4575:
4573:
4567:
4566:
4564:
4563:
4558:
4553:
4548:
4542:
4540:
4536:
4535:
4533:
4532:
4527:
4522:
4517:
4512:
4506:
4504:
4500:
4499:
4497:
4496:
4491:
4486:
4481:
4476:
4471:
4466:
4461:
4455:
4453:
4446:
4442:
4441:
4439:
4438:
4433:
4428:
4422:
4420:
4416:
4415:
4413:
4412:
4407:
4402:
4397:
4392:
4387:
4382:
4377:
4372:
4367:
4362:
4356:
4354:
4348:
4347:
4345:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4273:
4271:
4265:
4264:
4262:
4261:
4256:
4251:
4246:
4241:
4235:
4233:
4229:
4228:
4225:
4224:
4222:
4221:
4216:
4210:
4208:
4204:
4203:
4201:
4200:
4195:
4190:
4185:
4180:
4175:
4170:
4165:
4160:
4155:
4150:
4144:
4142:
4138:
4137:
4135:
4134:
4129:
4124:
4119:
4113:
4111:
4104:
4098:
4097:
4095:
4094:
4089:
4087:Riemann sphere
4084:
4079:
4074:
4068:
4066:
4060:
4059:
4053:
4051:
4050:
4043:
4036:
4028:
4021:
4020:
4013:
3973:
3938:(5): 483–487.
3917:
3916:
3915:
3911:
3910:
3904:
3891:
3855:(3): 827–869.
3844:
3827:
3816:
3783:
3765:(6): 564–631.
3752:
3712:
3682:
3679:
3678:
3677:
3672:
3667:
3662:
3657:
3650:
3647:
3639:
3638:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3605:
3600:
3595:
3590:
3586:
3581:
3578:
3575:
3572:
3567:
3563:
3535:
3530:
3525:
3521:
3504:
3501:
3497:
3496:
3485:
3482:
3479:
3476:
3473:
3469:
3465:
3462:
3459:
3454:
3450:
3435:
3434:
3423:
3420:
3417:
3414:
3411:
3406:
3399:
3376:
3373:
3369:
3365:
3362:
3359:
3356:
3346:Casimir energy
3336:
3335:
3332:
3329:
3316:
3312:
3300:
3299:
3296:
3293:
3280:
3276:
3264:
3263:
3260:
3257:
3244:
3240:
3228:
3227:
3224:
3221:
3208:
3204:
3192:
3191:
3188:
3185:
3172:
3168:
3156:
3155:
3152:
3149:
3136:
3132:
3120:
3119:
3116:
3113:
3100:
3096:
3084:
3083:
3080:
3077:
3064:
3060:
3048:
3047:
3044:
3041:
3028:
3024:
3012:
3011:
3008:
3005:
2992:
2988:
2976:
2975:
2972:
2969:
2956:
2952:
2940:
2939:
2936:
2933:
2891:
2867:
2864:
2860:
2859:
2848:
2845:
2842:
2837:
2833:
2829:
2826:
2823:
2820:
2815:
2811:
2807:
2804:
2801:
2798:
2793:
2789:
2785:
2782:
2779:
2776:
2771:
2767:
2763:
2760:
2732:
2721:
2720:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2684:
2679:
2675:
2671:
2668:
2665:
2662:
2658:
2655:
2652:
2649:
2646:
2643:
2638:
2634:
2629:
2624:
2620:
2614:
2610:
2606:
2603:
2600:
2596:
2593:
2590:
2587:
2582:
2578:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2533:
2529:
2525:
2520:
2516:
2512:
2507:
2503:
2499:
2494:
2490:
2486:
2483:
2479:
2476:
2472:
2469:
2465:
2462:
2459:
2445:
2444:
2438:
2432:
2426:
2407:
2404:
2403:
2402:
2391:
2388:
2382:
2377:
2372:
2369:
2366:
2363:
2358:
2355:
2352:
2349:
2346:
2343:
2335:
2330:
2325:
2320:
2317:
2314:
2308:
2305:
2297:
2293:
2288:
2285:
2282:
2279:
2276:
2273:
2265:
2262:
2237:
2233:
2212:
2209:
2206:
2201:
2197:
2192:
2189:
2186:
2181:
2177:
2172:
2169:
2166:
2161:
2157:
2153:
2130:
2127:
2124:
2119:
2114:
2111:
2108:
2105:
2102:
2097:
2094:
2091:
2088:
2085:
2082:
2077:
2074:
2069:
2065:
2044:
2041:
2038:
2033:
2029:
2024:
2021:
2018:
2013:
2009:
2004:
1998:
1995:
1989:
1984:
1980:
1976:
1949:
1945:
1933:
1932:
1921:
1918:
1914:
1908:
1903:
1897:
1894:
1888:
1884:
1881:
1874:
1870:
1865:
1862:
1859:
1856:
1853:
1850:
1845:
1842:
1837:
1833:
1814:
1813:
1802:
1799:
1794:
1789:
1786:
1783:
1780:
1757:
1729:
1718:
1717:
1706:
1703:
1698:
1693:
1690:
1687:
1684:
1681:
1676:
1673:
1670:
1667:
1664:
1661:
1656:
1653:
1648:
1644:
1618:
1614:
1587:
1575:
1574:
1563:
1560:
1557:
1552:
1547:
1544:
1541:
1538:
1533:
1530:
1527:
1524:
1521:
1518:
1513:
1510:
1505:
1501:
1478:
1477:
1466:
1463:
1458:
1454:
1448:
1445:
1440:
1436:
1430:
1426:
1420:
1417:
1412:
1408:
1402:
1399:
1394:
1390:
1384:
1380:
1374:
1371:
1366:
1362:
1356:
1352:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1288:
1285:
1280:
1273:
1270:
1259:
1258:
1247:
1242:
1234:
1229:
1226:
1223:
1216:
1212:
1209:
1206:
1199:
1195:
1191:
1188:
1183:
1178:
1175:
1172:
1169:
1168:
1161:
1157:
1154:
1151:
1144:
1140:
1137:
1132:
1127:
1124:
1121:
1118:
1114:
1109:
1106:
1103:
1102:
1100:
1095:
1090:
1086:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1033:
1032:
1021:
1015:
1008:
1004:
1001:
994:
988:
985:
978:
974:
970:
964:
960:
956:
953:
949:
945:
940:
936:
909:
906:
879:
856:
853:
850:
847:
844:
841:
838:
818:
815:
812:
809:
806:
803:
800:
780:
777:
774:
771:
768:
765:
762:
751:triangle group
738:
716:
712:
708:
703:
699:
693:
689:
668:
665:
660:
651:
647:
643:
638:
629:
625:
621:
616:
607:
603:
593:and relations
580:
576:
572:
567:
563:
559:
554:
550:
522:
519:
513:
507:
504:
498:
492:
489:
475:Fuchsian group
458:
455:
452:
449:
446:
443:
440:
408:
405:
388:
325:
299:
294:
281:
280:
269:
266:
261:
257:
253:
248:
244:
218:
213:
208:
205:
202:
199:
194:
190:
186:
161:
156:
131:
128:
125:
101:
98:
95:
90:
86:
82:
55:
15:
13:
10:
9:
6:
4:
3:
2:
4807:
4796:
4793:
4791:
4788:
4787:
4785:
4770:
4767:
4765:
4762:
4760:
4757:
4756:
4754:
4750:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4725:
4723:
4719:
4713:
4710:
4708:
4705:
4703:
4702:Bolza surface
4700:
4698:
4695:
4693:
4690:
4688:
4685:
4684:
4682:
4678:
4674:
4667:
4662:
4660:
4655:
4653:
4648:
4647:
4644:
4628:
4625:
4623:
4620:
4618:
4615:
4614:
4612:
4610:
4606:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4576:
4574:
4572:
4571:Singularities
4568:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4543:
4541:
4537:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4507:
4505:
4501:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4456:
4454:
4450:
4447:
4443:
4437:
4434:
4432:
4429:
4427:
4424:
4423:
4421:
4419:Constructions
4417:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4390:Klein quartic
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4370:Bolza surface
4368:
4366:
4365:Bring's curve
4363:
4361:
4358:
4357:
4355:
4353:
4349:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4297:Conic section
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4277:AF+BG theorem
4275:
4274:
4272:
4270:
4266:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4236:
4234:
4230:
4220:
4217:
4215:
4212:
4211:
4209:
4205:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4145:
4143:
4139:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4114:
4112:
4108:
4105:
4103:
4099:
4093:
4092:Twisted cubic
4090:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4069:
4067:
4065:
4061:
4057:
4049:
4044:
4042:
4037:
4035:
4030:
4029:
4026:
4016:
4014:9781470426651
4010:
4006:
4002:
3997:
3992:
3988:
3984:
3977:
3974:
3969:
3965:
3961:
3957:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3922:
3919:
3913:
3912:
3907:
3905:0-387-98386-4
3901:
3897:
3892:
3888:
3884:
3880:
3876:
3872:
3868:
3863:
3858:
3854:
3850:
3845:
3841:
3837:
3833:
3828:
3824:
3823:
3817:
3813:
3809:
3805:
3801:
3798:(1): 91–101.
3797:
3793:
3789:
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3759:
3753:
3749:
3745:
3741:
3737:
3732:
3727:
3724:(1): 95–107.
3723:
3720:
3719:
3713:
3710:
3706:
3702:
3698:
3694:
3690:
3685:
3684:
3680:
3676:
3673:
3671:
3668:
3666:
3665:Bring's curve
3663:
3661:
3660:Klein quartic
3658:
3656:
3653:
3652:
3648:
3646:
3644:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3603:
3598:
3593:
3588:
3584:
3579:
3576:
3573:
3570:
3565:
3561:
3553:
3552:
3551:
3549:
3528:
3510:
3502:
3500:
3483:
3480:
3477:
3471:
3467:
3463:
3460:
3448:
3440:
3439:
3438:
3421:
3418:
3404:
3390:
3389:
3388:
3371:
3367:
3363:
3360:
3354:
3347:
3343:
3333:
3330:
3314:
3310:
3302:
3301:
3297:
3294:
3278:
3274:
3266:
3265:
3261:
3258:
3242:
3238:
3230:
3229:
3225:
3222:
3206:
3202:
3194:
3193:
3189:
3186:
3170:
3166:
3158:
3157:
3153:
3150:
3134:
3130:
3122:
3121:
3117:
3114:
3098:
3094:
3086:
3085:
3081:
3078:
3062:
3058:
3050:
3049:
3045:
3042:
3026:
3022:
3014:
3013:
3009:
3006:
2990:
2986:
2978:
2977:
2973:
2970:
2954:
2950:
2942:
2941:
2938:Multiplicity
2937:
2934:
2931:
2930:
2924:
2922:
2917:
2913:
2912:perturbations
2909:
2905:
2877:
2872:
2865:
2863:
2862:as expected.
2846:
2843:
2835:
2831:
2824:
2821:
2813:
2809:
2802:
2799:
2791:
2787:
2780:
2777:
2769:
2765:
2758:
2751:
2750:
2749:
2746:
2730:
2707:
2701:
2698:
2695:
2692:
2689:
2686:
2682:
2677:
2673:
2669:
2666:
2663:
2660:
2656:
2653:
2650:
2647:
2644:
2641:
2636:
2632:
2627:
2622:
2618:
2612:
2608:
2604:
2601:
2598:
2594:
2591:
2588:
2585:
2580:
2576:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2531:
2527:
2523:
2518:
2514:
2510:
2505:
2501:
2497:
2492:
2488:
2484:
2481:
2477:
2474:
2470:
2467:
2463:
2460:
2450:
2449:
2448:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2420:
2419:
2412:
2405:
2389:
2386:
2375:
2370:
2367:
2361:
2333:
2323:
2318:
2315:
2306:
2303:
2295:
2291:
2263:
2260:
2253:
2252:
2251:
2235:
2231:
2207:
2204:
2199:
2195:
2190:
2187:
2184:
2179:
2175:
2170:
2167:
2164:
2159:
2155:
2142:
2128:
2125:
2117:
2112:
2109:
2106:
2100:
2075:
2072:
2067:
2063:
2039:
2036:
2031:
2027:
2022:
2019:
2016:
2011:
2007:
2002:
1996:
1993:
1987:
1982:
1978:
1965:
1947:
1943:
1919:
1916:
1912:
1906:
1901:
1895:
1892:
1886:
1882:
1879:
1872:
1868:
1843:
1840:
1835:
1831:
1823:
1822:
1821:
1819:
1800:
1792:
1787:
1784:
1781:
1771:
1770:
1769:
1755:
1747:
1743:
1727:
1704:
1696:
1691:
1688:
1685:
1679:
1654:
1651:
1646:
1642:
1634:
1633:
1632:
1616:
1612:
1585:
1561:
1558:
1550:
1545:
1542:
1536:
1511:
1508:
1503:
1499:
1491:
1490:
1489:
1487:
1483:
1464:
1461:
1456:
1452:
1446:
1443:
1438:
1434:
1428:
1424:
1418:
1415:
1410:
1406:
1400:
1397:
1392:
1388:
1382:
1378:
1372:
1369:
1364:
1360:
1354:
1350:
1342:
1341:
1340:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1286:
1283:
1278:
1271:
1268:
1245:
1240:
1232:
1227:
1224:
1214:
1210:
1207:
1204:
1197:
1193:
1189:
1181:
1176:
1173:
1159:
1155:
1152:
1149:
1142:
1138:
1130:
1125:
1122:
1112:
1107:
1104:
1098:
1093:
1088:
1084:
1076:
1075:
1074:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1019:
1013:
1006:
1002:
999:
992:
986:
983:
976:
972:
968:
962:
958:
954:
951:
947:
943:
938:
934:
926:
925:
924:
907:
904:
893:
892:Poincare disk
868:
851:
848:
845:
842:
839:
813:
810:
807:
804:
801:
775:
772:
769:
766:
763:
752:
714:
710:
706:
701:
697:
691:
687:
666:
663:
658:
649:
645:
641:
636:
627:
623:
619:
614:
605:
601:
578:
574:
570:
565:
561:
557:
552:
548:
539:
520:
517:
511:
505:
502:
496:
490:
487:
476:
472:
453:
450:
447:
444:
441:
425:
418:
413:
406:
404:
402:
386:
378:
374:
370:
366:
362:
358:
357:quantum chaos
353:
351:
347:
346:hyperelliptic
343:
339:
323:
315:
297:
267:
264:
259:
255:
251:
246:
242:
234:
233:
232:
216:
206:
200:
192:
188:
184:
177:
159:
145:
129:
126:
123:
115:
96:
88:
84:
80:
72:
69:
53:
46:
42:
38:
34:
30:
26:
25:Bolza surface
22:
4701:
4556:Prym variety
4530:Stable curve
4520:Hodge bundle
4510:ELSV formula
4369:
4312:Fermat curve
4269:Plane curves
4232:Higher genus
4207:Applications
4132:Modular form
3986:
3982:
3976:
3935:
3931:
3921:
3895:
3852:
3848:
3831:
3821:
3795:
3791:
3762:
3756:
3721:
3716:
3695:(1): 47–70,
3692:
3688:
3640:
3547:
3506:
3498:
3436:
3339:
2881:
2861:
2722:
2446:
2440:
2434:
2428:
2422:
2417:
2143:
1934:
1815:
1748:) and where
1719:
1576:
1485:
1479:
1260:
1034:
869:
867:group does.
430:
354:
342:Schmutz 1993
282:
144:finite field
28:
24:
18:
4431:Polar curve
2932:Eigenvalue
679:as well as
29:Bolza curve
21:mathematics
4784:Categories
4426:Dual curve
4054:Topics in
3996:1603.07356
3681:References
2908:Jenni 1981
373:eigenvalue
350:octahedron
4539:Morphisms
4287:Bitangent
3862:1110.2150
3779:120508826
3617:−
3574:−
3481:−
3478:≈
3461:−
3453:Δ
3449:ζ
3419:≈
3413:Δ
3405:ζ
3361:−
3355:ζ
3311:λ
3275:λ
3239:λ
3203:λ
3167:λ
3131:λ
3095:λ
3059:λ
3023:λ
2987:λ
2951:λ
2904:Cook 2018
2890:Δ
2876:FreeFEM++
2705:⟩
2485:∣
2458:⟨
2390:0.321281.
2387:≈
2362:
2292:
2232:ℓ
2196:ℓ
2176:ℓ
2156:ℓ
2126:≈
2101:
2064:ℓ
2028:ℓ
2008:ℓ
1979:ℓ
1944:ℓ
1917:≈
1893:π
1883:
1869:
1832:ℓ
1785:−
1680:
1643:ℓ
1613:ℓ
1559:≈
1537:
1500:ℓ
1444:−
1416:−
1398:−
1370:−
1321:…
1284:−
1269:α
1211:π
1198:−
1190:α
1156:π
1139:α
1055:…
1003:π
984:π
952:−
905:π
878:Γ
737:Γ
518:π
503:π
488:π
401:curvature
379:of genus
344:). As a
265:−
207:⋊
127:×
68:conformal
3968:20390243
3960:10039347
3914:Specific
3887:14305255
3840:45934169
3748:16510851
3649:See also
2055:, where
1920:3.05714.
1818:Formulae
1604:element
1562:3.05714.
4599:Tacnode
4584:Crunode
3940:Bibcode
3867:Bibcode
3800:Bibcode
3709:2369402
1486:systole
890:on the
367:of the
338:systole
35: (
4579:Acnode
4503:Moduli
4011:
3966:
3958:
3902:
3885:
3838:
3777:
3746:
3707:
2723:where
2129:4.8969
1720:where
1261:where
1035:where
363:. The
23:, the
3991:arXiv
3964:S2CID
3883:S2CID
3857:arXiv
3775:S2CID
3744:S2CID
3726:arXiv
3705:JSTOR
3643:order
538:index
45:genus
4589:Cusp
4009:ISBN
3956:PMID
3900:ISBN
3836:OCLC
3758:GAFA
3437:and
3344:and
3340:The
2906:), (
1577:The
1301:and
37:1887
4001:doi
3987:700
3948:doi
3875:doi
3853:317
3808:doi
3767:doi
3736:doi
3722:227
3697:doi
3548:i,j
3398:det
2745:GAP
1880:csc
283:in
116:of
43:of
19:In
4786::
4007:.
3999:.
3985:.
3962:.
3954:.
3946:.
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