4611:
4317:
6170:
5094:
32:
4606:{\displaystyle {\begin{aligned}\Gamma _{0}(N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv 0{\pmod {N}}\right\}\\\Gamma (N)&=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv b\equiv 0,a\equiv d\equiv 1{\pmod {N}}\right\}.\end{aligned}}}
1576:
5863:
are a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the
Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very
4921:
2867:
5479:. Such rings of modular forms are generated in weight at most 6 and the relations are generated in weight at most 12 when the congruence subgroup has nonzero odd weight modular forms, and the corresponding bounds are 5 and 10 when there are no nonzero odd weight modular forms.
2055:
2706:
859:
1318:
3021:
6100:
3383:
251:
5089:{\displaystyle \dim _{\mathbf {C} }M_{k}\left({\text{SL}}(2,\mathbf {Z} )\right)={\begin{cases}\left\lfloor k/12\right\rfloor &k\equiv 2{\pmod {12}}\\\left\lfloor k/12\right\rfloor +1&{\text{otherwise}}\end{cases}}}
2522:
1430:
2166:
1214:
3176:
2717:
5744:
1761:
1918:
6447:
3603:
3793:
5458:
1084:
319:
913:
571:
3685:
5654:
3933:
1371:
4322:
2722:
1952:
411:
5964:
4203:
985:
523:
2558:
1022:
674:
637:
5123:
6150:
4051:) of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on the
750:
445:
121:
6482:
3845:
6711:
50:
775:
6957:
1847:
1248:
6340:
5596:
1237:
1104:
349:
3477:
The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers by
3977:
5313:
vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(
770:
861:
The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication. In addition, it is called a
5566:
5546:
5526:
5147:
1938:
1124:
369:
2939:
6303:
5972:
3263:
184:
1571:{\displaystyle {\text{SL}}(2,\mathbf {Z} )=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|a,b,c,d\in \mathbf {Z} ,\ ad-bc=1\right\}}
7380:
3506:
2450:
5343:
Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
2862:{\displaystyle {\begin{aligned}G_{k}\left(-{\frac {1}{\tau }}\right)&=\tau ^{k}G_{k}(\tau ),\\G_{k}(\tau +1)&=G_{k}(\tau ).\end{aligned}}}
4040:. Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the
7109:
6282:. In 2001 all elliptic curves were proven to be modular over the rational numbers. In 2013 elliptic curves were proven to be modular over real
2066:
1132:
5153:
3093:
7472:
7069:
6950:
6588:
6286:. In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining the
5764:
is a modular form with a zero constant coefficient in its
Fourier series. It is called a cusp form because the form vanishes at all cusps.
5659:
6508:
5270:. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let
6930:
5482:
More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitrary
3248:
1666:
7538:
7160:
7059:
6782:
3482:
2926:
vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in
7528:
6719:
6371:
function can only have a finite number of negative-exponent terms in its
Laurent series, its q-expansion. It can only have at most a
1852:
6854:
6760:
6549:
6399:
6217:
3555:
68:
3707:
5396:
1043:
278:
7238:
6943:
5827:
variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a
5772:
There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory of
7568:
1374:
872:
530:
7385:
7296:
6195:
3615:
7306:
7233:
5601:
3859:
1817:
is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some
1330:
6983:
6191:
7203:
6245:
and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable)
2050:{\displaystyle S={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\qquad T={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}
7099:
7462:
7426:
4014:
be invariant with respect to a sub-group of the modular group of finite index. This is not adhered to in this article.
7125:
7038:
3236:
378:
6180:
5897:
4006:
Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that
1385:
A modular function is a function that is invariant with respect to the modular group, but without the condition that
2701:{\displaystyle G_{k}(\Lambda )=G_{k}(\tau )=\sum _{(0,0)\neq (m,n)\in \mathbf {Z} ^{2}}{\frac {1}{(m+n\tau )^{k}}},}
7573:
7436:
7074:
6846:
6806:
6624:
4253:
4237:
4153:
918:
132:
6279:
6199:
6184:
450:
7482:
6834:
5828:
3027:
2180:
7395:
7375:
7311:
7228:
7089:
4904:
1324:
7130:
6899:
990:
642:
7094:
5505:
5500:
576:
4036:
elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number
7286:
6372:
5173:
5102:
6108:
7563:
7079:
5278:
polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on
682:
7457:
7193:
6993:
6380:
6153:
5352:
5181:
3949:
3490:
3254:
424:
264:
100:
7155:
7104:
1404:: they are holomorphic on the complement of a set of isolated points, which are poles of the function.
854:{\textstyle \gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}_{2}(\mathbb {Z} ).\,}
6345:"DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions"
7533:
7405:
7316:
7064:
6912:
6263:
Taniyama and
Shimura identified a 1-to-1 matching between certain modular forms and elliptic curves.
5820:
5802:
5275:
5157:
3803:
3550:
3537:
3420:
3397:
2880:
1631:
1401:
1397:
373:
7370:
6680:
4990:
3991:
at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-
3485:. The crucial conceptual link between modular forms and number theory is furnished by the theory of
7248:
7213:
7170:
7150:
6908:
5834:
5806:
5333:
4734:
4309:
4230:
3224:
128:
6156:, a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.
4267:. Typically it is not compact, but can be compactified by adding a finite number of points called
7500:
7084:
6790:
6256:
5891:
5325:
5226:
The situation can be profitably compared to that which arises in the search for functions on the
5185:
4702:
3848:
3393:
3058:
has integer coordinates, either all even or all odd, and such that the sum of the coordinates of
2910:
1313:{\displaystyle X_{\Gamma }=\Gamma \backslash ({\mathcal {H}}\cup \mathbb {P} ^{1}(\mathbb {Q} ))}
155:
7291:
7271:
7243:
5805:. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's
7400:
7347:
7218:
7033:
7028:
6873:
6850:
6778:
6756:
6584:
6545:
6501:
6268:
6235:
5791:
4286:
2410:
2292:
2217:
1820:
178:
90:
6325:
6271:, which has become one of the most far-reaching and consequential research programs in math.
5575:
1222:
1089:
334:
7390:
7276:
7253:
6920:
6602:
6287:
6283:
6264:
5867:
5838:
5476:
5227:
4277:
3541:
3471:
3081:
1594:
414:
326:
170:
147:
94:
6814:
6598:
3955:
3509:
that the only modular forms are constant functions. However, relaxing the requirement that
7505:
7321:
7263:
7165:
6988:
6967:
6881:
6810:
6798:
6774:
6752:
6676:
6606:
6594:
6580:
6322:
Some authors use different conventions, allowing an additional constant depending only on
5850:
4298:
4264:
4101:
1582:
6820:
Provides an introduction to modular forms from the point of view of representation theory
6655:
4875:
to obtain further information about modular forms and functions. For example, the spaces
3016:{\displaystyle \vartheta _{L}(z)=\sum _{\lambda \in L}e^{\pi i\Vert \lambda \Vert ^{2}z}}
755:
16:
Analytic function on the upper half-plane with a certain behavior under the modular group
7188:
6916:
6642:, Publications of the Mathematical Society of Japan, vol. 11, Tokyo: Iwanami Shoten
2361:
remains bounded above as long as the absolute value of the smallest non-zero element in
7013:
6998:
6975:
6745:
6740:
6095:{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=\varepsilon (a,b,c,d)(cz+d)^{k}f(z).}
5881:
are meromorphic functions on the upper half plane of moderate growth at infinity which
5551:
5531:
5511:
5483:
5472:
5132:
5126:
4052:
4022:
4018:
3695:
3486:
3478:
3467:
3463:
3451:
2931:
2369:
The key idea in proving the equivalence of the two definitions is that such a function
2351:
2252:
2242:
2225:
1923:
1780:
1610:
1109:
354:
159:
3378:{\displaystyle \eta (z)=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}),\qquad q=e^{2\pi iz}.}
2373:
is determined, because of the second condition, by its values on lattices of the form
246:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}
19:"Modular function" redirects here. A distinct use of this term appears in relation to
7557:
7520:
7301:
7281:
7208:
7003:
6566:
5966:
which are used to generalise the modularity relation defining modular forms, so that
5794:
5337:
3606:
3416:
3221:
1422:
1240:
1037:
163:
151:
136:
3037:. It is not so easy to construct even unimodular lattices, but here is one way: Let
7467:
7441:
7431:
7421:
7223:
6579:, Grundlehren der Mathematischen Wissenschaften , vol. 244, Berlin, New York:
6275:
5773:
4134:
3455:
3415:
is a modular form of weight 12. The presence of 24 is related to the fact that the
1400:
in the upper half-plane (among other requirements). Instead, modular functions are
322:
20:
6526:
3087:. Because there is only one modular form of weight 8 up to scalar multiplication,
6574:
7342:
7180:
6826:
6368:
6291:
6249:
6242:
6169:
5860:
5787:
5370:
4041:
3459:
3228:
3207:
3077:
2517:{\displaystyle G_{k}(\Lambda )=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-k}.}
1033:
257:
82:
5320:
One might ask, since the homogeneous polynomials are not really functions on P(
256:
The term "modular form", as a systematic description, is usually attributed to
7337:
6894:
6887:
Chapter VII provides an elementary introduction to the theory of modular forms
6570:
5305:. On the one hand, these form a finite dimensional vector space for each
4033:
6935:
7198:
5871:
5798:
5761:
5755:
4092:
3424:
174:
5853:
in the same sense that classical modular forms (which are sometimes called
4903:
are finite-dimensional, and their dimensions can be computed thanks to the
5572:. These old forms can be constructed using the following observations: if
4777:. Again, modular forms that vanish at all cusps are called cusp forms for
2161:{\displaystyle f\left(-{\frac {1}{z}}\right)=z^{k}f(z),\qquad f(z+1)=f(z)}
1323:
The dimensions of these spaces of modular forms can be computed using the
1209:{\displaystyle f\in H^{0}(X_{\Gamma },\omega ^{\otimes k})=M_{k}(\Gamma )}
3232:
3171:{\displaystyle \vartheta _{L_{8}\times L_{8}}(z)=\vartheta _{L_{16}}(z),}
6864:
5340:
in this case). The situation with modular forms is precisely analogous.
5168:
is not identically 0, then it can be shown that the number of zeroes of
4138:
is a modular function whose poles and zeroes are confined to the cusps.
7510:
7495:
6924:
6255:
In the 1960s, as the needs of number theory and the formulation of the
7490:
6259:
in particular made it clear that modular forms are deeply implicated.
6152:
is called the nebentypus of the modular form. Functions such as the
5739:{\displaystyle M_{k}(\Gamma _{1}(M))\subseteq M_{k}(\Gamma _{1}(N))}
4841:), they are also referred to as modular/cusp forms and functions of
4205:
can be relaxed by requiring it only for matrices in smaller groups.
4017:
Another way to phrase the definition of modular functions is to use
3999:-expansion is bounded below, guaranteeing that it is meromorphic at
3489:, which also gives the link between the theory of modular forms and
169:
Modular form theory is a special case of the more general theory of
5841:
in the same way in which classical modular forms are associated to
3211:
2191:, the second condition above is equivalent to these two equations.
6344:
1756:{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=(cz+d)^{k}f(z)}
6805:, Annals of Mathematics Studies, vol. 83, Princeton, N.J.:
5301:). The solutions are then the homogeneous polynomials of degree
4301:, which allows one to speak of holo- and meromorphic functions.
1032:
Modular forms can also be interpreted as sections of a specific
6939:
6931:
Behold
Modular Forms, the ‘Fifth Fundamental Operation’ of Math
5528:
which cannot be constructed from modular forms of lower levels
1913:{\displaystyle \operatorname {Im} (z)>M\implies |f(z)|<D}
6640:
Introduction to the arithmetic theory of automorphic functions
6442:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
6163:
4289:±2) fixing the point. This yields a compact topological space
3598:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
25:
6897:(1988), "Jacobi forms and a certain space of modular forms",
5194:.It can be shown that the field of modular function of level
4765:
satisfying the above functional equation for all matrices in
3788:{\displaystyle f(z)=\sum _{n=-m}^{\infty }a_{n}e^{2i\pi nz}.}
6483:"Elliptic Curves Yield Their Secrets in a New Number System"
5453:{\displaystyle M(\Gamma )=\bigoplus _{k>0}M_{k}(\Gamma )}
3529:
is called modular if it satisfies the following properties:
3249:
Weierstrass's elliptic functions § Modular discriminant
1276:
1268:
1079:{\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}
390:
314:{\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}
177:
that transform nicely with respect to the action of certain
150:. The main importance of the theory is its connections with
107:
6230:
The theory of modular forms was developed in four periods:
5082:
1465:
6773:, Graduate Texts in Mathematics, vol. 228, New York:
4297:. What is more, it can be endowed with the structure of a
1806:, only the zero function can satisfy the second condition.
3026:
converges when Im(z) > 0, and as a consequence of the
2237:
A modular form can equivalently be defined as a function
6880:, Graduate Texts in Mathematics, vol. 7, New York:
6267:
built on this idea in the construction of his expansive
5857:
to emphasize the point) are related to elliptic curves.
3698:. The third condition is that this series is of the form
2343:
is a constant (typically a positive integer) called the
908:{\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )}
566:{\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )}
6747:
Modular functions and
Dirichlet Series in Number Theory
3680:{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=f(z)}
3435:
is expanded as a power series in q, the coefficient of
2313:
is the lattice obtained by multiplying each element of
46:
6408:
5649:{\displaystyle \Gamma _{1}(N)\subseteq \Gamma _{1}(M)}
4483:
4361:
3928:{\displaystyle f(z)=\sum _{n=-m}^{\infty }a_{n}q^{n}.}
3564:
3041:
be an integer divisible by 8 and consider all vectors
2409:
The simplest examples from this point of view are the
2016:
1967:
1472:
790:
778:
758:
685:
6683:
6402:
6328:
6111:
5975:
5900:
5662:
5604:
5578:
5554:
5534:
5514:
5399:
5234:): in that setting, one would ideally like functions
5135:
5105:
4924:
4817:, respectively. Similarly, a meromorphic function on
4320:
4156:
4010:
be meromorphic in the open upper half-plane and that
3958:
3862:
3806:
3710:
3618:
3558:
3266:
3096:
2942:
2720:
2561:
2453:
2069:
1955:
1926:
1855:
1823:
1669:
1433:
1366:{\displaystyle \Gamma ={\text{SL}}_{2}(\mathbb {Z} )}
1333:
1251:
1225:
1135:
1112:
1092:
1046:
993:
921:
875:
645:
579:
533:
453:
427:
381:
357:
337:
281:
187:
103:
3474:, which were shown to imply Ramanujan's conjecture.
3076:, this is the lattice generated by the roots in the
7519:
7481:
7450:
7414:
7363:
7356:
7330:
7262:
7179:
7143:
7118:
7052:
7021:
7012:
6974:
4859:, this gives back the afore-mentioned definitions.
4785:-vector spaces of modular and cusp forms of weight
4055:of isomorphism classes of complex elliptic curves.
3220:by these two lattices are consequently examples of
1809:The third condition is also phrased by saying that
41:
may be too technical for most readers to understand
6744:
6705:
6441:
6334:
6144:
6094:
5958:
5738:
5648:
5590:
5560:
5540:
5520:
5463:Rings of modular forms of congruence subgroups of
5452:
5156:of the Riemann surface, and hence form a field of
5141:
5117:
5088:
4605:
4197:
3971:
3927:
3839:
3787:
3679:
3597:
3377:
3170:
3015:
2861:
2700:
2516:
2233:Definition in terms of lattices or elliptic curves
2160:
2049:
1932:
1912:
1841:
1755:
1570:
1365:
1312:
1231:
1208:
1118:
1098:
1078:
1016:
979:
907:
853:
764:
744:
668:
631:
565:
517:
439:
405:
363:
343:
313:
245:
115:
5508:are a subspace of modular forms of a fixed level
4867:The theory of Riemann surfaces can be applied to
4304:Important examples are, for any positive integer
146:The theory of modular forms therefore belongs to
5864:analogous to the usual theory of modular forms.
4281:∪{∞}, such that there is a parabolic element of
865:if it satisfies the following growth condition:
406:{\displaystyle f:{\mathcal {H}}\to \mathbb {C} }
181:, generalizing the example of the modular group
6623:, Annals of Mathematics Studies, vol. 48,
5959:{\displaystyle \varepsilon (a,b,c,d)(cz+d)^{k}}
5385:is the vector space of modular forms of weight
4146:The functional equation, i.e., the behavior of
2905:II. Theta functions of even unimodular lattices
154:. Modular forms appear in other areas, such as
6769:Diamond, Fred; Shurman, Jerry Michael (2005),
5870:extend the notion of modular forms to general
5324:), what are they, geometrically speaking? The
6951:
6866:Lectures on Modular Forms and Hecke Operators
4198:{\displaystyle z\mapsto {\frac {az+b}{cz+d}}}
980:{\displaystyle (cz+d)^{-k}f(\gamma (z))\to 0}
8:
6789:Leads up to an overview of the proof of the
5656:giving a reverse inclusion of modular forms
5112:
5106:
3030:can be shown to be a modular form of weight
2999:
2992:
2901:, so that such series are identically zero.
1619:satisfying the following three conditions:
518:{\displaystyle f(\gamma (z))=(cz+d)^{k}f(z)}
6198:. Unsourced material may be challenged and
5242:which are polynomial in the coordinates of
7360:
7018:
6958:
6944:
6936:
5471:are finitely generated due to a result of
4729:exactly once and such that the closure of
1881:
1877:
6688:
6682:
6403:
6401:
6327:
6218:Learn how and when to remove this message
6110:
6071:
5983:
5974:
5950:
5899:
5718:
5705:
5680:
5667:
5661:
5631:
5609:
5603:
5577:
5553:
5533:
5513:
5435:
5419:
5398:
5134:
5104:
5074:
5053:
5025:
5001:
4985:
4969:
4955:
4944:
4930:
4929:
4923:
4575:
4531:
4517:
4478:
4429:
4409:
4395:
4356:
4329:
4321:
4319:
4163:
4155:
3979:are known as the Fourier coefficients of
3963:
3957:
3916:
3906:
3896:
3882:
3861:
3805:
3764:
3754:
3744:
3730:
3709:
3626:
3617:
3559:
3557:
3357:
3334:
3315:
3304:
3290:
3286:
3265:
3148:
3143:
3119:
3106:
3101:
3095:
3062:is an even integer. We call this lattice
3002:
2985:
2969:
2947:
2941:
2837:
2805:
2779:
2769:
2743:
2729:
2721:
2719:
2686:
2661:
2653:
2648:
2610:
2588:
2566:
2560:
2502:
2480:
2458:
2452:
2103:
2081:
2068:
2011:
1962:
1954:
1925:
1899:
1882:
1854:
1822:
1735:
1677:
1668:
1531:
1467:
1448:
1434:
1432:
1356:
1355:
1346:
1341:
1332:
1300:
1299:
1290:
1286:
1285:
1275:
1274:
1256:
1250:
1224:
1191:
1172:
1159:
1146:
1134:
1111:
1091:
1069:
1068:
1059:
1054:
1045:
1017:{\displaystyle {\text{im}}(z)\to \infty }
994:
992:
941:
920:
898:
897:
888:
883:
874:
850:
840:
839:
830:
825:
785:
777:
757:
719:
684:
669:{\displaystyle {\text{im}}(z)\to \infty }
646:
644:
599:
578:
556:
555:
546:
541:
532:
497:
452:
426:
399:
398:
389:
388:
380:
356:
336:
304:
303:
294:
289:
280:
236:
235:
226:
218:
207:
206:
197:
189:
186:
112:
106:
105:
104:
102:
69:Learn how and when to remove this message
53:, without removing the technical details.
632:{\displaystyle (cz+d)^{-k}f(\gamma (z))}
417:such that two conditions are satisfied:
6470:
6315:
7381:Clifford's theorem on special divisors
6304:Wiles's proof of Fermat's Last Theorem
5849:; in other words, they are related to
5784:determined by the conjugation action.
5118:{\displaystyle \lfloor \cdot \rfloor }
4271:. These are points at the boundary of
3513:be holomorphic leads to the notion of
3470:as a result of Deligne's proof of the
1940:is bounded above some horizontal line.
6527:"Modular Functions and Modular Forms"
6145:{\displaystyle \varepsilon (a,b,c,d)}
5152:The modular functions constitute the
4142:Modular forms for more general groups
1373:are sections of a line bundle on the
745:{\textstyle \gamma (z)=(az+b)/(cz+d)}
51:make it understandable to non-experts
7:
6476:
6474:
6196:adding citations to reliable sources
5809:. Groups which are not subgroups of
5389:, then the ring of modular forms of
3450:. This was confirmed by the work of
2255:which satisfies certain conditions:
6657:Modular Functions and Modular Forms
6514:from the original on 1 August 2020.
6502:"Cohomology of Automorphic Bundles"
5369:, the ring of modular forms is the
5202:≥ 1) is generated by the functions
5033:
5026:
4733:meets all orbits. For example, the
4583:
4437:
3987:is called the order of the pole of
263:Each modular form is attached to a
7539:Vector bundles on algebraic curves
7473:Weber's theorem (Algebraic curves)
7070:Hasse's theorem on elliptic curves
7060:Counting points on elliptic curves
6725:from the original on 31 July 2020.
6685:
5715:
5677:
5628:
5606:
5444:
5406:
5373:generated by the modular forms of
4454:
4326:
3995:coefficients are non-zero, so the
3897:
3745:
3690:The second condition implies that
3316:
2930:is an even integer. The so-called
2575:
2493:
2467:
1334:
1327:. The classical modular forms for
1265:
1257:
1239:is a canonical line bundle on the
1200:
1160:
1093:
1047:
1011:
663:
440:{\displaystyle \gamma \in \Gamma }
434:
338:
282:
222:
219:
193:
190:
116:{\displaystyle \,{\mathcal {H}}\,}
14:
6803:Automorphic Forms on Adèle Groups
6238:, in the early nineteenth century
6234:In connection with the theory of
4825:is called a modular function for
3694:is periodic, and therefore has a
3210:observed that the 16-dimensional
2306:is a non-zero complex number and
173:, which are functions defined on
6644:, Theorem 2.33, Proposition 2.26
6168:
4970:
4931:
4907:in terms of the geometry of the
4532:
4410:
4021:: every lattice Λ determines an
3940:This is also referred to as the
3800:It is often written in terms of
2649:
1799:is typically a positive integer.
1532:
1449:
30:
7161:Hurwitz's automorphisms theorem
6771:A First Course in Modular Forms
4576:
4430:
3840:{\displaystyle q=\exp(2\pi iz)}
3505:is zero, it can be shown using
3346:
2124:
2004:
1375:moduli stack of elliptic curves
1126:can be defined as an element of
7386:Gonality of an algebraic curve
7297:Differential of the first kind
6706:{\displaystyle \Gamma _{1}(N)}
6700:
6694:
6481:Van Wyk, Gerhard (July 2023).
6139:
6115:
6086:
6080:
6068:
6052:
6049:
6025:
5947:
5931:
5928:
5904:
5733:
5730:
5724:
5711:
5695:
5692:
5686:
5673:
5643:
5637:
5621:
5615:
5447:
5441:
5409:
5403:
5309:, and on the other, if we let
5037:
5027:
4974:
4960:
4701:can be understood by studying
4587:
4577:
4536:
4522:
4463:
4457:
4441:
4431:
4414:
4400:
4341:
4335:
4160:
3872:
3866:
3834:
3819:
3720:
3714:
3674:
3668:
3340:
3321:
3276:
3270:
3162:
3156:
3133:
3127:
2959:
2953:
2887:there is cancellation between
2849:
2843:
2823:
2811:
2791:
2785:
2683:
2667:
2641:
2629:
2623:
2611:
2600:
2594:
2578:
2572:
2470:
2464:
2155:
2149:
2140:
2128:
2118:
2112:
1900:
1896:
1890:
1883:
1878:
1868:
1862:
1750:
1744:
1732:
1716:
1453:
1439:
1360:
1352:
1307:
1304:
1296:
1271:
1203:
1197:
1181:
1152:
1073:
1065:
1008:
1005:
999:
971:
968:
965:
959:
953:
938:
922:
902:
894:
844:
836:
772:is identified with the matrix
739:
724:
716:
701:
695:
689:
660:
657:
651:
626:
623:
617:
611:
596:
580:
560:
552:
512:
506:
494:
478:
472:
469:
463:
457:
421:Automorphy condition: For any
395:
308:
300:
240:
232:
211:
203:
1:
7529:Birkhoff–Grothendieck theorem
7239:Nagata's conjecture on curves
7110:Schoof–Elkies–Atkin algorithm
6984:Five points determine a conic
5883:fail to be modular of weight
5568:. The other forms are called
4721:intersects each orbit of the
3243:III. The modular discriminant
1774:is required to be bounded as
275:In general, given a subgroup
7100:Supersingular elliptic curve
6540:Chandrasekharan, K. (1985).
6278:used modular forms to prove
5877:Modular integrals of weight
4118:is the order of the zero of
2534:is a modular form of weight
1028:As sections of a line bundle
869:Cuspidal condition: For any
7307:Riemann's existence theorem
7234:Hilbert's sixteenth problem
7126:Elliptic curve cryptography
7039:Fundamental pair of periods
6843:Modular forms and functions
6619:Gunning, Robert C. (1962),
3237:Hearing the shape of a drum
2259:If we consider the lattice
7592:
7437:Moduli of algebraic curves
6847:Cambridge University Press
6841:Rankin, Robert A. (1977),
6835:Vandenhoeck & Ruprecht
6807:Princeton University Press
6625:Princeton University Press
5894:are functions of the form
5753:
5498:
5350:
5172:is equal to the number of
4254:quotient topological space
3246:
2922:is a lattice generated by
2436:over all non-zero vectors
1408:Modular forms for SL(2, Z)
527:Growth condition: For any
18:
6621:Lectures on modular forms
5837:are associated to larger
5829:totally real number field
5250:and satisfy the equation
5164:). If a modular function
4769:, that is holomorphic on
4032:; two lattices determine
3181:even though the lattices
3028:Poisson summation formula
1417:A modular form of weight
7204:Cayley–Bacharach theorem
7131:Elliptic curve primality
6900:Inventiones Mathematicae
6677:"Atkin-Lehner Theory of
6294:of integers down to −5.
5888:by a rational function.
5328:answer is that they are
2413:. For each even integer
2276:generated by a constant
1943:The second condition for
1842:{\displaystyle M,D>0}
1086:a modular form of level
7463:Riemann–Hurwitz formula
7427:Gromov–Witten invariant
7287:Compact Riemann surface
7075:Mazur's torsion theorem
6379: = 0, not an
6335:{\displaystyle \gamma }
5591:{\displaystyle M\mid N}
3421:A celebrated conjecture
2911:even unimodular lattice
2365:is bounded away from 0.
1232:{\displaystyle \omega }
1099:{\displaystyle \Gamma }
344:{\displaystyle \Gamma }
142:and a growth condition.
7569:Analytic number theory
7080:Modular elliptic curve
6878:A Course in Arithmetic
6863:Ribet, K.; Stein, W.,
6707:
6638:Shimura, Goro (1971),
6443:
6336:
6146:
6096:
5960:
5855:elliptic modular forms
5740:
5650:
5592:
5562:
5542:
5522:
5454:
5347:Rings of modular forms
5336:(one could also say a
5143:
5119:
5090:
4607:
4199:
4079:, also paraphrased as
4003: = 0.
3973:
3929:
3901:
3841:
3789:
3749:
3681:
3599:
3379:
3320:
3172:
3017:
2863:
2702:
2518:
2162:
2051:
1934:
1914:
1843:
1757:
1572:
1367:
1321:
1314:
1233:
1217:
1210:
1120:
1100:
1080:
1018:
981:
909:
855:
766:
746:
670:
633:
567:
519:
441:
407:
365:
345:
331:modular form of level
315:
247:
117:
6994:Rational normal curve
6708:
6654:Milne, James (2010),
6444:
6381:essential singularity
6337:
6280:Fermat’s Last Theorem
6154:Dedekind eta function
6147:
6097:
5961:
5821:Hilbert modular forms
5741:
5651:
5593:
5563:
5543:
5523:
5455:
5377:. In other words, if
5353:Ring of modular forms
5144:
5120:
5091:
4608:
4263:can be shown to be a
4200:
3974:
3972:{\displaystyle a_{n}}
3950:q-expansion principle
3930:
3878:
3842:
3790:
3726:
3682:
3600:
3491:representation theory
3392:is the square of the
3380:
3300:
3255:Dedekind eta function
3247:Further information:
3214:obtained by dividing
3173:
3018:
2864:
2703:
2519:
2163:
2052:
1935:
1915:
1844:
1758:
1573:
1368:
1315:
1244:
1234:
1211:
1128:
1121:
1101:
1081:
1019:
982:
910:
856:
767:
747:
671:
634:
568:
520:
447:there is the equality
442:
408:
366:
346:
316:
265:Galois representation
248:
118:
7534:Stable vector bundle
7406:Weil reciprocity law
7396:Riemann–Roch theorem
7376:Brill–Noether theory
7312:Riemann–Roch theorem
7229:Genus–degree formula
7090:Mordell–Weil theorem
7065:Division polynomials
6681:
6400:
6326:
6192:improve this section
6109:
5973:
5898:
5835:Siegel modular forms
5807:mock theta functions
5660:
5602:
5576:
5552:
5532:
5512:
5397:
5274:be the ratio of two
5238:on the vector space
5158:transcendence degree
5133:
5103:
4922:
4905:Riemann–Roch theorem
4773:and at all cusps of
4318:
4310:congruence subgroups
4308:, either one of the
4209:The Riemann surface
4154:
3956:
3952:). The coefficients
3860:
3804:
3708:
3616:
3556:
3398:modular discriminant
3264:
3225:Riemannian manifolds
3094:
2940:
2718:
2559:
2451:
2405:I. Eisenstein series
2216:, modular forms are
2172:respectively. Since
2067:
1953:
1924:
1853:
1821:
1667:
1632:holomorphic function
1431:
1331:
1325:Riemann–Roch theorem
1249:
1223:
1133:
1110:
1090:
1044:
991:
919:
873:
776:
765:{\textstyle \gamma }
756:
683:
643:
577:
531:
451:
425:
379:
374:holomorphic function
355:
335:
279:
185:
131:with respect to the
101:
7357:Structure of curves
7249:Quartic plane curve
7171:Hyperelliptic curve
7151:De Franchis theorem
7095:Nagell–Lutz theorem
6917:1988InMat..94..113S
6831:Mathematische Werke
6799:Gelbart, Stephen S.
6544:. Springer-Verlag.
5892:Automorphic factors
5817:can be considered.
5776:, it is a function
5501:Atkin–Lehner theory
5393:is the graded ring
5266:) for all non-zero
4753:A modular form for
4703:fundamental domains
4244:in the same way as
3847:(the square of the
3507:Liouville's theorem
3443:has absolute value
3427:asserted that when
3419:has 24 dimensions.
1661:as above, we have:
1413:Standard definition
129:functional equation
7364:Divisors on curves
7156:Faltings's theorem
7105:Schoof's algorithm
7085:Modularity theorem
6925:10.1007/BF01394347
6874:Serre, Jean-Pierre
6791:modularity theorem
6703:
6542:Elliptic functions
6439:
6433:
6332:
6257:modularity theorem
6236:elliptic functions
6142:
6092:
5956:
5736:
5646:
5588:
5558:
5538:
5518:
5450:
5430:
5246: ≠ 0 in
5186:fundamental region
5154:field of functions
5139:
5115:
5086:
5081:
4603:
4601:
4508:
4386:
4229:that is of finite
4195:
3969:
3925:
3837:
3785:
3677:
3595:
3589:
3549:For every integer
3483:partition function
3375:
3168:
3013:
2980:
2859:
2857:
2698:
2660:
2514:
2497:
2224:, and thus have a
2218:periodic functions
2183:the modular group
2158:
2047:
2041:
1995:
1930:
1910:
1839:
1753:
1653:and any matrix in
1568:
1497:
1363:
1310:
1229:
1206:
1116:
1096:
1076:
1014:
977:
905:
851:
815:
762:
742:
666:
629:
563:
515:
437:
403:
361:
341:
311:
243:
179:discrete subgroups
156:algebraic topology
123:, that satisfies:
113:
7574:Special functions
7551:
7550:
7547:
7546:
7458:Hasse–Witt matrix
7401:Weierstrass point
7348:Smooth completion
7317:Teichmüller space
7219:Cubic plane curve
7139:
7138:
7053:Arithmetic theory
7034:Elliptic integral
7029:Elliptic function
6893:Skoruppa, N. P.;
6675:Mocanu, Andreea.
6590:978-0-387-90517-4
6567:Kubert, Daniel S.
6269:Langlands program
6228:
6227:
6220:
6013:
5868:Automorphic forms
5851:abelian varieties
5839:symplectic groups
5823:are functions in
5561:{\displaystyle N}
5541:{\displaystyle M}
5521:{\displaystyle N}
5415:
5326:algebro-geometric
5142:{\displaystyle k}
5077:
4958:
4761:is a function on
4745:can be computed.
4690:), respectively.
4520:
4398:
4221:be a subgroup of
4193:
4062:that vanishes at
3983:, and the number
3656:
3515:modular functions
3497:Modular functions
3206:are not similar.
2965:
2751:
2693:
2606:
2476:
2430:to be the sum of
2411:Eisenstein series
2293:analytic function
2089:
1933:{\displaystyle f}
1707:
1541:
1437:
1344:
1119:{\displaystyle k}
1057:
1038:modular varieties
997:
886:
828:
752:and the function
649:
544:
364:{\displaystyle k}
292:
171:automorphic forms
91:analytic function
79:
78:
71:
7581:
7391:Jacobian variety
7361:
7264:Riemann surfaces
7254:Real plane curve
7214:Cramer's paradox
7194:Bézout's theorem
7019:
6968:algebraic curves
6960:
6953:
6946:
6937:
6927:
6884:
6869:
6859:
6837:
6817:
6787:
6765:
6750:
6727:
6726:
6724:
6717:
6712:
6710:
6709:
6704:
6693:
6692:
6672:
6666:
6664:
6662:
6651:
6645:
6643:
6635:
6629:
6627:
6616:
6610:
6609:
6563:
6557:
6555:
6537:
6531:
6530:
6522:
6516:
6515:
6513:
6506:
6497:
6491:
6490:
6478:
6458:
6448:
6446:
6445:
6440:
6438:
6437:
6394:
6388:
6365:
6359:
6358:
6356:
6355:
6341:
6339:
6338:
6333:
6320:
6288:rational numbers
6284:quadratic fields
6265:Robert Langlands
6223:
6216:
6212:
6209:
6203:
6172:
6164:
6151:
6149:
6148:
6143:
6101:
6099:
6098:
6093:
6076:
6075:
6018:
6014:
6012:
5998:
5984:
5965:
5963:
5962:
5957:
5955:
5954:
5886:
5880:
5848:
5816:
5801:but need not be
5783:
5745:
5743:
5742:
5737:
5723:
5722:
5710:
5709:
5685:
5684:
5672:
5671:
5655:
5653:
5652:
5647:
5636:
5635:
5614:
5613:
5597:
5595:
5594:
5589:
5567:
5565:
5564:
5559:
5547:
5545:
5544:
5539:
5527:
5525:
5524:
5519:
5477:Michael Rapoport
5470:
5459:
5457:
5456:
5451:
5440:
5439:
5429:
5392:
5388:
5384:
5376:
5368:
5360:
5304:
5228:projective space
5148:
5146:
5145:
5140:
5124:
5122:
5121:
5116:
5095:
5093:
5092:
5087:
5085:
5084:
5078:
5075:
5065:
5061:
5057:
5040:
5013:
5009:
5005:
4981:
4977:
4973:
4959:
4956:
4949:
4948:
4936:
4935:
4934:
4910:
4902:
4888:
4858:
4828:
4816:
4802:
4780:
4776:
4768:
4756:
4724:
4693:The geometry of
4635:
4612:
4610:
4609:
4604:
4602:
4595:
4591:
4590:
4535:
4521:
4518:
4513:
4512:
4449:
4445:
4444:
4413:
4399:
4396:
4391:
4390:
4334:
4333:
4284:
4251:
4236:
4228:
4220:
4204:
4202:
4201:
4196:
4194:
4192:
4178:
4164:
4150:with respect to
4128:
4117:
4104:). The smallest
4089:
4078:
4068:
4039:
3978:
3976:
3975:
3970:
3968:
3967:
3934:
3932:
3931:
3926:
3921:
3920:
3911:
3910:
3900:
3895:
3846:
3844:
3843:
3838:
3794:
3792:
3791:
3786:
3781:
3780:
3759:
3758:
3748:
3743:
3686:
3684:
3683:
3678:
3661:
3657:
3655:
3641:
3627:
3610:
3604:
3602:
3601:
3596:
3594:
3593:
3542:upper half-plane
3501:When the weight
3472:Weil conjectures
3449:
3442:
3438:
3434:
3414:
3384:
3382:
3381:
3376:
3371:
3370:
3339:
3338:
3319:
3314:
3299:
3298:
3294:
3219:
3205:
3196:
3177:
3175:
3174:
3169:
3155:
3154:
3153:
3152:
3126:
3125:
3124:
3123:
3111:
3110:
3075:
3068:
3061:
3057:
3050:
3044:
3040:
3036:
3022:
3020:
3019:
3014:
3012:
3011:
3007:
3006:
2979:
2952:
2951:
2929:
2925:
2921:
2915:
2900:
2892:
2886:
2878:
2868:
2866:
2865:
2860:
2858:
2842:
2841:
2810:
2809:
2784:
2783:
2774:
2773:
2757:
2753:
2752:
2744:
2734:
2733:
2707:
2705:
2704:
2699:
2694:
2692:
2691:
2690:
2662:
2659:
2658:
2657:
2652:
2593:
2592:
2571:
2570:
2551:
2537:
2533:
2523:
2521:
2520:
2515:
2510:
2509:
2496:
2463:
2462:
2443:
2439:
2435:
2429:
2419:
2395:
2385:
2372:
2364:
2360:
2342:
2338:
2320:
2316:
2312:
2305:
2298:
2290:
2283:
2279:
2275:
2250:
2241:from the set of
2223:
2215:
2190:
2179:
2175:
2167:
2165:
2164:
2159:
2108:
2107:
2095:
2091:
2090:
2082:
2056:
2054:
2053:
2048:
2046:
2045:
2000:
1999:
1939:
1937:
1936:
1931:
1919:
1917:
1916:
1911:
1903:
1886:
1848:
1846:
1845:
1840:
1816:
1805:
1798:
1786:
1773:
1762:
1760:
1759:
1754:
1740:
1739:
1712:
1708:
1706:
1692:
1678:
1660:
1652:
1639:
1629:
1618:
1595:upper half-plane
1592:
1577:
1575:
1574:
1569:
1567:
1563:
1539:
1535:
1506:
1502:
1501:
1452:
1438:
1435:
1420:
1395:
1381:Modular function
1372:
1370:
1369:
1364:
1359:
1351:
1350:
1345:
1342:
1319:
1317:
1316:
1311:
1303:
1295:
1294:
1289:
1280:
1279:
1261:
1260:
1238:
1236:
1235:
1230:
1215:
1213:
1212:
1207:
1196:
1195:
1180:
1179:
1164:
1163:
1151:
1150:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1072:
1064:
1063:
1058:
1055:
1023:
1021:
1020:
1015:
998:
995:
986:
984:
983:
978:
949:
948:
914:
912:
911:
906:
901:
893:
892:
887:
884:
860:
858:
857:
852:
843:
835:
834:
829:
826:
820:
819:
771:
769:
768:
763:
751:
749:
748:
743:
723:
675:
673:
672:
667:
650:
647:
638:
636:
635:
630:
607:
606:
572:
570:
569:
564:
559:
551:
550:
545:
542:
524:
522:
521:
516:
502:
501:
446:
444:
443:
438:
415:upper half-plane
412:
410:
409:
404:
402:
394:
393:
370:
368:
367:
362:
350:
348:
347:
342:
327:arithmetic group
320:
318:
317:
312:
307:
299:
298:
293:
290:
252:
250:
249:
244:
239:
231:
230:
225:
210:
202:
201:
196:
148:complex analysis
122:
120:
119:
114:
111:
110:
95:upper half-plane
74:
67:
63:
60:
54:
34:
33:
26:
7591:
7590:
7584:
7583:
7582:
7580:
7579:
7578:
7554:
7553:
7552:
7543:
7515:
7506:Delta invariant
7477:
7446:
7410:
7371:Abel–Jacobi map
7352:
7326:
7322:Torelli theorem
7292:Dessin d'enfant
7272:Belyi's theorem
7258:
7244:Plücker formula
7175:
7166:Hurwitz surface
7135:
7114:
7048:
7022:Analytic theory
7014:Elliptic curves
7008:
6989:Projective line
6976:Rational curves
6970:
6964:
6892:
6882:Springer-Verlag
6872:
6862:
6857:
6840:
6825:
6797:
6785:
6775:Springer-Verlag
6768:
6763:
6753:Springer-Verlag
6741:Apostol, Tom M.
6739:
6736:
6731:
6730:
6722:
6715:
6713:-Modular Forms"
6684:
6679:
6678:
6674:
6673:
6669:
6660:
6653:
6652:
6648:
6637:
6636:
6632:
6618:
6617:
6613:
6591:
6581:Springer-Verlag
6565:
6564:
6560:
6552:
6539:
6538:
6534:
6524:
6523:
6519:
6511:
6504:
6499:
6498:
6494:
6480:
6479:
6472:
6467:
6462:
6461:
6432:
6431:
6426:
6420:
6419:
6414:
6404:
6398:
6397:
6396:Here, a matrix
6395:
6391:
6366:
6362:
6353:
6351:
6343:
6324:
6323:
6321:
6317:
6312:
6300:
6252:from about 1925
6224:
6213:
6207:
6204:
6189:
6173:
6162:
6107:
6106:
6067:
5999:
5985:
5979:
5971:
5970:
5946:
5896:
5895:
5884:
5878:
5842:
5810:
5777:
5770:
5768:Generalizations
5758:
5752:
5714:
5701:
5676:
5663:
5658:
5657:
5627:
5605:
5600:
5599:
5574:
5573:
5550:
5549:
5530:
5529:
5510:
5509:
5503:
5497:
5492:
5484:Fuchsian groups
5464:
5431:
5395:
5394:
5390:
5386:
5382:
5378:
5374:
5362:
5358:
5357:For a subgroup
5355:
5349:
5302:
5224:
5193:
5131:
5130:
5101:
5100:
5080:
5079:
5072:
5049:
5045:
5042:
5041:
5014:
4997:
4993:
4986:
4954:
4950:
4940:
4925:
4920:
4919:
4915:. For example,
4908:
4895:
4890:
4881:
4876:
4865:
4853:= Γ(1) = SL(2,
4849:
4836:
4826:
4809:
4804:
4795:
4790:
4778:
4774:
4766:
4754:
4751:
4722:
4709:, i.e. subsets
4669:
4658:
4629:
4623:
4600:
4599:
4507:
4506:
4501:
4495:
4494:
4489:
4479:
4477:
4473:
4466:
4451:
4450:
4385:
4384:
4379:
4373:
4372:
4367:
4357:
4355:
4351:
4344:
4325:
4316:
4315:
4299:Riemann surface
4285:(a matrix with
4282:
4265:Hausdorff space
4245:
4234:
4233:. Such a group
4222:
4218:
4215:
4179:
4165:
4152:
4151:
4144:
4123:
4114:
4109:
4080:
4076:
4070:
4069:(equivalently,
4063:
4058:A modular form
4037:
4019:elliptic curves
3959:
3954:
3953:
3912:
3902:
3858:
3857:
3802:
3801:
3760:
3750:
3706:
3705:
3642:
3628:
3622:
3614:
3613:
3608:
3588:
3587:
3582:
3576:
3575:
3570:
3560:
3554:
3553:
3499:
3487:Hecke operators
3479:quadratic forms
3444:
3440:
3436:
3428:
3400:
3353:
3330:
3282:
3262:
3261:
3251:
3215:
3204:
3198:
3195:
3188:
3182:
3144:
3139:
3115:
3102:
3097:
3092:
3091:
3085:
3070:
3067:
3063:
3059:
3052:
3046:
3042:
3038:
3031:
2998:
2981:
2943:
2938:
2937:
2927:
2923:
2917:
2913:
2894:
2888:
2884:
2873:
2856:
2855:
2833:
2826:
2801:
2798:
2797:
2775:
2765:
2758:
2739:
2735:
2725:
2716:
2715:
2682:
2666:
2647:
2584:
2562:
2557:
2556:
2539:
2535:
2532:
2528:
2498:
2454:
2449:
2448:
2441:
2437:
2431:
2426:
2421:
2414:
2402:
2387:
2374:
2370:
2362:
2355:
2340:
2322:
2318:
2314:
2307:
2303:
2296:
2285:
2281:
2280:and a variable
2277:
2260:
2253:complex numbers
2246:
2235:
2221:
2197:
2184:
2177:
2173:
2099:
2077:
2073:
2065:
2064:
2040:
2039:
2034:
2028:
2027:
2022:
2012:
1994:
1993:
1988:
1982:
1981:
1973:
1963:
1951:
1950:
1922:
1921:
1851:
1850:
1819:
1818:
1810:
1803:
1796:
1775:
1767:
1731:
1693:
1679:
1673:
1665:
1664:
1654:
1644:
1635:
1623:
1597:
1586:
1496:
1495:
1490:
1484:
1483:
1478:
1468:
1464:
1463:
1459:
1429:
1428:
1418:
1415:
1410:
1386:
1383:
1340:
1329:
1328:
1284:
1252:
1247:
1246:
1221:
1220:
1187:
1168:
1155:
1142:
1131:
1130:
1108:
1107:
1088:
1087:
1053:
1042:
1041:
1030:
989:
988:
937:
917:
916:
882:
871:
870:
824:
814:
813:
808:
802:
801:
796:
786:
774:
773:
754:
753:
681:
680:
641:
640:
639:is bounded for
595:
575:
574:
540:
529:
528:
493:
449:
448:
423:
422:
377:
376:
353:
352:
333:
332:
288:
277:
276:
273:
217:
188:
183:
182:
99:
98:
89:is a (complex)
75:
64:
58:
55:
47:help improve it
44:
35:
31:
24:
17:
12:
11:
5:
7589:
7588:
7585:
7577:
7576:
7571:
7566:
7556:
7555:
7549:
7548:
7545:
7544:
7542:
7541:
7536:
7531:
7525:
7523:
7521:Vector bundles
7517:
7516:
7514:
7513:
7508:
7503:
7498:
7493:
7487:
7485:
7479:
7478:
7476:
7475:
7470:
7465:
7460:
7454:
7452:
7448:
7447:
7445:
7444:
7439:
7434:
7429:
7424:
7418:
7416:
7412:
7411:
7409:
7408:
7403:
7398:
7393:
7388:
7383:
7378:
7373:
7367:
7365:
7358:
7354:
7353:
7351:
7350:
7345:
7340:
7334:
7332:
7328:
7327:
7325:
7324:
7319:
7314:
7309:
7304:
7299:
7294:
7289:
7284:
7279:
7274:
7268:
7266:
7260:
7259:
7257:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7211:
7206:
7201:
7196:
7191:
7185:
7183:
7177:
7176:
7174:
7173:
7168:
7163:
7158:
7153:
7147:
7145:
7141:
7140:
7137:
7136:
7134:
7133:
7128:
7122:
7120:
7116:
7115:
7113:
7112:
7107:
7102:
7097:
7092:
7087:
7082:
7077:
7072:
7067:
7062:
7056:
7054:
7050:
7049:
7047:
7046:
7041:
7036:
7031:
7025:
7023:
7016:
7010:
7009:
7007:
7006:
7001:
6999:Riemann sphere
6996:
6991:
6986:
6980:
6978:
6972:
6971:
6965:
6963:
6962:
6955:
6948:
6940:
6934:
6933:
6928:
6890:
6870:
6860:
6855:
6838:
6823:
6795:
6784:978-0387232294
6783:
6766:
6761:
6735:
6732:
6729:
6728:
6702:
6699:
6696:
6691:
6687:
6667:
6665:, Theorem 6.1.
6646:
6630:
6611:
6589:
6583:, p. 24,
6558:
6550:
6532:
6517:
6500:Lan, Kai-Wen.
6492:
6469:
6468:
6466:
6463:
6460:
6459:
6436:
6430:
6427:
6425:
6422:
6421:
6418:
6415:
6413:
6410:
6409:
6407:
6389:
6360:
6331:
6314:
6313:
6311:
6308:
6307:
6306:
6299:
6296:
6261:
6260:
6253:
6246:
6239:
6226:
6225:
6176:
6174:
6167:
6161:
6158:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6120:
6117:
6114:
6103:
6102:
6091:
6088:
6085:
6082:
6079:
6074:
6070:
6066:
6063:
6060:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6021:
6017:
6011:
6008:
6005:
6002:
5997:
5994:
5991:
5988:
5982:
5978:
5953:
5949:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5903:
5795:eigenfunctions
5769:
5766:
5754:Main article:
5751:
5748:
5735:
5732:
5729:
5726:
5721:
5717:
5713:
5708:
5704:
5700:
5697:
5694:
5691:
5688:
5683:
5679:
5675:
5670:
5666:
5645:
5642:
5639:
5634:
5630:
5626:
5623:
5620:
5617:
5612:
5608:
5587:
5584:
5581:
5557:
5537:
5517:
5499:Main article:
5496:
5493:
5491:
5488:
5473:Pierre Deligne
5449:
5446:
5443:
5438:
5434:
5428:
5425:
5422:
5418:
5414:
5411:
5408:
5405:
5402:
5380:
5351:Main article:
5348:
5345:
5290:) =
5258:) =
5223:
5220:
5191:
5138:
5127:floor function
5114:
5111:
5108:
5097:
5096:
5083:
5073:
5071:
5068:
5064:
5060:
5056:
5052:
5048:
5044:
5043:
5039:
5036:
5032:
5029:
5024:
5021:
5018:
5015:
5012:
5008:
5004:
5000:
4996:
4992:
4991:
4989:
4984:
4980:
4976:
4972:
4968:
4965:
4962:
4953:
4947:
4943:
4939:
4933:
4928:
4893:
4879:
4864:
4861:
4834:
4807:
4793:
4750:
4747:
4667:
4656:
4621:
4614:
4613:
4598:
4594:
4589:
4586:
4582:
4579:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4534:
4530:
4527:
4524:
4516:
4511:
4505:
4502:
4500:
4497:
4496:
4493:
4490:
4488:
4485:
4484:
4482:
4476:
4472:
4469:
4467:
4465:
4462:
4459:
4456:
4453:
4452:
4448:
4443:
4440:
4436:
4433:
4428:
4425:
4422:
4419:
4416:
4412:
4408:
4405:
4402:
4394:
4389:
4383:
4380:
4378:
4375:
4374:
4371:
4368:
4366:
4363:
4362:
4360:
4354:
4350:
4347:
4345:
4343:
4340:
4337:
4332:
4328:
4324:
4323:
4214:
4207:
4191:
4188:
4185:
4182:
4177:
4174:
4171:
4168:
4162:
4159:
4143:
4140:
4112:
4090:) is called a
4074:
4023:elliptic curve
3966:
3962:
3944:-expansion of
3938:
3937:
3936:
3935:
3924:
3919:
3915:
3909:
3905:
3899:
3894:
3891:
3888:
3885:
3881:
3877:
3874:
3871:
3868:
3865:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3798:
3797:
3796:
3795:
3784:
3779:
3776:
3773:
3770:
3767:
3763:
3757:
3753:
3747:
3742:
3739:
3736:
3733:
3729:
3725:
3722:
3719:
3716:
3713:
3700:
3699:
3696:Fourier series
3688:
3676:
3673:
3670:
3667:
3664:
3660:
3654:
3651:
3648:
3645:
3640:
3637:
3634:
3631:
3625:
3621:
3607:modular group
3592:
3586:
3583:
3581:
3578:
3577:
3574:
3571:
3569:
3566:
3565:
3563:
3547:
3498:
3495:
3468:Pierre Deligne
3439:for any prime
3386:
3385:
3374:
3369:
3366:
3363:
3360:
3356:
3352:
3349:
3345:
3342:
3337:
3333:
3329:
3326:
3323:
3318:
3313:
3310:
3307:
3303:
3297:
3293:
3289:
3285:
3281:
3278:
3275:
3272:
3269:
3257:is defined as
3202:
3193:
3186:
3179:
3178:
3167:
3164:
3161:
3158:
3151:
3147:
3142:
3138:
3135:
3132:
3129:
3122:
3118:
3114:
3109:
3105:
3100:
3083:
3065:
3024:
3023:
3010:
3005:
3001:
2997:
2994:
2991:
2988:
2984:
2978:
2975:
2972:
2968:
2964:
2961:
2958:
2955:
2950:
2946:
2932:theta function
2879:is needed for
2872:The condition
2870:
2869:
2854:
2851:
2848:
2845:
2840:
2836:
2832:
2829:
2827:
2825:
2822:
2819:
2816:
2813:
2808:
2804:
2800:
2799:
2796:
2793:
2790:
2787:
2782:
2778:
2772:
2768:
2764:
2761:
2759:
2756:
2750:
2747:
2742:
2738:
2732:
2728:
2724:
2723:
2709:
2708:
2697:
2689:
2685:
2681:
2678:
2675:
2672:
2669:
2665:
2656:
2651:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2609:
2605:
2602:
2599:
2596:
2591:
2587:
2583:
2580:
2577:
2574:
2569:
2565:
2530:
2525:
2524:
2513:
2508:
2505:
2501:
2495:
2492:
2489:
2486:
2483:
2479:
2475:
2472:
2469:
2466:
2461:
2457:
2424:
2401:
2398:
2367:
2366:
2352:absolute value
2348:
2300:
2251:to the set of
2234:
2231:
2230:
2229:
2226:Fourier series
2220:, with period
2206:+ 1) =
2193:
2192:
2170:
2169:
2168:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2123:
2120:
2117:
2114:
2111:
2106:
2102:
2098:
2094:
2088:
2085:
2080:
2076:
2072:
2059:
2058:
2057:
2044:
2038:
2035:
2033:
2030:
2029:
2026:
2023:
2021:
2018:
2017:
2015:
2010:
2007:
2003:
1998:
1992:
1989:
1987:
1984:
1983:
1980:
1977:
1974:
1972:
1969:
1968:
1966:
1961:
1958:
1945:
1944:
1941:
1929:
1909:
1906:
1902:
1898:
1895:
1892:
1889:
1885:
1880:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1838:
1835:
1832:
1829:
1826:
1807:
1800:
1789:
1788:
1765:
1764:
1763:
1752:
1749:
1746:
1743:
1738:
1734:
1730:
1727:
1724:
1721:
1718:
1715:
1711:
1705:
1702:
1699:
1696:
1691:
1688:
1685:
1682:
1676:
1672:
1641:
1583:complex-valued
1579:
1578:
1566:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1538:
1534:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1505:
1500:
1494:
1491:
1489:
1486:
1485:
1482:
1479:
1477:
1474:
1473:
1471:
1466:
1462:
1458:
1455:
1451:
1447:
1444:
1441:
1414:
1411:
1409:
1406:
1382:
1379:
1362:
1358:
1354:
1349:
1339:
1336:
1309:
1306:
1302:
1298:
1293:
1288:
1283:
1278:
1273:
1270:
1267:
1264:
1259:
1255:
1228:
1205:
1202:
1199:
1194:
1190:
1186:
1183:
1178:
1175:
1171:
1167:
1162:
1158:
1154:
1149:
1145:
1141:
1138:
1115:
1095:
1075:
1071:
1067:
1062:
1052:
1049:
1029:
1026:
1025:
1024:
1013:
1010:
1007:
1004:
1001:
976:
973:
970:
967:
964:
961:
958:
955:
952:
947:
944:
940:
936:
933:
930:
927:
924:
904:
900:
896:
891:
881:
878:
849:
846:
842:
838:
833:
823:
818:
812:
809:
807:
804:
803:
800:
797:
795:
792:
791:
789:
784:
781:
761:
741:
738:
735:
732:
729:
726:
722:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
677:
676:
665:
662:
659:
656:
653:
628:
625:
622:
619:
616:
613:
610:
605:
602:
598:
594:
591:
588:
585:
582:
562:
558:
554:
549:
539:
536:
525:
514:
511:
508:
505:
500:
496:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
436:
433:
430:
401:
397:
392:
387:
384:
360:
340:
310:
306:
302:
297:
287:
284:
272:
269:
242:
238:
234:
229:
224:
221:
216:
213:
209:
205:
200:
195:
192:
160:sphere packing
144:
143:
140:
109:
77:
76:
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
7587:
7586:
7575:
7572:
7570:
7567:
7565:
7564:Modular forms
7562:
7561:
7559:
7540:
7537:
7535:
7532:
7530:
7527:
7526:
7524:
7522:
7518:
7512:
7509:
7507:
7504:
7502:
7499:
7497:
7494:
7492:
7489:
7488:
7486:
7484:
7483:Singularities
7480:
7474:
7471:
7469:
7466:
7464:
7461:
7459:
7456:
7455:
7453:
7449:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7419:
7417:
7413:
7407:
7404:
7402:
7399:
7397:
7394:
7392:
7389:
7387:
7384:
7382:
7379:
7377:
7374:
7372:
7369:
7368:
7366:
7362:
7359:
7355:
7349:
7346:
7344:
7341:
7339:
7336:
7335:
7333:
7331:Constructions
7329:
7323:
7320:
7318:
7315:
7313:
7310:
7308:
7305:
7303:
7302:Klein quartic
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7282:Bolza surface
7280:
7278:
7277:Bring's curve
7275:
7273:
7270:
7269:
7267:
7265:
7261:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7210:
7209:Conic section
7207:
7205:
7202:
7200:
7197:
7195:
7192:
7190:
7189:AF+BG theorem
7187:
7186:
7184:
7182:
7178:
7172:
7169:
7167:
7164:
7162:
7159:
7157:
7154:
7152:
7149:
7148:
7146:
7142:
7132:
7129:
7127:
7124:
7123:
7121:
7117:
7111:
7108:
7106:
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7086:
7083:
7081:
7078:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7057:
7055:
7051:
7045:
7042:
7040:
7037:
7035:
7032:
7030:
7027:
7026:
7024:
7020:
7017:
7015:
7011:
7005:
7004:Twisted cubic
7002:
7000:
6997:
6995:
6992:
6990:
6987:
6985:
6982:
6981:
6979:
6977:
6973:
6969:
6961:
6956:
6954:
6949:
6947:
6942:
6941:
6938:
6932:
6929:
6926:
6922:
6918:
6914:
6910:
6906:
6902:
6901:
6896:
6891:
6888:
6883:
6879:
6875:
6871:
6868:
6867:
6861:
6858:
6856:0-521-21212-X
6852:
6848:
6845:, Cambridge:
6844:
6839:
6836:
6833:, Göttingen:
6832:
6828:
6824:
6821:
6816:
6812:
6808:
6804:
6800:
6796:
6793:
6792:
6786:
6780:
6776:
6772:
6767:
6764:
6762:0-387-97127-0
6758:
6754:
6749:
6748:
6742:
6738:
6737:
6733:
6721:
6714:
6697:
6689:
6671:
6668:
6659:
6658:
6650:
6647:
6641:
6634:
6631:
6626:
6622:
6615:
6612:
6608:
6604:
6600:
6596:
6592:
6586:
6582:
6578:
6577:
6576:Modular units
6572:
6568:
6562:
6559:
6553:
6551:3-540-15295-4
6547:
6543:
6536:
6533:
6529:. p. 51.
6528:
6521:
6518:
6510:
6503:
6496:
6493:
6488:
6484:
6477:
6475:
6471:
6464:
6456:
6452:
6434:
6428:
6423:
6416:
6411:
6405:
6393:
6390:
6386:
6382:
6378:
6374:
6370:
6364:
6361:
6350:
6349:dlmf.nist.gov
6346:
6329:
6319:
6316:
6309:
6305:
6302:
6301:
6297:
6295:
6293:
6289:
6285:
6281:
6277:
6272:
6270:
6266:
6258:
6254:
6251:
6247:
6244:
6240:
6237:
6233:
6232:
6231:
6222:
6219:
6211:
6201:
6197:
6193:
6187:
6186:
6182:
6177:This section
6175:
6171:
6166:
6165:
6159:
6157:
6155:
6136:
6133:
6130:
6127:
6124:
6121:
6118:
6112:
6105:The function
6089:
6083:
6077:
6072:
6064:
6061:
6058:
6055:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6022:
6019:
6015:
6009:
6006:
6003:
6000:
5995:
5992:
5989:
5986:
5980:
5976:
5969:
5968:
5967:
5951:
5943:
5940:
5937:
5934:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5901:
5893:
5889:
5887:
5875:
5873:
5869:
5865:
5862:
5858:
5856:
5852:
5846:
5840:
5836:
5832:
5830:
5826:
5822:
5818:
5814:
5808:
5804:
5800:
5796:
5793:
5792:real-analytic
5789:
5785:
5781:
5775:
5774:Haar measures
5767:
5765:
5763:
5757:
5749:
5747:
5727:
5719:
5706:
5702:
5698:
5689:
5681:
5668:
5664:
5640:
5632:
5624:
5618:
5610:
5585:
5582:
5579:
5571:
5555:
5535:
5515:
5507:
5502:
5494:
5489:
5487:
5485:
5480:
5478:
5474:
5468:
5461:
5436:
5432:
5426:
5423:
5420:
5416:
5412:
5400:
5372:
5366:
5354:
5346:
5344:
5341:
5339:
5335:
5331:
5327:
5323:
5318:
5316:
5312:
5308:
5300:
5296:
5293:
5289:
5285:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5253:
5249:
5245:
5241:
5237:
5233:
5229:
5221:
5219:
5217:
5213:
5209:
5205:
5201:
5197:
5190:
5187:
5183:
5179:
5175:
5171:
5167:
5163:
5159:
5155:
5150:
5136:
5128:
5109:
5069:
5066:
5062:
5058:
5054:
5050:
5046:
5034:
5030:
5022:
5019:
5016:
5010:
5006:
5002:
4998:
4994:
4987:
4982:
4978:
4966:
4963:
4951:
4945:
4941:
4937:
4926:
4918:
4917:
4916:
4914:
4906:
4900:
4896:
4886:
4882:
4874:
4870:
4862:
4860:
4856:
4852:
4847:
4844:
4840:
4832:
4824:
4820:
4814:
4810:
4800:
4796:
4788:
4784:
4772:
4764:
4760:
4748:
4746:
4744:
4740:
4736:
4732:
4728:
4720:
4716:
4712:
4708:
4704:
4700:
4696:
4691:
4689:
4685:
4681:
4677:
4673:
4666:
4662:
4655:
4651:
4647:
4643:
4639:
4636:, the spaces
4633:
4627:
4619:
4596:
4592:
4584:
4580:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4528:
4525:
4514:
4509:
4503:
4498:
4491:
4486:
4480:
4474:
4470:
4468:
4460:
4446:
4438:
4434:
4426:
4423:
4420:
4417:
4406:
4403:
4392:
4387:
4381:
4376:
4369:
4364:
4358:
4352:
4348:
4346:
4338:
4330:
4314:
4313:
4312:
4311:
4307:
4302:
4300:
4296:
4292:
4288:
4280:
4279:
4274:
4270:
4266:
4262:
4258:
4255:
4249:
4243:
4239:
4232:
4226:
4212:
4208:
4206:
4189:
4186:
4183:
4180:
4175:
4172:
4169:
4166:
4157:
4149:
4141:
4139:
4137:
4136:
4130:
4126:
4121:
4115:
4107:
4103:
4099:
4095:
4094:
4087:
4083:
4073:
4066:
4061:
4056:
4054:
4050:
4046:
4043:
4035:
4031:
4027:
4024:
4020:
4015:
4013:
4009:
4004:
4002:
3998:
3994:
3990:
3986:
3982:
3964:
3960:
3951:
3947:
3943:
3922:
3917:
3913:
3907:
3903:
3892:
3889:
3886:
3883:
3879:
3875:
3869:
3863:
3856:
3855:
3854:
3853:
3852:
3850:
3831:
3828:
3825:
3822:
3816:
3813:
3810:
3807:
3782:
3777:
3774:
3771:
3768:
3765:
3761:
3755:
3751:
3740:
3737:
3734:
3731:
3727:
3723:
3717:
3711:
3704:
3703:
3702:
3701:
3697:
3693:
3689:
3671:
3665:
3662:
3658:
3652:
3649:
3646:
3643:
3638:
3635:
3632:
3629:
3623:
3619:
3611:
3590:
3584:
3579:
3572:
3567:
3561:
3552:
3548:
3546:
3543:
3539:
3535:
3532:
3531:
3530:
3528:
3524:
3520:
3517:. A function
3516:
3512:
3508:
3504:
3496:
3494:
3492:
3488:
3484:
3480:
3475:
3473:
3469:
3465:
3461:
3457:
3453:
3448:
3432:
3426:
3422:
3418:
3417:Leech lattice
3412:
3408:
3404:
3399:
3395:
3391:
3372:
3367:
3364:
3361:
3358:
3354:
3350:
3347:
3343:
3335:
3331:
3327:
3324:
3311:
3308:
3305:
3301:
3295:
3291:
3287:
3283:
3279:
3273:
3267:
3260:
3259:
3258:
3256:
3250:
3245:
3244:
3240:
3238:
3234:
3230:
3226:
3223:
3218:
3213:
3209:
3201:
3192:
3185:
3165:
3159:
3149:
3145:
3140:
3136:
3130:
3120:
3116:
3112:
3107:
3103:
3098:
3090:
3089:
3088:
3086:
3079:
3073:
3056:
3049:
3034:
3029:
3008:
3003:
2995:
2989:
2986:
2982:
2976:
2973:
2970:
2966:
2962:
2956:
2948:
2944:
2936:
2935:
2934:
2933:
2920:
2912:
2907:
2906:
2902:
2898:
2891:
2882:
2876:
2852:
2846:
2838:
2834:
2830:
2828:
2820:
2817:
2814:
2806:
2802:
2794:
2788:
2780:
2776:
2770:
2766:
2762:
2760:
2754:
2748:
2745:
2740:
2736:
2730:
2726:
2714:
2713:
2712:
2695:
2687:
2679:
2676:
2673:
2670:
2663:
2654:
2644:
2638:
2635:
2632:
2626:
2620:
2617:
2614:
2607:
2603:
2597:
2589:
2585:
2581:
2567:
2563:
2555:
2554:
2553:
2550:
2547:
2543:
2511:
2506:
2503:
2499:
2490:
2487:
2484:
2481:
2477:
2473:
2459:
2455:
2447:
2446:
2445:
2434:
2427:
2417:
2412:
2407:
2406:
2399:
2397:
2394:
2390:
2384:
2381:
2377:
2358:
2353:
2349:
2346:
2336:
2333:
2329:
2325:
2310:
2301:
2294:
2288:
2274:
2271:
2267:
2264:
2258:
2257:
2256:
2254:
2249:
2244:
2240:
2232:
2227:
2219:
2213:
2209:
2205:
2201:
2195:
2194:
2188:
2182:
2171:
2152:
2146:
2143:
2137:
2134:
2131:
2125:
2121:
2115:
2109:
2104:
2100:
2096:
2092:
2086:
2083:
2078:
2074:
2070:
2063:
2062:
2060:
2042:
2036:
2031:
2024:
2019:
2013:
2008:
2005:
2001:
1996:
1990:
1985:
1978:
1975:
1970:
1964:
1959:
1956:
1949:
1948:
1947:
1946:
1942:
1927:
1907:
1904:
1893:
1887:
1874:
1871:
1865:
1859:
1856:
1836:
1833:
1830:
1827:
1824:
1814:
1808:
1801:
1794:
1793:
1792:
1784:
1783:
1778:
1771:
1766:
1747:
1741:
1736:
1728:
1725:
1722:
1719:
1713:
1709:
1703:
1700:
1697:
1694:
1689:
1686:
1683:
1680:
1674:
1670:
1663:
1662:
1658:
1651:
1647:
1642:
1638:
1633:
1627:
1622:
1621:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1590:
1584:
1564:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1536:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1503:
1498:
1492:
1487:
1480:
1475:
1469:
1460:
1456:
1445:
1442:
1427:
1426:
1425:
1424:
1423:modular group
1412:
1407:
1405:
1403:
1399:
1393:
1389:
1380:
1378:
1376:
1347:
1337:
1326:
1320:
1291:
1281:
1262:
1253:
1243:
1242:
1241:modular curve
1226:
1216:
1192:
1188:
1184:
1176:
1173:
1169:
1165:
1156:
1147:
1143:
1139:
1136:
1127:
1113:
1060:
1050:
1039:
1035:
1027:
1002:
974:
962:
956:
950:
945:
942:
934:
931:
928:
925:
915:the function
889:
879:
876:
868:
867:
866:
864:
847:
831:
821:
816:
810:
805:
798:
793:
787:
782:
779:
759:
736:
733:
730:
727:
720:
713:
710:
707:
704:
698:
692:
686:
654:
620:
614:
608:
603:
600:
592:
589:
586:
583:
573:the function
547:
537:
534:
526:
509:
503:
498:
490:
487:
484:
481:
475:
466:
460:
454:
431:
428:
420:
419:
418:
416:
385:
382:
375:
371:
358:
328:
324:
295:
285:
270:
268:
266:
261:
259:
254:
227:
214:
198:
180:
176:
172:
167:
165:
164:string theory
161:
157:
153:
152:number theory
149:
141:
138:
137:modular group
134:
130:
126:
125:
124:
96:
92:
88:
84:
73:
70:
62:
59:February 2024
52:
48:
42:
39:This article
37:
28:
27:
22:
7468:Prym variety
7442:Stable curve
7432:Hodge bundle
7422:ELSV formula
7224:Fermat curve
7181:Plane curves
7144:Higher genus
7119:Applications
7044:Modular form
7043:
6904:
6898:
6886:
6877:
6865:
6842:
6830:
6827:Hecke, Erich
6819:
6802:
6788:
6770:
6751:, New York:
6746:
6670:
6663:, p. 88
6656:
6649:
6639:
6633:
6620:
6614:
6575:
6561:
6541:
6535:
6520:
6495:
6486:
6454:
6450:
6392:
6384:
6376:
6363:
6352:. Retrieved
6348:
6318:
6276:Andrew Wiles
6273:
6262:
6229:
6214:
6208:October 2019
6205:
6190:Please help
6178:
6104:
5890:
5882:
5876:
5866:
5861:Jacobi forms
5859:
5854:
5844:
5833:
5824:
5819:
5812:
5786:
5779:
5771:
5759:
5569:
5504:
5481:
5466:
5462:
5364:
5356:
5342:
5329:
5321:
5319:
5314:
5310:
5306:
5298:
5294:
5291:
5287:
5283:
5279:
5271:
5267:
5263:
5259:
5255:
5251:
5247:
5243:
5239:
5235:
5231:
5225:
5222:Line bundles
5215:
5211:
5207:
5203:
5199:
5195:
5188:
5177:
5169:
5165:
5161:
5151:
5125:denotes the
5098:
4912:
4898:
4891:
4884:
4877:
4872:
4868:
4866:
4863:Consequences
4854:
4850:
4845:
4842:
4838:
4830:
4822:
4818:
4812:
4805:
4798:
4791:
4789:are denoted
4786:
4782:
4770:
4762:
4758:
4752:
4742:
4738:
4730:
4726:
4718:
4714:
4710:
4706:
4698:
4694:
4692:
4687:
4683:
4679:
4675:
4671:
4664:
4660:
4653:
4652:are denoted
4649:
4645:
4641:
4637:
4631:
4625:
4617:
4615:
4305:
4303:
4294:
4290:
4276:
4272:
4268:
4260:
4256:
4247:
4241:
4224:
4216:
4210:
4147:
4145:
4135:modular unit
4133:
4131:
4124:
4119:
4110:
4105:
4097:
4091:
4085:
4081:
4071:
4064:
4059:
4057:
4053:moduli space
4048:
4044:
4029:
4025:
4016:
4011:
4007:
4005:
4000:
3996:
3992:
3988:
3984:
3980:
3945:
3941:
3939:
3799:
3691:
3544:
3540:in the open
3533:
3526:
3522:
3518:
3514:
3510:
3502:
3500:
3476:
3446:
3430:
3410:
3406:
3402:
3389:
3387:
3252:
3242:
3241:
3216:
3199:
3190:
3183:
3180:
3071:
3054:
3047:
3032:
3025:
2918:
2908:
2904:
2903:
2896:
2889:
2874:
2871:
2710:
2548:
2545:
2541:
2526:
2432:
2422:
2420:, we define
2415:
2408:
2404:
2403:
2392:
2388:
2382:
2379:
2375:
2368:
2356:
2347:of the form.
2344:
2334:
2331:
2327:
2323:
2308:
2286:
2272:
2269:
2265:
2262:
2247:
2238:
2236:
2211:
2207:
2203:
2199:
2186:
1812:
1790:
1781:
1776:
1769:
1656:
1649:
1645:
1636:
1625:
1614:
1606:
1602:
1598:
1588:
1580:
1416:
1391:
1387:
1384:
1322:
1245:
1218:
1129:
1031:
862:
678:
330:
325:, called an
323:finite index
274:
262:
255:
168:
145:
133:group action
87:modular form
86:
80:
65:
56:
40:
21:Haar measure
7343:Polar curve
6571:Lang, Serge
6449:sends ∞ to
6369:meromorphic
6342:, see e.g.
6292:square root
6250:Erich Hecke
6243:Felix Klein
5803:holomorphic
5788:Maass forms
5371:graded ring
5338:line bundle
5276:homogeneous
4911:-action on
4725:-action on
4098:Spitzenform
4042:j-invariant
3538:meromorphic
3396:. Then the
3229:isospectral
3208:John Milnor
3078:root system
2881:convergence
1795:The weight
1402:meromorphic
1398:holomorphic
1106:and weight
1034:line bundle
351:and weight
258:Erich Hecke
83:mathematics
7558:Categories
7338:Dual curve
6966:Topics in
6895:Zagier, D.
6734:References
6607:0492.12002
6354:2023-07-07
5872:Lie groups
5750:Cusp forms
5282:, letting
5160:one (over
4829:. In case
4757:of weight
4749:Definition
4717:such that
4275:, i.e. in
4108:such that
4034:isomorphic
3227:which are
3051:such that
2883:; for odd
1920:, meaning
1849:such that
1791:Remarks:
1617:) > 0},
271:Definition
175:Lie groups
127:a kind of
7451:Morphisms
7199:Bitangent
6686:Γ
6465:Citations
6383:as exp(1/
6330:γ
6290:with the
6179:does not
6113:ε
6023:ε
5902:ε
5799:Laplacian
5762:cusp form
5756:Cusp form
5716:Γ
5699:⊆
5678:Γ
5629:Γ
5625:⊆
5607:Γ
5583:∣
5570:old forms
5548:dividing
5506:New forms
5495:New forms
5445:Γ
5417:⨁
5407:Γ
5149:is even.
5113:⌋
5110:⋅
5107:⌊
5076:otherwise
5020:≡
4938:
4570:≡
4564:≡
4552:≡
4546:≡
4515:∈
4455:Γ
4424:≡
4393:∈
4327:Γ
4161:↦
4093:cusp form
3898:∞
3890:−
3880:∑
3826:π
3817:
3772:π
3746:∞
3738:−
3728:∑
3425:Ramanujan
3405:) = (2π)
3362:π
3328:−
3317:∞
3302:∏
3268:η
3233:isometric
3141:ϑ
3113:×
3099:ϑ
3000:‖
2996:λ
2993:‖
2987:π
2974:∈
2971:λ
2967:∑
2945:ϑ
2847:τ
2815:τ
2789:τ
2767:τ
2749:τ
2741:−
2680:τ
2645:∈
2627:≠
2608:∑
2598:τ
2576:Λ
2504:−
2500:λ
2494:Λ
2491:∈
2488:λ
2485:≠
2478:∑
2468:Λ
2079:−
1976:−
1879:⟹
1860:
1585:function
1549:−
1529:∈
1335:Γ
1282:∪
1269:∖
1266:Γ
1258:Γ
1227:ω
1201:Γ
1174:⊗
1170:ω
1161:Γ
1140:∈
1094:Γ
1051:⊂
1048:Γ
1012:∞
1009:→
972:→
957:γ
943:−
880:∈
877:γ
863:cusp form
822:∈
780:γ
760:γ
687:γ
664:∞
661:→
615:γ
601:−
538:∈
535:γ
461:γ
435:Γ
432:∈
429:γ
413:from the
396:→
339:Γ
286:⊂
283:Γ
215:⊂
6909:Springer
6876:(1973),
6829:(1970),
6801:(1975),
6743:(1990),
6720:Archived
6573:(1981),
6509:Archived
6298:See also
6274:In 1994
5330:sections
5063:⌋
5047:⌊
5011:⌋
4995:⌊
4028:/Λ over
3521: :
3481:and the
3231:but not
2552:we have
2400:Examples
2386:, where
2243:lattices
2210: (
2202: (
2181:generate
1802:For odd
1643:For any
1421:for the
1390: (
7511:Tacnode
7496:Crunode
6913:Bibcode
6911:: 113,
6815:0379375
6628:, p. 13
6599:0648603
6525:Milne.
6200:removed
6185:sources
6160:History
5797:of the
5361:of the
5184:of the
5182:closure
5180:in the
3851:), as:
3605:in the
3456:Shimura
3452:Eichler
3222:compact
3080:called
3069:. When
2321:, then
2284:, then
2198:
1815:
1811:
1772:
1768:
1628:
1624:
1593:on the
1591:
1587:
135:of the
93:on the
45:Please
7491:Acnode
7415:Moduli
6853:
6813:
6781:
6759:
6605:
6597:
6587:
6548:
6487:Quanta
6387:) has.
5843:SL(2,
5811:SL(2,
5465:SL(2,
5363:SL(2,
5210:) and
5099:where
4848:. For
4781:. The
4674:) and
4663:) and
4252:. The
4246:SL(2,
4223:SL(2,
4102:German
3551:matrix
3466:, and
3388:where
2877:> 2
2538:. For
2418:> 2
2345:weight
2339:where
2291:is an
2196:Since
2185:SL(2,
2061:reads
1655:SL(2,
1540:
1219:where
1040:. For
679:where
162:, and
6723:(PDF)
6716:(PDF)
6661:(PDF)
6556:p. 15
6512:(PDF)
6505:(PDF)
6310:Notes
5598:then
5490:Types
5334:sheaf
5332:of a
5174:poles
4843:level
4735:genus
4628:) or
4287:trace
4269:cusps
4231:index
3464:Ihara
3235:(see
2527:Then
2330:Λ) =
1630:is a
1581:is a
372:is a
7501:Cusp
6851:ISBN
6779:ISBN
6757:ISBN
6585:ISBN
6546:ISBN
6373:pole
6183:any
6181:cite
5790:are
5475:and
5424:>
5129:and
4889:and
4803:and
4705:for
4644:and
4616:For
4238:acts
4217:Let
3849:nome
3460:Kuga
3394:nome
3253:The
3212:tori
3197:and
2893:and
2711:and
2540:Λ =
2350:The
2261:Λ =
2176:and
1905:<
1872:>
1834:>
329:, a
85:, a
6921:doi
6603:Zbl
6375:at
6248:By
6241:By
6194:by
5383:(Γ)
5317:).
5218:).
5176:of
5031:mod
4927:dim
4833:= Γ
4737:of
4682:),
4620:= Γ
4581:mod
4435:mod
4240:on
4122:at
4116:≠ 0
4100:in
4077:= 0
4067:= 0
3814:exp
3536:is
3445:≤ 2
3423:of
3239:.)
3074:= 8
3045:in
2916:in
2909:An
2440:of
2428:(Λ)
2359:(Λ)
2354:of
2337:(Λ)
2317:by
2302:If
2295:of
2289:(Λ)
2245:in
1634:on
1601:= {
1396:be
1036:on
987:as
321:of
81:In
49:to
7560::
6919:,
6907:,
6905:94
6903:,
6885:.
6849:,
6818:.
6811:MR
6809:,
6777:,
6755:,
6718:.
6601:,
6595:MR
6593:,
6569:;
6507:.
6485:.
6473:^
6367:A
6347:.
5874:.
5831:.
5778:Δ(
5760:A
5746:.
5486:.
5460:.
5288:cv
5256:cv
5230:P(
5216:Nz
5059:12
5035:12
5007:12
4957:SL
4713:⊂
4630:Γ(
4519:SL
4397:SL
4213:\H
4132:A
4129:.
4084:=
3612:,
3525:→
3493:.
3462:,
3458:,
3454:,
3429:Δ(
3401:Δ(
3296:24
3203:16
3189:×
3150:16
3035:/2
2895:(−
2544:+
2444::
2396:.
2391:∈
2378:+
2268:+
1857:Im
1779:→
1648:∈
1611:Im
1609:,
1605:∈
1436:SL
1377:.
1343:SL
1056:SL
996:im
885:SL
827:SL
648:im
543:SL
291:SL
267:.
260:.
253:.
166:.
158:,
97:,
6959:e
6952:t
6945:v
6923::
6915::
6889:.
6822:.
6794:.
6701:)
6698:N
6695:(
6690:1
6554:.
6489:.
6457:.
6455:c
6453:/
6451:a
6435:)
6429:d
6424:c
6417:b
6412:a
6406:(
6385:q
6377:q
6357:.
6221:)
6215:(
6210:)
6206:(
6202:.
6188:.
6140:)
6137:d
6134:,
6131:c
6128:,
6125:b
6122:,
6119:a
6116:(
6090:.
6087:)
6084:z
6081:(
6078:f
6073:k
6069:)
6065:d
6062:+
6059:z
6056:c
6053:(
6050:)
6047:d
6044:,
6041:c
6038:,
6035:b
6032:,
6029:a
6026:(
6020:=
6016:)
6010:d
6007:+
6004:z
6001:c
5996:b
5993:+
5990:z
5987:a
5981:(
5977:f
5952:k
5948:)
5944:d
5941:+
5938:z
5935:c
5932:(
5929:)
5926:d
5923:,
5920:c
5917:,
5914:b
5911:,
5908:a
5905:(
5885:k
5879:k
5847:)
5845:R
5825:n
5815:)
5813:Z
5782:)
5780:g
5734:)
5731:)
5728:N
5725:(
5720:1
5712:(
5707:k
5703:M
5696:)
5693:)
5690:M
5687:(
5682:1
5674:(
5669:k
5665:M
5644:)
5641:M
5638:(
5633:1
5622:)
5619:N
5616:(
5611:1
5586:N
5580:M
5556:N
5536:M
5516:N
5469:)
5467:Z
5448:)
5442:(
5437:k
5433:M
5427:0
5421:k
5413:=
5410:)
5404:(
5401:M
5391:Γ
5387:k
5381:k
5379:M
5375:Γ
5367:)
5365:Z
5359:Γ
5322:V
5315:V
5311:k
5307:k
5303:k
5299:v
5297:(
5295:F
5292:c
5286:(
5284:F
5280:c
5272:F
5268:c
5264:v
5262:(
5260:F
5254:(
5252:F
5248:V
5244:v
5240:V
5236:F
5232:V
5214:(
5212:j
5208:z
5206:(
5204:j
5200:N
5198:(
5196:N
5192:Γ
5189:R
5178:f
5170:f
5166:f
5162:C
5137:k
5070:1
5067:+
5055:/
5051:k
5038:)
5028:(
5023:2
5017:k
5003:/
4999:k
4988:{
4983:=
4979:)
4975:)
4971:Z
4967:,
4964:2
4961:(
4952:(
4946:k
4942:M
4932:C
4913:H
4909:G
4901:)
4899:G
4897:(
4894:k
4892:S
4887:)
4885:G
4883:(
4880:k
4878:M
4873:H
4871:\
4869:G
4857:)
4855:Z
4851:G
4846:N
4839:N
4837:(
4835:0
4831:G
4827:G
4823:H
4821:\
4819:G
4815:)
4813:G
4811:(
4808:k
4806:S
4801:)
4799:G
4797:(
4794:k
4792:M
4787:k
4783:C
4779:G
4775:G
4771:H
4767:G
4763:H
4759:k
4755:G
4743:H
4741:\
4739:G
4731:D
4727:H
4723:G
4719:D
4715:H
4711:D
4707:G
4699:H
4697:\
4695:G
4688:N
4686:(
4684:X
4680:N
4678:(
4676:Y
4672:N
4670:(
4668:0
4665:X
4661:N
4659:(
4657:0
4654:Y
4650:H
4648:\
4646:G
4642:H
4640:\
4638:G
4634:)
4632:N
4626:N
4624:(
4622:0
4618:G
4597:.
4593:}
4588:)
4585:N
4578:(
4573:1
4567:d
4561:a
4558:,
4555:0
4549:b
4543:c
4540::
4537:)
4533:Z
4529:,
4526:2
4523:(
4510:)
4504:d
4499:c
4492:b
4487:a
4481:(
4475:{
4471:=
4464:)
4461:N
4458:(
4447:}
4442:)
4439:N
4432:(
4427:0
4421:c
4418::
4415:)
4411:Z
4407:,
4404:2
4401:(
4388:)
4382:d
4377:c
4370:b
4365:a
4359:(
4353:{
4349:=
4342:)
4339:N
4336:(
4331:0
4306:N
4295:H
4293:\
4291:G
4283:G
4278:Q
4273:H
4261:H
4259:\
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