4622:
6181:
4328:
43:
5105:
4617:{\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}}}
1587:
5874:
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
4932:
2878:
5490:. 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.
2066:
2717:
870:
1329:
3032:
6111:
3394:
262:
5100:{\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}}}
2533:
1441:
2177:
1225:
3187:
2728:
5755:
1772:
1929:
6458:
3614:
3804:
5469:
1095:
330:
924:
582:
3696:
5665:
3944:
1382:
4333:
2733:
1963:
422:
5975:
4214:
996:
534:
2569:
1033:
685:
648:
5134:
6161:
4062:) 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
761:
456:
132:
6493:
3856:
6722:
786:
1858:
1259:
6351:
5607:
1248:
1115:
360:
3488:
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
3988:
5324:
vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(
781:
872:
The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication. In addition, it is called a
5577:
5557:
5537:
5158:
1949:
1135:
380:
2950:
5983:
3274:
195:
1582:{\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\}}
2461:
6968:
5354:
Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
61:
2873:{\displaystyle {\begin{aligned}G_{k}\left(-{\frac {1}{\tau }}\right)&=\tau ^{k}G_{k}(\tau ),\\G_{k}(\tau +1)&=G_{k}(\tau ).\end{aligned}}}
4051:. Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the
6293:. 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
2077:
1143:
3104:
3259:
6297:. In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining the
6314:
5775:
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.
5670:
6519:
5281:. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let
6941:
5493:
More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitrary
1677:
2937:
vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in
7391:
6730:
6382:
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
3517:
1863:
6410:
3566:
3718:
5407:
1054:
289:
17:
5838:
variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a
5783:
There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory of
5164:
883:
541:
7483:
7120:
7080:
6961:
6599:
3626:
5612:
3870:
1828:
is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some
1341:
6256:
and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable)
2061:{\displaystyle S={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\qquad T={\begin{pmatrix}1&1\\0&1\end{pmatrix}}}
7549:
7171:
7070:
6793:
4025:
be invariant with respect to a sub-group of the modular group of finite index. This is not adhered to in this article.
3493:
389:
7539:
6865:
6771:
6560:
6228:
5908:
4017:
Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that
1396:
A modular function is a function that is invariant with respect to the modular group, but without the condition that
79:
6210:
2712:{\displaystyle G_{k}(\Lambda )=G_{k}(\tau )=\sum _{(0,0)\neq (m,n)\in \mathbf {Z} ^{2}}{\frac {1}{(m+n\tau )^{k}}},}
7249:
6954:
4164:
929:
461:
7579:
1385:
7396:
7317:
7307:
7244:
6206:
1001:
653:
587:
6994:
6202:
4047:
elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number
5113:
7214:
7110:
6119:
5289:
polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on
693:
7473:
7437:
7136:
7049:
3247:
435:
111:
7584:
7447:
7085:
6857:
6817:
6635:
4264:
4248:
1415:: they are holomorphic on the complement of a set of isolated points, which are poles of the function.
865:{\textstyle \gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}_{2}(\mathbb {Z} ).\,}
143:
6356:"DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions"
7493:
6845:
6290:
6274:
Taniyama and
Shimura identified a 1-to-1 matching between certain modular forms and elliptic curves.
5839:
3814:
3038:
2191:
6691:
5001:
4002:
at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-
3496:. The crucial conceptual link between modular forms and number theory is furnished by the theory of
7406:
7386:
7322:
7239:
7141:
7100:
6910:
6191:
4915:
1335:
6167:, a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.
4278:. Typically it is not compact, but can be compactified by adding a finite number of points called
7297:
7105:
6383:
6195:
5516:
5511:
5237:
The situation can be profitably compared to that which arises in the search for functions on the
5184:
3069:
has integer coordinates, either all even or all odd, and such that the sum of the coordinates of
1324:{\displaystyle X_{\Gamma }=\Gamma \backslash ({\mathcal {H}}\cup \mathbb {P} ^{1}(\mathbb {Q} ))}
5816:. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's
31:
7574:
7090:
7204:
6512:
1831:
7468:
7166:
7115:
7004:
6391:
6336:
6282:, which has become one of the most far-reaching and consequential research programs in math.
6164:
5586:
5363:
5192:
3960:
3501:
3265:
1233:
1100:
345:
275:
7544:
7416:
7075:
6923:
6825:
6609:
5831:
5813:
5286:
5168:
3966:
3561:
3548:
3520:
that the only modular forms are constant functions. However, relaxing the requirement that
3431:
3408:
2891:
1642:
1412:
1408:
384:
7327:
6687:
6617:
6333:
Some authors use different conventions, allowing an additional constant depending only on
8:
7381:
7259:
7224:
7181:
7161:
6919:
6831:
Provides an introduction to modular forms from the point of view of representation theory
6666:
5845:
5817:
5344:
4886:
to obtain further information about modular forms and functions. For example, the spaces
4745:
4320:
4241:
3235:
3027:{\displaystyle \vartheta _{L}(z)=\sum _{\lambda \in L}e^{\pi i\Vert \lambda \Vert ^{2}z}}
766:
139:
27:
Analytic function on the upper half-plane with a certain behavior under the modular group
6927:
6653:, Publications of the Mathematical Society of Japan, vol. 11, Tokyo: Iwanami Shoten
2372:
remains bounded above as long as the absolute value of the smallest non-zero element in
7511:
7302:
7282:
7095:
6801:
6756:
6267:
6106:{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=\varepsilon (a,b,c,d)(cz+d)^{k}f(z).}
5902:
5892:
are meromorphic functions on the upper half plane of moderate growth at infinity which
5562:
5542:
5522:
5336:
5196:
5143:
4713:
3859:
3404:
2921:
2380:
The key idea in proving the equivalence of the two definitions is that such a function
1934:
1120:
365:
166:
7254:
3389:{\displaystyle \eta (z)=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}),\qquad q=e^{2\pi iz}.}
2384:
is determined, because of the second condition, by its values on lattices of the form
257:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}
57:
30:"Modular function" redirects here. A distinct use of this term appears in relation to
7411:
7358:
7229:
7044:
7039:
6884:
6861:
6789:
6767:
6595:
6556:
6279:
6246:
5977:
which are used to generalise the modularity relation defining modular forms, so that
5802:
4297:
2421:
2303:
2228:
189:
101:
3048:. It is not so easy to construct even unimodular lattices, but here is one way: Let
7401:
7287:
7264:
6931:
6613:
6590:, Grundlehren der Mathematischen Wissenschaften , vol. 244, Berlin, New York:
6298:
6294:
6275:
5878:
5849:
5487:
5238:
4288:
3552:
3482:
3426:
is a modular form of weight 12. The presence of 24 is related to the fact that the
3092:
1605:
1411:
in the upper half-plane (among other requirements). Instead, modular functions are
425:
337:
181:
158:
105:
6537:
3098:. Because there is only one modular form of weight 8 up to scalar multiplication,
7516:
7332:
7274:
7176:
6999:
6978:
6892:
6821:
6809:
6785:
6763:
6605:
6591:
6585:
5861:
4309:
4275:
4112:
1593:
2528:{\displaystyle G_{k}(\Lambda )=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-k}.}
7199:
7024:
7009:
6986:
6751:
5494:
5483:
5331:
One might ask, since the homogeneous polynomials are not really functions on P(
5137:
4063:
4033:
4029:
3706:
3497:
3489:
3478:
3474:
3462:
2942:
2362:
2263:
2253:
2236:
1791:
1621:
267:
The term "modular form", as a systematic description, is usually attributed to
170:
6898:
Chapter VII provides an elementary introduction to the theory of modular forms
5316:. On the one hand, these form a finite dimensional vector space for each
7568:
7531:
7312:
7292:
7219:
7014:
6946:
6577:
5805:
5348:
3617:
3427:
3232:
1433:
1251:
1048:
174:
162:
147:
7478:
7452:
7442:
7432:
7234:
6286:
5864:
in the same sense that classical modular forms (which are sometimes called
5784:
4914:
are finite-dimensional, and their dimensions can be computed thanks to the
4145:
3466:
333:
5583:. These old forms can be constructed using the following observations: if
4788:. Again, modular forms that vanish at all cusps are called cusp forms for
2172:{\displaystyle f\left(-{\frac {1}{z}}\right)=z^{k}f(z),\qquad f(z+1)=f(z)}
1334:
The dimensions of these spaces of modular forms can be computed using the
1220:{\displaystyle f\in H^{0}(X_{\Gamma },\omega ^{\otimes k})=M_{k}(\Gamma )}
7353:
7191:
6837:
6379:
6302:
6260:
6253:
5871:
5798:
5381:
4052:
3470:
3239:
3218:
3182:{\displaystyle \vartheta _{L_{8}\times L_{8}}(z)=\vartheta _{L_{16}}(z),}
3088:
1044:
268:
93:
6875:
5351:
in this case). The situation with modular forms is precisely analogous.
5179:
is not identically 0, then it can be shown that the number of zeroes of
4149:
is a modular function whose poles and zeroes are confined to the cusps.
7348:
6935:
6905:
6581:
6266:
In the 1960s, as the needs of number theory and the formulation of the
4044:
7209:
6270:
in particular made it clear that modular forms are deeply implicated.
5882:
5809:
5772:
5766:
4103:
3435:
2243:
185:
6180:
6163:
is called the nebentypus of the modular form. Functions such as the
5750:{\displaystyle M_{k}(\Gamma _{1}(M))\subseteq M_{k}(\Gamma _{1}(N))}
4852:), they are also referred to as modular/cusp forms and functions of
4216:
can be relaxed by requiring it only for matrices in smaller groups.
4028:
Another way to phrase the definition of modular functions is to use
4010:-expansion is bounded below, guaranteeing that it is meromorphic at
3500:, which also gives the link between the theory of modular forms and
180:
Modular form theory is a special case of the more general theory of
5852:
in the same way in which classical modular forms are associated to
3243:
2202:, the second condition above is equivalent to these two equations.
7521:
7506:
6355:
1767:{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=(cz+d)^{k}f(z)}
6816:, Annals of Mathematics Studies, vol. 83, Princeton, N.J.:
5312:). The solutions are then the homogeneous polynomials of degree
4312:, which allows one to speak of holo- and meromorphic functions.
1043:
Modular forms can also be interpreted as sections of a specific
7501:
6942:
Behold
Modular Forms, the ‘Fifth Fundamental Operation’ of Math
5539:
which cannot be constructed from modular forms of lower levels
1924:{\displaystyle \operatorname {Im} (z)>M\implies |f(z)|<D}
6651:
Introduction to the arithmetic theory of automorphic functions
6453:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
4300:±2) fixing the point. This yields a compact topological space
3609:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
6908:(1988), "Jacobi forms and a certain space of modular forms",
5205:.It can be shown that the field of modular function of level
4776:
satisfying the above functional equation for all matrices in
3799:{\displaystyle f(z)=\sum _{n=-m}^{\infty }a_{n}e^{2i\pi nz}.}
3222:
6494:"Elliptic Curves Yield Their Secrets in a New Number System"
5464:{\displaystyle M(\Gamma )=\bigoplus _{k>0}M_{k}(\Gamma )}
3540:
is called modular if it satisfies the following properties:
3260:
Weierstrass's elliptic functions § Modular discriminant
1090:{\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}
325:{\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {Z} )}
188:
that transform nicely with respect to the action of certain
161:. The main importance of the theory is its connections with
6241:
The theory of modular forms was developed in four periods:
5093:
1476:
6784:, Graduate Texts in Mathematics, vol. 228, New York:
4308:. What is more, it can be endowed with the structure of a
1817:, only the zero function can satisfy the second condition.
3037:
converges when Im(z) > 0, and as a consequence of the
2248:
A modular form can equivalently be defined as a function
6891:, Graduate Texts in Mathematics, vol. 7, New York:
6278:
built on this idea in the construction of his expansive
5868:
to emphasize the point) are related to elliptic curves.
4152:
3709:. The third condition is that this series is of the form
2354:
is a constant (typically a positive integer) called the
919:{\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )}
577:{\displaystyle \gamma \in {\text{SL}}_{2}(\mathbb {Z} )}
6758:
Modular functions and
Dirichlet Series in Number Theory
3691:{\displaystyle f\left({\frac {az+b}{cz+d}}\right)=f(z)}
3446:
is expanded as a power series in q, the coefficient of
2324:
is the lattice obtained by multiplying each element of
6419:
5660:{\displaystyle \Gamma _{1}(N)\subseteq \Gamma _{1}(M)}
4494:
4372:
3939:{\displaystyle f(z)=\sum _{n=-m}^{\infty }a_{n}q^{n}.}
3575:
3052:
be an integer divisible by 8 and consider all vectors
2420:
The simplest examples from this point of view are the
2027:
1978:
1483:
1279:
801:
789:
769:
696:
6694:
6413:
6339:
6122:
5986:
5911:
5673:
5615:
5589:
5565:
5545:
5525:
5410:
5245:): in that setting, one would ideally like functions
5146:
5116:
4935:
4828:, respectively. Similarly, a meromorphic function on
4331:
4167:
4021:
be meromorphic in the open upper half-plane and that
3969:
3873:
3817:
3721:
3629:
3569:
3277:
3107:
2953:
2731:
2572:
2464:
2080:
1966:
1937:
1866:
1834:
1680:
1444:
1377:{\displaystyle \Gamma ={\text{SL}}_{2}(\mathbb {Z} )}
1344:
1262:
1236:
1146:
1123:
1103:
1057:
1004:
932:
886:
656:
590:
544:
464:
438:
392:
368:
348:
292:
198:
114:
3485:, which were shown to imply Ramanujan's conjecture.
3087:, this is the lattice generated by the roots in the
4870:, this gives back the afore-mentioned definitions.
4796:-vector spaces of modular and cusp forms of weight
4066:of isomorphism classes of complex elliptic curves.
3231:by these two lattices are consequently examples of
1820:The third condition is also phrased by saying that
52:
may be too technical for most readers to understand
6755:
6716:
6452:
6345:
6155:
6105:
5969:
5749:
5659:
5601:
5571:
5551:
5531:
5474:Rings of modular forms of congruence subgroups of
5463:
5167:of the Riemann surface, and hence form a field of
5152:
5128:
5099:
4616:
4208:
3982:
3938:
3850:
3798:
3690:
3608:
3388:
3181:
3026:
2872:
2711:
2527:
2244:Definition in terms of lattices or elliptic curves
2171:
2060:
1943:
1923:
1852:
1766:
1581:
1376:
1323:
1242:
1219:
1129:
1109:
1089:
1027:
990:
918:
864:
775:
755:
679:
642:
576:
528:
450:
416:
374:
354:
324:
256:
126:
5519:are a subspace of modular forms of a fixed level
4878:The theory of Riemann surfaces can be applied to
4315:Important examples are, for any positive integer
157:The theory of modular forms therefore belongs to
7566:
5875:analogous to the usual theory of modular forms.
4292:∪{∞}, such that there is a parabolic element of
876:if it satisfies the following growth condition:
417:{\displaystyle f:{\mathcal {H}}\to \mathbb {C} }
192:, generalizing the example of the modular group
6634:, Annals of Mathematics Studies, vol. 48,
6550:
5970:{\displaystyle \varepsilon (a,b,c,d)(cz+d)^{k}}
5396:is the vector space of modular forms of weight
4157:The functional equation, i.e., the behavior of
2916:II. Theta functions of even unimodular lattices
1038:
165:. Modular forms appear in other areas, such as
6976:
6903:
6780:Diamond, Fred; Shurman, Jerry Michael (2005),
6779:
5881:extend the notion of modular forms to general
5335:), what are they, geometrically speaking? The
6962:
6877:Lectures on Modular Forms and Hecke Operators
4209:{\displaystyle z\mapsto {\frac {az+b}{cz+d}}}
1418:
991:{\displaystyle (cz+d)^{-k}f(\gamma (z))\to 0}
6800:Leads up to an overview of the proof of the
5667:giving a reverse inclusion of modular forms
5123:
5117:
3041:can be shown to be a modular form of weight
3010:
3003:
2912:, so that such series are identically zero.
1630:satisfying the following three conditions:
529:{\displaystyle f(\gamma (z))=(cz+d)^{k}f(z)}
6209:. Unsourced material may be challenged and
5253:which are polynomial in the coordinates of
4219:
6969:
6955:
6576:
5482:are finitely generated due to a result of
4740:exactly once and such that the closure of
1892:
1888:
6229:Learn how and when to remove this message
5357:
3990:are known as the Fourier coefficients of
3073:is an even integer. We call this lattice
1367:
1311:
1297:
1080:
1028:{\displaystyle {\text{im}}(z)\to \infty }
909:
861:
851:
680:{\displaystyle {\text{im}}(z)\to \infty }
567:
410:
315:
247:
218:
123:
115:
80:Learn how and when to remove this message
64:, without removing the technical details.
643:{\displaystyle (cz+d)^{-k}f(\gamma (z))}
428:such that two conditions are satisfied:
6808:
6750:
6648:
6629:
6491:
14:
7567:
7392:Clifford's theorem on special divisors
6873:
6851:
6315:Wiles's proof of Fermat's Last Theorem
5860:; in other words, they are related to
5795:determined by the conjugation action.
5129:{\displaystyle \lfloor \cdot \rfloor }
4282:. These are points at the boundary of
3524:be holomorphic leads to the notion of
3481:as a result of Deligne's proof of the
1951:is bounded above some horizontal line.
1423:
6950:
6883:
6836:
6664:
6538:"Modular Functions and Modular Forms"
6156:{\displaystyle \varepsilon (a,b,c,d)}
5163:The modular functions constitute the
4153:Modular forms for more general groups
1384:are sections of a line bundle on the
756:{\textstyle \gamma (z)=(az+b)/(cz+d)}
62:make it understandable to non-experts
6487:
6485:
6207:adding citations to reliable sources
6174:
5820:. Groups which are not subgroups of
5400:, then the ring of modular forms of
3507:
3461:. This was confirmed by the work of
2266:which satisfies certain conditions:
36:
6668:Modular Functions and Modular Forms
6525:from the original on 1 August 2020.
6513:"Cohomology of Automorphic Bundles"
5380:, the ring of modular forms is the
5213:≥ 1) is generated by the functions
5044:
5037:
4744:meets all orbits. For example, the
4594:
4448:
3998:is called the order of the pole of
1391:
274:Each modular form is attached to a
24:
7550:Vector bundles on algebraic curves
7484:Weber's theorem (Algebraic curves)
7081:Hasse's theorem on elliptic curves
7071:Counting points on elliptic curves
6736:from the original on 31 July 2020.
6696:
6685:
5778:
5726:
5688:
5639:
5617:
5455:
5417:
5384:generated by the modular forms of
4465:
4337:
4006:coefficients are non-zero, so the
3908:
3756:
3701:The second condition implies that
3327:
2941:is an even integer. The so-called
2586:
2504:
2478:
1345:
1338:. The classical modular forms for
1287:
1276:
1268:
1250:is a canonical line bundle on the
1211:
1171:
1104:
1058:
1022:
674:
451:{\displaystyle \gamma \in \Gamma }
445:
401:
349:
293:
233:
230:
204:
201:
127:{\displaystyle \,{\mathcal {H}}\,}
118:
25:
7596:
6814:Automorphic Forms on Adèle Groups
6535:
6482:
6249:, in the early nineteenth century
6245:In connection with the theory of
4836:is called a modular function for
3705:is periodic, and therefore has a
3221:observed that the 16-dimensional
2317:is a non-zero complex number and
184:, which are functions defined on
6655:, Theorem 2.33, Proposition 2.26
6179:
4981:
4942:
4918:in terms of the geometry of the
4543:
4421:
4032:: every lattice Λ determines an
3951:This is also referred to as the
3811:It is often written in terms of
2660:
1810:is typically a positive integer.
1543:
1460:
41:
7172:Hurwitz's automorphisms theorem
6782:A First Course in Modular Forms
6679:
6658:
6642:
6510:
5232:
4873:
4759:
4587:
4441:
3851:{\displaystyle q=\exp(2\pi iz)}
3516:is zero, it can be shown using
3357:
2135:
2015:
1386:moduli stack of elliptic curves
1137:can be defined as an element of
7397:Gonality of an algebraic curve
7308:Differential of the first kind
6717:{\displaystyle \Gamma _{1}(N)}
6711:
6705:
6623:
6570:
6544:
6529:
6504:
6492:Van Wyk, Gerhard (July 2023).
6401:
6372:
6327:
6150:
6126:
6097:
6091:
6079:
6063:
6060:
6036:
5958:
5942:
5939:
5915:
5744:
5741:
5735:
5722:
5706:
5703:
5697:
5684:
5654:
5648:
5632:
5626:
5458:
5452:
5420:
5414:
5320:, and on the other, if we let
5048:
5038:
4985:
4971:
4712:can be understood by studying
4598:
4588:
4547:
4533:
4474:
4468:
4452:
4442:
4425:
4411:
4352:
4346:
4171:
3883:
3877:
3845:
3830:
3731:
3725:
3685:
3679:
3351:
3332:
3287:
3281:
3173:
3167:
3144:
3138:
2970:
2964:
2898:there is cancellation between
2860:
2854:
2834:
2822:
2802:
2796:
2694:
2678:
2652:
2640:
2634:
2622:
2611:
2605:
2589:
2583:
2481:
2475:
2166:
2160:
2151:
2139:
2129:
2123:
1911:
1907:
1901:
1894:
1889:
1879:
1873:
1761:
1755:
1743:
1727:
1464:
1450:
1371:
1363:
1318:
1315:
1307:
1282:
1214:
1208:
1192:
1163:
1084:
1076:
1019:
1016:
1010:
982:
979:
976:
970:
964:
949:
933:
913:
905:
855:
847:
783:is identified with the matrix
750:
735:
727:
712:
706:
700:
671:
668:
662:
637:
634:
628:
622:
607:
591:
571:
563:
523:
517:
505:
489:
483:
480:
474:
468:
432:Automorphy condition: For any
406:
319:
311:
251:
243:
222:
214:
13:
1:
7540:Birkhoff–Grothendieck theorem
7250:Nagata's conjecture on curves
7121:Schoof–Elkies–Atkin algorithm
6995:Five points determine a conic
6744:
5894:fail to be modular of weight
5760:
5579:. The other forms are called
4732:intersects each orbit of the
3254:III. The modular discriminant
1785:is required to be bounded as
286:In general, given a subgroup
281:
7111:Supersingular elliptic curve
6551:Chandrasekharan, K. (1985).
6475:
6289:used modular forms to prove
5888:Modular integrals of weight
5505:
4129:is the order of the zero of
2545:is a modular form of weight
1039:As sections of a line bundle
880:Cuspidal condition: For any
7:
7318:Riemann's existence theorem
7245:Hilbert's sixteenth problem
7137:Elliptic curve cryptography
7050:Fundamental pair of periods
6854:Modular forms and functions
6630:Gunning, Robert C. (1962),
6308:
3248:Hearing the shape of a drum
2410:
2270:If we consider the lattice
10:
7601:
7448:Moduli of algebraic curves
6858:Cambridge University Press
6852:Rankin, Robert A. (1977),
6846:Vandenhoeck & Ruprecht
6818:Princeton University Press
6636:Princeton University Press
6170:
5905:are functions of the form
5764:
5509:
5361:
5183:is equal to the number of
4265:quotient topological space
3257:
2933:is a lattice generated by
2447:over all non-zero vectors
1419:Modular forms for SL(2, Z)
538:Growth condition: For any
29:
7530:
7492:
7461:
7425:
7374:
7367:
7341:
7273:
7190:
7154:
7129:
7063:
7032:
7023:
6985:
6632:Lectures on modular forms
5848:are associated to larger
5840:totally real number field
5261:and satisfy the equation
5175:). If a modular function
4780:, that is holomorphic on
4043:; two lattices determine
3192:even though the lattices
3039:Poisson summation formula
1428:A modular form of weight
7215:Cayley–Bacharach theorem
7142:Elliptic curve primality
6911:Inventiones Mathematicae
6688:"Atkin-Lehner Theory of
6320:
6305:of integers down to −5.
5899:by a rational function.
5500:
5339:answer is that they are
2424:. For each even integer
2287:generated by a constant
1954:The second condition for
1853:{\displaystyle M,D>0}
1097:a modular form of level
7474:Riemann–Hurwitz formula
7438:Gromov–Witten invariant
7298:Compact Riemann surface
7086:Mazur's torsion theorem
6390: = 0, not an
6346:{\displaystyle \gamma }
5602:{\displaystyle M\mid N}
3432:A celebrated conjecture
2922:even unimodular lattice
2376:is bounded away from 0.
1243:{\displaystyle \omega }
1110:{\displaystyle \Gamma }
355:{\displaystyle \Gamma }
153:and a growth condition.
18:Level of a modular form
7580:Analytic number theory
7091:Modular elliptic curve
6889:A Course in Arithmetic
6874:Ribet, K.; Stein, W.,
6718:
6649:Shimura, Goro (1971),
6454:
6347:
6157:
6107:
5971:
5866:elliptic modular forms
5751:
5661:
5603:
5573:
5553:
5533:
5465:
5358:Rings of modular forms
5347:(one could also say a
5154:
5130:
5101:
4618:
4210:
4090:, also paraphrased as
4014: = 0.
3984:
3940:
3912:
3852:
3800:
3760:
3692:
3610:
3390:
3331:
3183:
3028:
2874:
2713:
2529:
2173:
2062:
1945:
1925:
1854:
1768:
1583:
1378:
1332:
1325:
1244:
1228:
1221:
1131:
1111:
1091:
1029:
992:
920:
866:
777:
757:
681:
644:
578:
530:
452:
418:
376:
356:
342:modular form of level
326:
258:
128:
7005:Rational normal curve
6719:
6665:Milne, James (2010),
6455:
6392:essential singularity
6348:
6291:Fermat’s Last Theorem
6165:Dedekind eta function
6158:
6108:
5972:
5832:Hilbert modular forms
5752:
5662:
5604:
5574:
5554:
5534:
5466:
5388:. In other words, if
5364:Ring of modular forms
5155:
5131:
5102:
4619:
4274:can be shown to be a
4211:
3985:
3983:{\displaystyle a_{n}}
3961:q-expansion principle
3941:
3889:
3853:
3801:
3737:
3693:
3611:
3502:representation theory
3403:is the square of the
3391:
3311:
3266:Dedekind eta function
3258:Further information:
3225:obtained by dividing
3184:
3029:
2875:
2714:
2530:
2174:
2063:
1946:
1926:
1855:
1769:
1584:
1379:
1326:
1255:
1245:
1222:
1139:
1132:
1112:
1092:
1030:
993:
921:
867:
778:
758:
682:
645:
579:
531:
458:there is the equality
453:
419:
377:
357:
327:
276:Galois representation
259:
129:
7545:Stable vector bundle
7417:Weil reciprocity law
7407:Riemann–Roch theorem
7387:Brill–Noether theory
7323:Riemann–Roch theorem
7240:Genus–degree formula
7101:Mordell–Weil theorem
7076:Division polynomials
6692:
6411:
6337:
6203:improve this section
6120:
5984:
5909:
5846:Siegel modular forms
5818:mock theta functions
5671:
5613:
5587:
5563:
5543:
5523:
5408:
5285:be the ratio of two
5249:on the vector space
5169:transcendence degree
5144:
5114:
4933:
4916:Riemann–Roch theorem
4784:and at all cusps of
4329:
4321:congruence subgroups
4319:, either one of the
4220:The Riemann surface
4165:
3967:
3963:). The coefficients
3871:
3815:
3719:
3627:
3567:
3409:modular discriminant
3275:
3236:Riemannian manifolds
3105:
2951:
2729:
2570:
2462:
2416:I. Eisenstein series
2227:, modular forms are
2183:respectively. Since
2078:
1964:
1935:
1864:
1832:
1678:
1643:holomorphic function
1442:
1342:
1336:Riemann–Roch theorem
1260:
1234:
1144:
1121:
1101:
1055:
1002:
930:
884:
787:
776:{\textstyle \gamma }
767:
694:
654:
588:
542:
462:
436:
390:
385:holomorphic function
366:
346:
290:
196:
142:with respect to the
112:
7368:Structure of curves
7260:Quartic plane curve
7182:Hyperelliptic curve
7162:De Franchis theorem
7106:Nagell–Lutz theorem
6928:1988InMat..94..113S
6842:Mathematische Werke
6810:Gelbart, Stephen S.
6555:. Springer-Verlag.
5903:Automorphic factors
5828:can be considered.
5787:, it is a function
5512:Atkin–Lehner theory
5404:is the graded ring
5277:) for all non-zero
4764:A modular form for
4714:fundamental domains
4255:in the same way as
3858:(the square of the
3518:Liouville's theorem
3454:has absolute value
3438:asserted that when
3430:has 24 dimensions.
1672:as above, we have:
1424:Standard definition
140:functional equation
7375:Divisors on curves
7167:Faltings's theorem
7116:Schoof's algorithm
7096:Modularity theorem
6936:10.1007/BF01394347
6885:Serre, Jean-Pierre
6802:modularity theorem
6714:
6553:Elliptic functions
6450:
6444:
6343:
6268:modularity theorem
6247:elliptic functions
6153:
6103:
5967:
5747:
5657:
5599:
5569:
5549:
5529:
5461:
5441:
5257: ≠ 0 in
5197:fundamental region
5165:field of functions
5150:
5126:
5097:
5092:
4614:
4612:
4519:
4397:
4240:that is of finite
4206:
3980:
3936:
3848:
3796:
3688:
3606:
3600:
3560:For every integer
3494:partition function
3386:
3179:
3024:
2991:
2870:
2868:
2709:
2671:
2525:
2508:
2235:, and thus have a
2229:periodic functions
2194:the modular group
2169:
2058:
2052:
2006:
1941:
1921:
1850:
1764:
1664:and any matrix in
1579:
1508:
1374:
1321:
1240:
1217:
1127:
1107:
1087:
1025:
988:
916:
862:
826:
773:
753:
677:
640:
574:
526:
448:
414:
372:
352:
322:
254:
190:discrete subgroups
167:algebraic topology
134:, that satisfies:
124:
7585:Special functions
7562:
7561:
7558:
7557:
7469:Hasse–Witt matrix
7412:Weierstrass point
7359:Smooth completion
7328:Teichmüller space
7230:Cubic plane curve
7150:
7149:
7064:Arithmetic theory
7045:Elliptic integral
7040:Elliptic function
6904:Skoruppa, N. P.;
6686:Mocanu, Andreea.
6601:978-0-387-90517-4
6578:Kubert, Daniel S.
6280:Langlands program
6239:
6238:
6231:
6024:
5879:Automorphic forms
5862:abelian varieties
5850:symplectic groups
5834:are functions in
5572:{\displaystyle N}
5552:{\displaystyle M}
5532:{\displaystyle N}
5426:
5337:algebro-geometric
5153:{\displaystyle k}
5088:
4969:
4772:is a function on
4756:can be computed.
4701:), respectively.
4531:
4409:
4232:be a subgroup of
4204:
4073:that vanishes at
3994:, and the number
3667:
3526:modular functions
3508:Modular functions
3217:are not similar.
2976:
2762:
2704:
2617:
2487:
2441:to be the sum of
2422:Eisenstein series
2304:analytic function
2100:
1944:{\displaystyle f}
1718:
1552:
1448:
1355:
1130:{\displaystyle k}
1068:
1049:modular varieties
1008:
897:
839:
763:and the function
660:
555:
375:{\displaystyle k}
303:
182:automorphic forms
102:analytic function
90:
89:
82:
16:(Redirected from
7592:
7402:Jacobian variety
7372:
7371:
7275:Riemann surfaces
7265:Real plane curve
7225:Cramer's paradox
7205:Bézout's theorem
7030:
7029:
6979:algebraic curves
6971:
6964:
6957:
6948:
6947:
6938:
6895:
6880:
6870:
6848:
6828:
6798:
6776:
6761:
6738:
6737:
6735:
6728:
6723:
6721:
6720:
6715:
6704:
6703:
6683:
6677:
6675:
6673:
6662:
6656:
6654:
6646:
6640:
6638:
6627:
6621:
6620:
6574:
6568:
6566:
6548:
6542:
6541:
6533:
6527:
6526:
6524:
6517:
6508:
6502:
6501:
6489:
6469:
6459:
6457:
6456:
6451:
6449:
6448:
6405:
6399:
6376:
6370:
6369:
6367:
6366:
6352:
6350:
6349:
6344:
6331:
6299:rational numbers
6295:quadratic fields
6276:Robert Langlands
6234:
6227:
6223:
6220:
6214:
6183:
6175:
6162:
6160:
6159:
6154:
6112:
6110:
6109:
6104:
6087:
6086:
6029:
6025:
6023:
6009:
5995:
5976:
5974:
5973:
5968:
5966:
5965:
5897:
5891:
5859:
5827:
5812:but need not be
5794:
5756:
5754:
5753:
5748:
5734:
5733:
5721:
5720:
5696:
5695:
5683:
5682:
5666:
5664:
5663:
5658:
5647:
5646:
5625:
5624:
5608:
5606:
5605:
5600:
5578:
5576:
5575:
5570:
5558:
5556:
5555:
5550:
5538:
5536:
5535:
5530:
5488:Michael Rapoport
5481:
5470:
5468:
5467:
5462:
5451:
5450:
5440:
5403:
5399:
5395:
5387:
5379:
5371:
5315:
5239:projective space
5159:
5157:
5156:
5151:
5135:
5133:
5132:
5127:
5106:
5104:
5103:
5098:
5096:
5095:
5089:
5086:
5076:
5072:
5068:
5051:
5024:
5020:
5016:
4992:
4988:
4984:
4970:
4967:
4960:
4959:
4947:
4946:
4945:
4921:
4913:
4899:
4869:
4839:
4827:
4813:
4791:
4787:
4779:
4767:
4735:
4704:The geometry of
4646:
4623:
4621:
4620:
4615:
4613:
4606:
4602:
4601:
4546:
4532:
4529:
4524:
4523:
4460:
4456:
4455:
4424:
4410:
4407:
4402:
4401:
4345:
4344:
4295:
4262:
4247:
4239:
4231:
4215:
4213:
4212:
4207:
4205:
4203:
4189:
4175:
4161:with respect to
4139:
4128:
4115:). The smallest
4100:
4089:
4079:
4050:
3989:
3987:
3986:
3981:
3979:
3978:
3945:
3943:
3942:
3937:
3932:
3931:
3922:
3921:
3911:
3906:
3857:
3855:
3854:
3849:
3805:
3803:
3802:
3797:
3792:
3791:
3770:
3769:
3759:
3754:
3697:
3695:
3694:
3689:
3672:
3668:
3666:
3652:
3638:
3621:
3615:
3613:
3612:
3607:
3605:
3604:
3553:upper half-plane
3512:When the weight
3483:Weil conjectures
3460:
3453:
3449:
3445:
3425:
3395:
3393:
3392:
3387:
3382:
3381:
3350:
3349:
3330:
3325:
3310:
3309:
3305:
3230:
3216:
3207:
3188:
3186:
3185:
3180:
3166:
3165:
3164:
3163:
3137:
3136:
3135:
3134:
3122:
3121:
3086:
3079:
3072:
3068:
3061:
3055:
3051:
3047:
3033:
3031:
3030:
3025:
3023:
3022:
3018:
3017:
2990:
2963:
2962:
2940:
2936:
2932:
2926:
2911:
2903:
2897:
2889:
2879:
2877:
2876:
2871:
2869:
2853:
2852:
2821:
2820:
2795:
2794:
2785:
2784:
2768:
2764:
2763:
2755:
2745:
2744:
2718:
2716:
2715:
2710:
2705:
2703:
2702:
2701:
2673:
2670:
2669:
2668:
2663:
2604:
2603:
2582:
2581:
2562:
2548:
2544:
2534:
2532:
2531:
2526:
2521:
2520:
2507:
2474:
2473:
2454:
2450:
2446:
2440:
2430:
2406:
2396:
2383:
2375:
2371:
2353:
2349:
2331:
2327:
2323:
2316:
2309:
2301:
2294:
2290:
2286:
2261:
2252:from the set of
2234:
2226:
2201:
2190:
2186:
2178:
2176:
2175:
2170:
2119:
2118:
2106:
2102:
2101:
2093:
2067:
2065:
2064:
2059:
2057:
2056:
2011:
2010:
1950:
1948:
1947:
1942:
1930:
1928:
1927:
1922:
1914:
1897:
1859:
1857:
1856:
1851:
1827:
1816:
1809:
1797:
1784:
1773:
1771:
1770:
1765:
1751:
1750:
1723:
1719:
1717:
1703:
1689:
1671:
1663:
1650:
1640:
1629:
1606:upper half-plane
1603:
1588:
1586:
1585:
1580:
1578:
1574:
1550:
1546:
1517:
1513:
1512:
1463:
1449:
1446:
1431:
1406:
1392:Modular function
1383:
1381:
1380:
1375:
1370:
1362:
1361:
1356:
1353:
1330:
1328:
1327:
1322:
1314:
1306:
1305:
1300:
1291:
1290:
1272:
1271:
1249:
1247:
1246:
1241:
1226:
1224:
1223:
1218:
1207:
1206:
1191:
1190:
1175:
1174:
1162:
1161:
1136:
1134:
1133:
1128:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1083:
1075:
1074:
1069:
1066:
1034:
1032:
1031:
1026:
1009:
1006:
997:
995:
994:
989:
960:
959:
925:
923:
922:
917:
912:
904:
903:
898:
895:
871:
869:
868:
863:
854:
846:
845:
840:
837:
831:
830:
782:
780:
779:
774:
762:
760:
759:
754:
734:
686:
684:
683:
678:
661:
658:
649:
647:
646:
641:
618:
617:
583:
581:
580:
575:
570:
562:
561:
556:
553:
535:
533:
532:
527:
513:
512:
457:
455:
454:
449:
426:upper half-plane
423:
421:
420:
415:
413:
405:
404:
381:
379:
378:
373:
361:
359:
358:
353:
338:arithmetic group
331:
329:
328:
323:
318:
310:
309:
304:
301:
263:
261:
260:
255:
250:
242:
241:
236:
221:
213:
212:
207:
159:complex analysis
133:
131:
130:
125:
122:
121:
106:upper half-plane
85:
78:
74:
71:
65:
45:
44:
37:
21:
7600:
7599:
7595:
7594:
7593:
7591:
7590:
7589:
7565:
7564:
7563:
7554:
7526:
7517:Delta invariant
7488:
7457:
7421:
7382:Abel–Jacobi map
7363:
7337:
7333:Torelli theorem
7303:Dessin d'enfant
7283:Belyi's theorem
7269:
7255:Plücker formula
7186:
7177:Hurwitz surface
7146:
7125:
7059:
7033:Analytic theory
7025:Elliptic curves
7019:
7000:Projective line
6987:Rational curves
6981:
6975:
6893:Springer-Verlag
6868:
6796:
6786:Springer-Verlag
6774:
6764:Springer-Verlag
6752:Apostol, Tom M.
6747:
6742:
6741:
6733:
6726:
6724:-Modular Forms"
6699:
6695:
6693:
6690:
6689:
6684:
6680:
6671:
6663:
6659:
6647:
6643:
6628:
6624:
6602:
6592:Springer-Verlag
6575:
6571:
6563:
6549:
6545:
6534:
6530:
6522:
6515:
6509:
6505:
6490:
6483:
6478:
6473:
6472:
6443:
6442:
6437:
6431:
6430:
6425:
6415:
6414:
6412:
6409:
6408:
6407:Here, a matrix
6406:
6402:
6377:
6373:
6364:
6362:
6354:
6338:
6335:
6334:
6332:
6328:
6323:
6311:
6263:from about 1925
6235:
6224:
6218:
6215:
6200:
6184:
6173:
6121:
6118:
6117:
6082:
6078:
6010:
5996:
5994:
5990:
5985:
5982:
5981:
5961:
5957:
5910:
5907:
5906:
5895:
5889:
5853:
5821:
5788:
5781:
5779:Generalizations
5769:
5763:
5729:
5725:
5716:
5712:
5691:
5687:
5678:
5674:
5672:
5669:
5668:
5642:
5638:
5620:
5616:
5614:
5611:
5610:
5588:
5585:
5584:
5564:
5561:
5560:
5544:
5541:
5540:
5524:
5521:
5520:
5514:
5508:
5503:
5495:Fuchsian groups
5475:
5446:
5442:
5430:
5409:
5406:
5405:
5401:
5397:
5393:
5389:
5385:
5373:
5369:
5368:For a subgroup
5366:
5360:
5313:
5235:
5204:
5145:
5142:
5141:
5115:
5112:
5111:
5091:
5090:
5085:
5083:
5064:
5060:
5056:
5053:
5052:
5036:
5025:
5012:
5008:
5004:
4997:
4996:
4980:
4966:
4965:
4961:
4955:
4951:
4941:
4940:
4936:
4934:
4931:
4930:
4926:. For example,
4919:
4906:
4901:
4892:
4887:
4876:
4864:= Γ(1) = SL(2,
4860:
4847:
4837:
4820:
4815:
4806:
4801:
4789:
4785:
4777:
4765:
4762:
4733:
4720:, i.e. subsets
4680:
4669:
4640:
4634:
4611:
4610:
4586:
4542:
4528:
4518:
4517:
4512:
4506:
4505:
4500:
4490:
4489:
4488:
4484:
4477:
4462:
4461:
4440:
4420:
4406:
4396:
4395:
4390:
4384:
4383:
4378:
4368:
4367:
4366:
4362:
4355:
4340:
4336:
4332:
4330:
4327:
4326:
4310:Riemann surface
4296:(a matrix with
4293:
4276:Hausdorff space
4256:
4245:
4244:. Such a group
4233:
4229:
4226:
4190:
4176:
4174:
4166:
4163:
4162:
4155:
4134:
4125:
4120:
4091:
4087:
4081:
4080:(equivalently,
4074:
4069:A modular form
4048:
4030:elliptic curves
3974:
3970:
3968:
3965:
3964:
3927:
3923:
3917:
3913:
3907:
3893:
3872:
3869:
3868:
3816:
3813:
3812:
3775:
3771:
3765:
3761:
3755:
3741:
3720:
3717:
3716:
3653:
3639:
3637:
3633:
3628:
3625:
3624:
3619:
3599:
3598:
3593:
3587:
3586:
3581:
3571:
3570:
3568:
3565:
3564:
3510:
3498:Hecke operators
3490:quadratic forms
3455:
3451:
3447:
3439:
3411:
3368:
3364:
3345:
3341:
3326:
3315:
3301:
3297:
3293:
3276:
3273:
3272:
3262:
3226:
3215:
3209:
3206:
3199:
3193:
3159:
3155:
3154:
3150:
3130:
3126:
3117:
3113:
3112:
3108:
3106:
3103:
3102:
3096:
3081:
3078:
3074:
3070:
3063:
3057:
3053:
3049:
3042:
3013:
3009:
2996:
2992:
2980:
2958:
2954:
2952:
2949:
2948:
2938:
2934:
2928:
2924:
2905:
2899:
2895:
2884:
2867:
2866:
2848:
2844:
2837:
2816:
2812:
2809:
2808:
2790:
2786:
2780:
2776:
2769:
2754:
2750:
2746:
2740:
2736:
2732:
2730:
2727:
2726:
2697:
2693:
2677:
2672:
2664:
2659:
2658:
2621:
2599:
2595:
2577:
2573:
2571:
2568:
2567:
2550:
2546:
2543:
2539:
2513:
2509:
2491:
2469:
2465:
2463:
2460:
2459:
2452:
2448:
2442:
2437:
2432:
2425:
2413:
2398:
2385:
2381:
2373:
2366:
2351:
2333:
2329:
2325:
2318:
2314:
2307:
2296:
2292:
2291:and a variable
2288:
2271:
2264:complex numbers
2257:
2246:
2232:
2208:
2195:
2188:
2184:
2114:
2110:
2092:
2088:
2084:
2079:
2076:
2075:
2051:
2050:
2045:
2039:
2038:
2033:
2023:
2022:
2005:
2004:
1999:
1993:
1992:
1984:
1974:
1973:
1965:
1962:
1961:
1936:
1933:
1932:
1910:
1893:
1865:
1862:
1861:
1833:
1830:
1829:
1821:
1814:
1807:
1786:
1778:
1746:
1742:
1704:
1690:
1688:
1684:
1679:
1676:
1675:
1665:
1655:
1646:
1634:
1608:
1597:
1542:
1507:
1506:
1501:
1495:
1494:
1489:
1479:
1478:
1475:
1474:
1470:
1459:
1445:
1443:
1440:
1439:
1429:
1426:
1421:
1397:
1394:
1366:
1357:
1352:
1351:
1343:
1340:
1339:
1310:
1301:
1296:
1295:
1286:
1285:
1267:
1263:
1261:
1258:
1257:
1235:
1232:
1231:
1202:
1198:
1183:
1179:
1170:
1166:
1157:
1153:
1145:
1142:
1141:
1122:
1119:
1118:
1102:
1099:
1098:
1079:
1070:
1065:
1064:
1056:
1053:
1052:
1041:
1005:
1003:
1000:
999:
952:
948:
931:
928:
927:
908:
899:
894:
893:
885:
882:
881:
850:
841:
836:
835:
825:
824:
819:
813:
812:
807:
797:
796:
788:
785:
784:
768:
765:
764:
730:
695:
692:
691:
657:
655:
652:
651:
650:is bounded for
610:
606:
589:
586:
585:
566:
557:
552:
551:
543:
540:
539:
508:
504:
463:
460:
459:
437:
434:
433:
409:
400:
399:
391:
388:
387:
367:
364:
363:
347:
344:
343:
314:
305:
300:
299:
291:
288:
287:
284:
246:
237:
229:
228:
217:
208:
200:
199:
197:
194:
193:
117:
116:
113:
110:
109:
100:is a (complex)
86:
75:
69:
66:
58:help improve it
55:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
7598:
7588:
7587:
7582:
7577:
7560:
7559:
7556:
7555:
7553:
7552:
7547:
7542:
7536:
7534:
7532:Vector bundles
7528:
7527:
7525:
7524:
7519:
7514:
7509:
7504:
7498:
7496:
7490:
7489:
7487:
7486:
7481:
7476:
7471:
7465:
7463:
7459:
7458:
7456:
7455:
7450:
7445:
7440:
7435:
7429:
7427:
7423:
7422:
7420:
7419:
7414:
7409:
7404:
7399:
7394:
7389:
7384:
7378:
7376:
7369:
7365:
7364:
7362:
7361:
7356:
7351:
7345:
7343:
7339:
7338:
7336:
7335:
7330:
7325:
7320:
7315:
7310:
7305:
7300:
7295:
7290:
7285:
7279:
7277:
7271:
7270:
7268:
7267:
7262:
7257:
7252:
7247:
7242:
7237:
7232:
7227:
7222:
7217:
7212:
7207:
7202:
7196:
7194:
7188:
7187:
7185:
7184:
7179:
7174:
7169:
7164:
7158:
7156:
7152:
7151:
7148:
7147:
7145:
7144:
7139:
7133:
7131:
7127:
7126:
7124:
7123:
7118:
7113:
7108:
7103:
7098:
7093:
7088:
7083:
7078:
7073:
7067:
7065:
7061:
7060:
7058:
7057:
7052:
7047:
7042:
7036:
7034:
7027:
7021:
7020:
7018:
7017:
7012:
7010:Riemann sphere
7007:
7002:
6997:
6991:
6989:
6983:
6982:
6974:
6973:
6966:
6959:
6951:
6945:
6944:
6939:
6901:
6881:
6871:
6866:
6849:
6834:
6806:
6795:978-0387232294
6794:
6777:
6772:
6746:
6743:
6740:
6739:
6713:
6710:
6707:
6702:
6698:
6678:
6676:, Theorem 6.1.
6657:
6641:
6622:
6600:
6594:, p. 24,
6569:
6561:
6543:
6528:
6511:Lan, Kai-Wen.
6503:
6480:
6479:
6477:
6474:
6471:
6470:
6447:
6441:
6438:
6436:
6433:
6432:
6429:
6426:
6424:
6421:
6420:
6418:
6400:
6371:
6342:
6325:
6324:
6322:
6319:
6318:
6317:
6310:
6307:
6272:
6271:
6264:
6257:
6250:
6237:
6236:
6187:
6185:
6178:
6172:
6169:
6152:
6149:
6146:
6143:
6140:
6137:
6134:
6131:
6128:
6125:
6114:
6113:
6102:
6099:
6096:
6093:
6090:
6085:
6081:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6028:
6022:
6019:
6016:
6013:
6008:
6005:
6002:
5999:
5993:
5989:
5964:
5960:
5956:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5920:
5917:
5914:
5806:eigenfunctions
5780:
5777:
5765:Main article:
5762:
5759:
5746:
5743:
5740:
5737:
5732:
5728:
5724:
5719:
5715:
5711:
5708:
5705:
5702:
5699:
5694:
5690:
5686:
5681:
5677:
5656:
5653:
5650:
5645:
5641:
5637:
5634:
5631:
5628:
5623:
5619:
5598:
5595:
5592:
5568:
5548:
5528:
5510:Main article:
5507:
5504:
5502:
5499:
5484:Pierre Deligne
5460:
5457:
5454:
5449:
5445:
5439:
5436:
5433:
5429:
5425:
5422:
5419:
5416:
5413:
5391:
5362:Main article:
5359:
5356:
5301:) =
5269:) =
5234:
5231:
5202:
5149:
5138:floor function
5125:
5122:
5119:
5108:
5107:
5094:
5084:
5082:
5079:
5075:
5071:
5067:
5063:
5059:
5055:
5054:
5050:
5047:
5043:
5040:
5035:
5032:
5029:
5026:
5023:
5019:
5015:
5011:
5007:
5003:
5002:
5000:
4995:
4991:
4987:
4983:
4979:
4976:
4973:
4964:
4958:
4954:
4950:
4944:
4939:
4904:
4890:
4875:
4872:
4845:
4818:
4804:
4761:
4758:
4678:
4667:
4632:
4625:
4624:
4609:
4605:
4600:
4597:
4593:
4590:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4545:
4541:
4538:
4535:
4527:
4522:
4516:
4513:
4511:
4508:
4507:
4504:
4501:
4499:
4496:
4495:
4493:
4487:
4483:
4480:
4478:
4476:
4473:
4470:
4467:
4464:
4463:
4459:
4454:
4451:
4447:
4444:
4439:
4436:
4433:
4430:
4427:
4423:
4419:
4416:
4413:
4405:
4400:
4394:
4391:
4389:
4386:
4385:
4382:
4379:
4377:
4374:
4373:
4371:
4365:
4361:
4358:
4356:
4354:
4351:
4348:
4343:
4339:
4335:
4334:
4225:
4218:
4202:
4199:
4196:
4193:
4188:
4185:
4182:
4179:
4173:
4170:
4154:
4151:
4123:
4101:) is called a
4085:
4034:elliptic curve
3977:
3973:
3955:-expansion of
3949:
3948:
3947:
3946:
3935:
3930:
3926:
3920:
3916:
3910:
3905:
3902:
3899:
3896:
3892:
3888:
3885:
3882:
3879:
3876:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3809:
3808:
3807:
3806:
3795:
3790:
3787:
3784:
3781:
3778:
3774:
3768:
3764:
3758:
3753:
3750:
3747:
3744:
3740:
3736:
3733:
3730:
3727:
3724:
3711:
3710:
3707:Fourier series
3699:
3687:
3684:
3681:
3678:
3675:
3671:
3665:
3662:
3659:
3656:
3651:
3648:
3645:
3642:
3636:
3632:
3618:modular group
3603:
3597:
3594:
3592:
3589:
3588:
3585:
3582:
3580:
3577:
3576:
3574:
3558:
3509:
3506:
3479:Pierre Deligne
3450:for any prime
3397:
3396:
3385:
3380:
3377:
3374:
3371:
3367:
3363:
3360:
3356:
3353:
3348:
3344:
3340:
3337:
3334:
3329:
3324:
3321:
3318:
3314:
3308:
3304:
3300:
3296:
3292:
3289:
3286:
3283:
3280:
3268:is defined as
3213:
3204:
3197:
3190:
3189:
3178:
3175:
3172:
3169:
3162:
3158:
3153:
3149:
3146:
3143:
3140:
3133:
3129:
3125:
3120:
3116:
3111:
3094:
3076:
3035:
3034:
3021:
3016:
3012:
3008:
3005:
3002:
2999:
2995:
2989:
2986:
2983:
2979:
2975:
2972:
2969:
2966:
2961:
2957:
2943:theta function
2890:is needed for
2883:The condition
2881:
2880:
2865:
2862:
2859:
2856:
2851:
2847:
2843:
2840:
2838:
2836:
2833:
2830:
2827:
2824:
2819:
2815:
2811:
2810:
2807:
2804:
2801:
2798:
2793:
2789:
2783:
2779:
2775:
2772:
2770:
2767:
2761:
2758:
2753:
2749:
2743:
2739:
2735:
2734:
2720:
2719:
2708:
2700:
2696:
2692:
2689:
2686:
2683:
2680:
2676:
2667:
2662:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2620:
2616:
2613:
2610:
2607:
2602:
2598:
2594:
2591:
2588:
2585:
2580:
2576:
2541:
2536:
2535:
2524:
2519:
2516:
2512:
2506:
2503:
2500:
2497:
2494:
2490:
2486:
2483:
2480:
2477:
2472:
2468:
2435:
2412:
2409:
2378:
2377:
2363:absolute value
2359:
2311:
2262:to the set of
2245:
2242:
2241:
2240:
2237:Fourier series
2231:, with period
2217:+ 1) =
2204:
2203:
2181:
2180:
2179:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2134:
2131:
2128:
2125:
2122:
2117:
2113:
2109:
2105:
2099:
2096:
2091:
2087:
2083:
2070:
2069:
2068:
2055:
2049:
2046:
2044:
2041:
2040:
2037:
2034:
2032:
2029:
2028:
2026:
2021:
2018:
2014:
2009:
2003:
2000:
1998:
1995:
1994:
1991:
1988:
1985:
1983:
1980:
1979:
1977:
1972:
1969:
1956:
1955:
1952:
1940:
1920:
1917:
1913:
1909:
1906:
1903:
1900:
1896:
1891:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1849:
1846:
1843:
1840:
1837:
1818:
1811:
1800:
1799:
1776:
1775:
1774:
1763:
1760:
1757:
1754:
1749:
1745:
1741:
1738:
1735:
1732:
1729:
1726:
1722:
1716:
1713:
1710:
1707:
1702:
1699:
1696:
1693:
1687:
1683:
1652:
1594:complex-valued
1590:
1589:
1577:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1549:
1545:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1516:
1511:
1505:
1502:
1500:
1497:
1496:
1493:
1490:
1488:
1485:
1484:
1482:
1477:
1473:
1469:
1466:
1462:
1458:
1455:
1452:
1425:
1422:
1420:
1417:
1393:
1390:
1373:
1369:
1365:
1360:
1350:
1347:
1320:
1317:
1313:
1309:
1304:
1299:
1294:
1289:
1284:
1281:
1278:
1275:
1270:
1266:
1239:
1216:
1213:
1210:
1205:
1201:
1197:
1194:
1189:
1186:
1182:
1178:
1173:
1169:
1165:
1160:
1156:
1152:
1149:
1126:
1106:
1086:
1082:
1078:
1073:
1063:
1060:
1040:
1037:
1036:
1035:
1024:
1021:
1018:
1015:
1012:
987:
984:
981:
978:
975:
972:
969:
966:
963:
958:
955:
951:
947:
944:
941:
938:
935:
915:
911:
907:
902:
892:
889:
860:
857:
853:
849:
844:
834:
829:
823:
820:
818:
815:
814:
811:
808:
806:
803:
802:
800:
795:
792:
772:
752:
749:
746:
743:
740:
737:
733:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
688:
687:
676:
673:
670:
667:
664:
639:
636:
633:
630:
627:
624:
621:
616:
613:
609:
605:
602:
599:
596:
593:
573:
569:
565:
560:
550:
547:
536:
525:
522:
519:
516:
511:
507:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
447:
444:
441:
412:
408:
403:
398:
395:
371:
351:
321:
317:
313:
308:
298:
295:
283:
280:
253:
249:
245:
240:
235:
232:
227:
224:
220:
216:
211:
206:
203:
171:sphere packing
155:
154:
151:
120:
88:
87:
49:
47:
40:
26:
9:
6:
4:
3:
2:
7597:
7586:
7583:
7581:
7578:
7576:
7575:Modular forms
7573:
7572:
7570:
7551:
7548:
7546:
7543:
7541:
7538:
7537:
7535:
7533:
7529:
7523:
7520:
7518:
7515:
7513:
7510:
7508:
7505:
7503:
7500:
7499:
7497:
7495:
7494:Singularities
7491:
7485:
7482:
7480:
7477:
7475:
7472:
7470:
7467:
7466:
7464:
7460:
7454:
7451:
7449:
7446:
7444:
7441:
7439:
7436:
7434:
7431:
7430:
7428:
7424:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7393:
7390:
7388:
7385:
7383:
7380:
7379:
7377:
7373:
7370:
7366:
7360:
7357:
7355:
7352:
7350:
7347:
7346:
7344:
7342:Constructions
7340:
7334:
7331:
7329:
7326:
7324:
7321:
7319:
7316:
7314:
7313:Klein quartic
7311:
7309:
7306:
7304:
7301:
7299:
7296:
7294:
7293:Bolza surface
7291:
7289:
7288:Bring's curve
7286:
7284:
7281:
7280:
7278:
7276:
7272:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7248:
7246:
7243:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7221:
7220:Conic section
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7200:AF+BG theorem
7198:
7197:
7195:
7193:
7189:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7163:
7160:
7159:
7157:
7153:
7143:
7140:
7138:
7135:
7134:
7132:
7128:
7122:
7119:
7117:
7114:
7112:
7109:
7107:
7104:
7102:
7099:
7097:
7094:
7092:
7089:
7087:
7084:
7082:
7079:
7077:
7074:
7072:
7069:
7068:
7066:
7062:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7037:
7035:
7031:
7028:
7026:
7022:
7016:
7015:Twisted cubic
7013:
7011:
7008:
7006:
7003:
7001:
6998:
6996:
6993:
6992:
6990:
6988:
6984:
6980:
6972:
6967:
6965:
6960:
6958:
6953:
6952:
6949:
6943:
6940:
6937:
6933:
6929:
6925:
6921:
6917:
6913:
6912:
6907:
6902:
6899:
6894:
6890:
6886:
6882:
6879:
6878:
6872:
6869:
6867:0-521-21212-X
6863:
6859:
6856:, Cambridge:
6855:
6850:
6847:
6844:, Göttingen:
6843:
6839:
6835:
6832:
6827:
6823:
6819:
6815:
6811:
6807:
6804:
6803:
6797:
6791:
6787:
6783:
6778:
6775:
6773:0-387-97127-0
6769:
6765:
6760:
6759:
6753:
6749:
6748:
6732:
6725:
6708:
6700:
6682:
6670:
6669:
6661:
6652:
6645:
6637:
6633:
6626:
6619:
6615:
6611:
6607:
6603:
6597:
6593:
6589:
6588:
6587:Modular units
6583:
6579:
6573:
6564:
6562:3-540-15295-4
6558:
6554:
6547:
6540:. p. 51.
6539:
6532:
6521:
6514:
6507:
6499:
6495:
6488:
6486:
6481:
6467:
6463:
6445:
6439:
6434:
6427:
6422:
6416:
6404:
6397:
6393:
6389:
6385:
6381:
6375:
6361:
6360:dlmf.nist.gov
6357:
6340:
6330:
6326:
6316:
6313:
6312:
6306:
6304:
6300:
6296:
6292:
6288:
6283:
6281:
6277:
6269:
6265:
6262:
6258:
6255:
6251:
6248:
6244:
6243:
6242:
6233:
6230:
6222:
6212:
6208:
6204:
6198:
6197:
6193:
6188:This section
6186:
6182:
6177:
6176:
6168:
6166:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6123:
6116:The function
6100:
6094:
6088:
6083:
6075:
6072:
6069:
6066:
6057:
6054:
6051:
6048:
6045:
6042:
6039:
6033:
6030:
6026:
6020:
6017:
6014:
6011:
6006:
6003:
6000:
5997:
5991:
5987:
5980:
5979:
5978:
5962:
5954:
5951:
5948:
5945:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5912:
5904:
5900:
5898:
5886:
5884:
5880:
5876:
5873:
5869:
5867:
5863:
5857:
5851:
5847:
5843:
5841:
5837:
5833:
5829:
5825:
5819:
5815:
5811:
5807:
5804:
5803:real-analytic
5800:
5796:
5792:
5786:
5785:Haar measures
5776:
5774:
5768:
5758:
5738:
5730:
5717:
5713:
5709:
5700:
5692:
5679:
5675:
5651:
5643:
5635:
5629:
5621:
5596:
5593:
5590:
5582:
5566:
5546:
5526:
5518:
5513:
5498:
5496:
5491:
5489:
5485:
5479:
5472:
5447:
5443:
5437:
5434:
5431:
5427:
5423:
5411:
5383:
5377:
5365:
5355:
5352:
5350:
5346:
5342:
5338:
5334:
5329:
5327:
5323:
5319:
5311:
5307:
5304:
5300:
5296:
5292:
5288:
5284:
5280:
5276:
5272:
5268:
5264:
5260:
5256:
5252:
5248:
5244:
5240:
5230:
5228:
5224:
5220:
5216:
5212:
5208:
5201:
5198:
5194:
5190:
5186:
5182:
5178:
5174:
5170:
5166:
5161:
5147:
5139:
5120:
5080:
5077:
5073:
5069:
5065:
5061:
5057:
5045:
5041:
5033:
5030:
5027:
5021:
5017:
5013:
5009:
5005:
4998:
4993:
4989:
4977:
4974:
4962:
4956:
4952:
4948:
4937:
4929:
4928:
4927:
4925:
4917:
4911:
4907:
4897:
4893:
4885:
4881:
4871:
4867:
4863:
4858:
4855:
4851:
4843:
4835:
4831:
4825:
4821:
4811:
4807:
4799:
4795:
4783:
4775:
4771:
4757:
4755:
4751:
4747:
4743:
4739:
4731:
4727:
4723:
4719:
4715:
4711:
4707:
4702:
4700:
4696:
4692:
4688:
4684:
4677:
4673:
4666:
4662:
4658:
4654:
4650:
4647:, the spaces
4644:
4638:
4630:
4607:
4603:
4595:
4591:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4550:
4539:
4536:
4525:
4520:
4514:
4509:
4502:
4497:
4491:
4485:
4481:
4479:
4471:
4457:
4449:
4445:
4437:
4434:
4431:
4428:
4417:
4414:
4403:
4398:
4392:
4387:
4380:
4375:
4369:
4363:
4359:
4357:
4349:
4341:
4325:
4324:
4323:
4322:
4318:
4313:
4311:
4307:
4303:
4299:
4291:
4290:
4285:
4281:
4277:
4273:
4269:
4266:
4260:
4254:
4250:
4243:
4237:
4223:
4217:
4200:
4197:
4194:
4191:
4186:
4183:
4180:
4177:
4168:
4160:
4150:
4148:
4147:
4141:
4137:
4132:
4126:
4118:
4114:
4110:
4106:
4105:
4098:
4094:
4084:
4077:
4072:
4067:
4065:
4061:
4057:
4054:
4046:
4042:
4038:
4035:
4031:
4026:
4024:
4020:
4015:
4013:
4009:
4005:
4001:
3997:
3993:
3975:
3971:
3962:
3958:
3954:
3933:
3928:
3924:
3918:
3914:
3903:
3900:
3897:
3894:
3890:
3886:
3880:
3874:
3867:
3866:
3865:
3864:
3863:
3861:
3842:
3839:
3836:
3833:
3827:
3824:
3821:
3818:
3793:
3788:
3785:
3782:
3779:
3776:
3772:
3766:
3762:
3751:
3748:
3745:
3742:
3738:
3734:
3728:
3722:
3715:
3714:
3713:
3712:
3708:
3704:
3700:
3682:
3676:
3673:
3669:
3663:
3660:
3657:
3654:
3649:
3646:
3643:
3640:
3634:
3630:
3622:
3601:
3595:
3590:
3583:
3578:
3572:
3563:
3559:
3557:
3554:
3550:
3546:
3543:
3542:
3541:
3539:
3535:
3531:
3528:. A function
3527:
3523:
3519:
3515:
3505:
3503:
3499:
3495:
3491:
3486:
3484:
3480:
3476:
3472:
3468:
3464:
3459:
3443:
3437:
3433:
3429:
3428:Leech lattice
3423:
3419:
3415:
3410:
3406:
3402:
3383:
3378:
3375:
3372:
3369:
3365:
3361:
3358:
3354:
3346:
3342:
3338:
3335:
3322:
3319:
3316:
3312:
3306:
3302:
3298:
3294:
3290:
3284:
3278:
3271:
3270:
3269:
3267:
3261:
3256:
3255:
3251:
3249:
3245:
3241:
3237:
3234:
3229:
3224:
3220:
3212:
3203:
3196:
3176:
3170:
3160:
3156:
3151:
3147:
3141:
3131:
3127:
3123:
3118:
3114:
3109:
3101:
3100:
3099:
3097:
3090:
3084:
3067:
3060:
3045:
3040:
3019:
3014:
3006:
3000:
2997:
2993:
2987:
2984:
2981:
2977:
2973:
2967:
2959:
2955:
2947:
2946:
2945:
2944:
2931:
2923:
2918:
2917:
2913:
2909:
2902:
2893:
2887:
2863:
2857:
2849:
2845:
2841:
2839:
2831:
2828:
2825:
2817:
2813:
2805:
2799:
2791:
2787:
2781:
2777:
2773:
2771:
2765:
2759:
2756:
2751:
2747:
2741:
2737:
2725:
2724:
2723:
2706:
2698:
2690:
2687:
2684:
2681:
2674:
2665:
2655:
2649:
2646:
2643:
2637:
2631:
2628:
2625:
2618:
2614:
2608:
2600:
2596:
2592:
2578:
2574:
2566:
2565:
2564:
2561:
2558:
2554:
2522:
2517:
2514:
2510:
2501:
2498:
2495:
2492:
2488:
2484:
2470:
2466:
2458:
2457:
2456:
2445:
2438:
2428:
2423:
2418:
2417:
2408:
2405:
2401:
2395:
2392:
2388:
2369:
2364:
2360:
2357:
2347:
2344:
2340:
2336:
2321:
2312:
2305:
2299:
2285:
2282:
2278:
2275:
2269:
2268:
2267:
2265:
2260:
2255:
2251:
2238:
2230:
2224:
2220:
2216:
2212:
2206:
2205:
2199:
2193:
2182:
2163:
2157:
2154:
2148:
2145:
2142:
2136:
2132:
2126:
2120:
2115:
2111:
2107:
2103:
2097:
2094:
2089:
2085:
2081:
2074:
2073:
2071:
2053:
2047:
2042:
2035:
2030:
2024:
2019:
2016:
2012:
2007:
2001:
1996:
1989:
1986:
1981:
1975:
1970:
1967:
1960:
1959:
1958:
1957:
1953:
1938:
1918:
1915:
1904:
1898:
1885:
1882:
1876:
1870:
1867:
1847:
1844:
1841:
1838:
1835:
1825:
1819:
1812:
1805:
1804:
1803:
1795:
1794:
1789:
1782:
1777:
1758:
1752:
1747:
1739:
1736:
1733:
1730:
1724:
1720:
1714:
1711:
1708:
1705:
1700:
1697:
1694:
1691:
1685:
1681:
1674:
1673:
1669:
1662:
1658:
1653:
1649:
1644:
1638:
1633:
1632:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1601:
1595:
1575:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1547:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1514:
1509:
1503:
1498:
1491:
1486:
1480:
1471:
1467:
1456:
1453:
1438:
1437:
1436:
1435:
1434:modular group
1416:
1414:
1410:
1404:
1400:
1389:
1387:
1358:
1348:
1337:
1331:
1302:
1292:
1273:
1264:
1254:
1253:
1252:modular curve
1237:
1227:
1203:
1199:
1195:
1187:
1184:
1180:
1176:
1167:
1158:
1154:
1150:
1147:
1138:
1124:
1071:
1061:
1050:
1046:
1013:
985:
973:
967:
961:
956:
953:
945:
942:
939:
936:
926:the function
900:
890:
887:
879:
878:
877:
875:
858:
842:
832:
827:
821:
816:
809:
804:
798:
793:
790:
770:
747:
744:
741:
738:
731:
724:
721:
718:
715:
709:
703:
697:
665:
631:
625:
619:
614:
611:
603:
600:
597:
594:
584:the function
558:
548:
545:
537:
520:
514:
509:
501:
498:
495:
492:
486:
477:
471:
465:
442:
439:
431:
430:
429:
427:
396:
393:
386:
382:
369:
339:
335:
306:
296:
279:
277:
272:
270:
265:
238:
225:
209:
191:
187:
183:
178:
176:
175:string theory
172:
168:
164:
163:number theory
160:
152:
149:
148:modular group
145:
141:
137:
136:
135:
107:
103:
99:
95:
84:
81:
73:
70:February 2024
63:
59:
53:
50:This article
48:
39:
38:
33:
19:
7479:Prym variety
7453:Stable curve
7443:Hodge bundle
7433:ELSV formula
7235:Fermat curve
7192:Plane curves
7155:Higher genus
7130:Applications
7055:Modular form
7054:
6915:
6909:
6897:
6888:
6876:
6853:
6841:
6838:Hecke, Erich
6830:
6813:
6799:
6781:
6762:, New York:
6757:
6681:
6674:, p. 88
6667:
6660:
6650:
6644:
6631:
6625:
6586:
6572:
6552:
6546:
6531:
6506:
6497:
6465:
6461:
6403:
6395:
6387:
6374:
6363:. Retrieved
6359:
6329:
6287:Andrew Wiles
6284:
6273:
6240:
6225:
6219:October 2019
6216:
6201:Please help
6189:
6115:
5901:
5893:
5887:
5877:
5872:Jacobi forms
5870:
5865:
5855:
5844:
5835:
5830:
5823:
5797:
5790:
5782:
5770:
5580:
5515:
5492:
5477:
5473:
5375:
5367:
5353:
5340:
5332:
5330:
5325:
5321:
5317:
5309:
5305:
5302:
5298:
5294:
5290:
5282:
5278:
5274:
5270:
5266:
5262:
5258:
5254:
5250:
5246:
5242:
5236:
5233:Line bundles
5226:
5222:
5218:
5214:
5210:
5206:
5199:
5188:
5180:
5176:
5172:
5162:
5136:denotes the
5109:
4923:
4909:
4902:
4895:
4888:
4883:
4879:
4877:
4874:Consequences
4865:
4861:
4856:
4853:
4849:
4841:
4833:
4829:
4823:
4816:
4809:
4802:
4800:are denoted
4797:
4793:
4781:
4773:
4769:
4763:
4753:
4749:
4741:
4737:
4729:
4725:
4721:
4717:
4709:
4705:
4703:
4698:
4694:
4690:
4686:
4682:
4675:
4671:
4664:
4663:are denoted
4660:
4656:
4652:
4648:
4642:
4636:
4628:
4626:
4316:
4314:
4305:
4301:
4287:
4283:
4279:
4271:
4267:
4258:
4252:
4235:
4227:
4221:
4158:
4156:
4146:modular unit
4144:
4142:
4135:
4130:
4121:
4116:
4108:
4102:
4096:
4092:
4082:
4075:
4070:
4068:
4064:moduli space
4059:
4055:
4040:
4036:
4027:
4022:
4018:
4016:
4011:
4007:
4003:
3999:
3995:
3991:
3956:
3952:
3950:
3810:
3702:
3555:
3551:in the open
3544:
3537:
3533:
3529:
3525:
3521:
3513:
3511:
3487:
3457:
3441:
3421:
3417:
3413:
3400:
3398:
3263:
3253:
3252:
3227:
3210:
3201:
3194:
3191:
3082:
3065:
3058:
3043:
3036:
2929:
2919:
2915:
2914:
2907:
2900:
2885:
2882:
2721:
2559:
2556:
2552:
2537:
2443:
2433:
2431:, we define
2426:
2419:
2415:
2414:
2403:
2399:
2393:
2390:
2386:
2379:
2367:
2358:of the form.
2355:
2345:
2342:
2338:
2334:
2319:
2297:
2283:
2280:
2276:
2273:
2258:
2249:
2247:
2222:
2218:
2214:
2210:
2197:
1823:
1801:
1792:
1787:
1780:
1667:
1660:
1656:
1647:
1636:
1625:
1617:
1613:
1609:
1599:
1591:
1427:
1402:
1398:
1395:
1333:
1256:
1229:
1140:
1042:
873:
689:
341:
336:, called an
334:finite index
285:
273:
266:
179:
156:
144:group action
98:modular form
97:
91:
76:
67:
51:
32:Haar measure
7354:Polar curve
6582:Lang, Serge
6460:sends ∞ to
6380:meromorphic
6353:, see e.g.
6303:square root
6261:Erich Hecke
6254:Felix Klein
5814:holomorphic
5799:Maass forms
5382:graded ring
5349:line bundle
5287:homogeneous
4922:-action on
4736:-action on
4109:Spitzenform
4053:j-invariant
3549:meromorphic
3407:. Then the
3240:isospectral
3219:John Milnor
3089:root system
2892:convergence
1806:The weight
1413:meromorphic
1409:holomorphic
1117:and weight
1045:line bundle
362:and weight
269:Erich Hecke
94:mathematics
7569:Categories
7349:Dual curve
6977:Topics in
6906:Zagier, D.
6745:References
6618:0492.12002
6365:2023-07-07
5883:Lie groups
5761:Cusp forms
5293:, letting
5171:one (over
4840:. In case
4768:of weight
4760:Definition
4728:such that
4286:, i.e. in
4119:such that
4045:isomorphic
3238:which are
3062:such that
2894:; for odd
1931:, meaning
1860:such that
1802:Remarks:
1628:) > 0},
282:Definition
186:Lie groups
138:a kind of
7462:Morphisms
7210:Bitangent
6697:Γ
6476:Citations
6394:as exp(1/
6341:γ
6301:with the
6190:does not
6124:ε
6034:ε
5913:ε
5810:Laplacian
5773:cusp form
5767:Cusp form
5727:Γ
5710:⊆
5689:Γ
5640:Γ
5636:⊆
5618:Γ
5594:∣
5581:old forms
5559:dividing
5517:New forms
5506:New forms
5456:Γ
5428:⨁
5418:Γ
5160:is even.
5124:⌋
5121:⋅
5118:⌊
5087:otherwise
5031:≡
4949:
4581:≡
4575:≡
4563:≡
4557:≡
4526:∈
4466:Γ
4435:≡
4404:∈
4338:Γ
4172:↦
4104:cusp form
3909:∞
3901:−
3891:∑
3837:π
3828:
3783:π
3757:∞
3749:−
3739:∑
3436:Ramanujan
3416:) = (2π)
3373:π
3339:−
3328:∞
3313:∏
3279:η
3244:isometric
3152:ϑ
3124:×
3110:ϑ
3011:‖
3007:λ
3004:‖
2998:π
2985:∈
2982:λ
2978:∑
2956:ϑ
2858:τ
2826:τ
2800:τ
2778:τ
2760:τ
2752:−
2691:τ
2656:∈
2638:≠
2619:∑
2609:τ
2587:Λ
2515:−
2511:λ
2505:Λ
2502:∈
2499:λ
2496:≠
2489:∑
2479:Λ
2090:−
1987:−
1890:⟹
1871:
1596:function
1560:−
1540:∈
1346:Γ
1293:∪
1280:∖
1277:Γ
1269:Γ
1238:ω
1212:Γ
1185:⊗
1181:ω
1172:Γ
1151:∈
1105:Γ
1062:⊂
1059:Γ
1023:∞
1020:→
983:→
968:γ
954:−
891:∈
888:γ
874:cusp form
833:∈
791:γ
771:γ
698:γ
675:∞
672:→
626:γ
612:−
549:∈
546:γ
472:γ
446:Γ
443:∈
440:γ
424:from the
407:→
350:Γ
297:⊂
294:Γ
226:⊂
6920:Springer
6887:(1973),
6840:(1970),
6812:(1975),
6754:(1990),
6731:Archived
6584:(1981),
6520:Archived
6309:See also
6285:In 1994
5341:sections
5074:⌋
5058:⌊
5022:⌋
5006:⌊
4039:/Λ over
3532: :
3492:and the
3242:but not
2563:we have
2411:Examples
2397:, where
2254:lattices
2221: (
2213: (
2192:generate
1813:For odd
1654:For any
1432:for the
1401: (
7522:Tacnode
7507:Crunode
6924:Bibcode
6922:: 113,
6826:0379375
6639:, p. 13
6610:0648603
6536:Milne.
6211:removed
6196:sources
6171:History
5808:of the
5372:of the
5195:of the
5193:closure
5191:in the
3862:), as:
3616:in the
3467:Shimura
3463:Eichler
3233:compact
3091:called
3080:. When
2332:, then
2295:, then
2209:
1826:
1822:
1783:
1779:
1639:
1635:
1604:on the
1602:
1598:
146:of the
104:on the
56:Please
7502:Acnode
7426:Moduli
6864:
6824:
6792:
6770:
6616:
6608:
6598:
6559:
6498:Quanta
6398:) has.
5854:SL(2,
5822:SL(2,
5476:SL(2,
5374:SL(2,
5221:) and
5110:where
4859:. For
4792:. The
4685:) and
4674:) and
4263:. The
4257:SL(2,
4234:SL(2,
4113:German
3562:matrix
3477:, and
3399:where
2888:> 2
2549:. For
2429:> 2
2356:weight
2350:where
2302:is an
2207:Since
2196:SL(2,
2072:reads
1666:SL(2,
1551:
1230:where
1051:. For
690:where
173:, and
6734:(PDF)
6727:(PDF)
6672:(PDF)
6567:p. 15
6523:(PDF)
6516:(PDF)
6321:Notes
5609:then
5501:Types
5345:sheaf
5343:of a
5185:poles
4854:level
4746:genus
4639:) or
4298:trace
4280:cusps
4242:index
3475:Ihara
3246:(see
2538:Then
2341:Λ) =
1641:is a
1592:is a
383:is a
7512:Cusp
6862:ISBN
6790:ISBN
6768:ISBN
6596:ISBN
6557:ISBN
6384:pole
6194:any
6192:cite
5801:are
5486:and
5435:>
5140:and
4900:and
4814:and
4716:for
4655:and
4627:For
4249:acts
4228:Let
3860:nome
3471:Kuga
3405:nome
3264:The
3223:tori
3208:and
2904:and
2722:and
2551:Λ =
2361:The
2272:Λ =
2187:and
1916:<
1883:>
1845:>
340:, a
96:, a
6932:doi
6614:Zbl
6386:at
6259:By
6252:By
6205:by
5394:(Γ)
5328:).
5229:).
5187:of
5042:mod
4938:dim
4844:= Γ
4748:of
4693:),
4631:= Γ
4592:mod
4446:mod
4251:on
4133:at
4127:≠ 0
4111:in
4088:= 0
4078:= 0
3825:exp
3547:is
3456:≤ 2
3434:of
3250:.)
3085:= 8
3056:in
2927:in
2920:An
2451:of
2439:(Λ)
2370:(Λ)
2365:of
2348:(Λ)
2328:by
2313:If
2306:of
2300:(Λ)
2256:in
1645:on
1612:= {
1407:be
1047:on
998:as
332:of
92:In
60:to
7571::
6930:,
6918:,
6916:94
6914:,
6896:.
6860:,
6829:.
6822:MR
6820:,
6788:,
6766:,
6729:.
6612:,
6606:MR
6604:,
6580:;
6518:.
6496:.
6484:^
6378:A
6358:.
5885:.
5842:.
5789:Δ(
5771:A
5757:.
5497:.
5471:.
5299:cv
5267:cv
5241:P(
5227:Nz
5070:12
5046:12
5018:12
4968:SL
4724:⊂
4641:Γ(
4530:SL
4408:SL
4224:\H
4143:A
4140:.
4095:=
3623:,
3536:→
3504:.
3473:,
3469:,
3465:,
3440:Δ(
3412:Δ(
3307:24
3214:16
3200:×
3161:16
3046:/2
2906:(−
2555:+
2455::
2407:.
2402:∈
2389:+
2279:+
1868:Im
1790:→
1659:∈
1622:Im
1620:,
1616:∈
1447:SL
1388:.
1354:SL
1067:SL
1007:im
896:SL
838:SL
659:im
554:SL
302:SL
278:.
271:.
264:.
177:.
169:,
108:,
6970:e
6963:t
6956:v
6934::
6926::
6900:.
6833:.
6805:.
6712:)
6709:N
6706:(
6701:1
6565:.
6500:.
6468:.
6466:c
6464:/
6462:a
6446:)
6440:d
6435:c
6428:b
6423:a
6417:(
6396:q
6388:q
6368:.
6232:)
6226:(
6221:)
6217:(
6213:.
6199:.
6151:)
6148:d
6145:,
6142:c
6139:,
6136:b
6133:,
6130:a
6127:(
6101:.
6098:)
6095:z
6092:(
6089:f
6084:k
6080:)
6076:d
6073:+
6070:z
6067:c
6064:(
6061:)
6058:d
6055:,
6052:c
6049:,
6046:b
6043:,
6040:a
6037:(
6031:=
6027:)
6021:d
6018:+
6015:z
6012:c
6007:b
6004:+
6001:z
5998:a
5992:(
5988:f
5963:k
5959:)
5955:d
5952:+
5949:z
5946:c
5943:(
5940:)
5937:d
5934:,
5931:c
5928:,
5925:b
5922:,
5919:a
5916:(
5896:k
5890:k
5858:)
5856:R
5836:n
5826:)
5824:Z
5793:)
5791:g
5745:)
5742:)
5739:N
5736:(
5731:1
5723:(
5718:k
5714:M
5707:)
5704:)
5701:M
5698:(
5693:1
5685:(
5680:k
5676:M
5655:)
5652:M
5649:(
5644:1
5633:)
5630:N
5627:(
5622:1
5597:N
5591:M
5567:N
5547:M
5527:N
5480:)
5478:Z
5459:)
5453:(
5448:k
5444:M
5438:0
5432:k
5424:=
5421:)
5415:(
5412:M
5402:Γ
5398:k
5392:k
5390:M
5386:Γ
5378:)
5376:Z
5370:Γ
5333:V
5326:V
5322:k
5318:k
5314:k
5310:v
5308:(
5306:F
5303:c
5297:(
5295:F
5291:c
5283:F
5279:c
5275:v
5273:(
5271:F
5265:(
5263:F
5259:V
5255:v
5251:V
5247:F
5243:V
5225:(
5223:j
5219:z
5217:(
5215:j
5211:N
5209:(
5207:N
5203:Γ
5200:R
5189:f
5181:f
5177:f
5173:C
5148:k
5081:1
5078:+
5066:/
5062:k
5049:)
5039:(
5034:2
5028:k
5014:/
5010:k
4999:{
4994:=
4990:)
4986:)
4982:Z
4978:,
4975:2
4972:(
4963:(
4957:k
4953:M
4943:C
4924:H
4920:G
4912:)
4910:G
4908:(
4905:k
4903:S
4898:)
4896:G
4894:(
4891:k
4889:M
4884:H
4882:\
4880:G
4868:)
4866:Z
4862:G
4857:N
4850:N
4848:(
4846:0
4842:G
4838:G
4834:H
4832:\
4830:G
4826:)
4824:G
4822:(
4819:k
4817:S
4812:)
4810:G
4808:(
4805:k
4803:M
4798:k
4794:C
4790:G
4786:G
4782:H
4778:G
4774:H
4770:k
4766:G
4754:H
4752:\
4750:G
4742:D
4738:H
4734:G
4730:D
4726:H
4722:D
4718:G
4710:H
4708:\
4706:G
4699:N
4697:(
4695:X
4691:N
4689:(
4687:Y
4683:N
4681:(
4679:0
4676:X
4672:N
4670:(
4668:0
4665:Y
4661:H
4659:\
4657:G
4653:H
4651:\
4649:G
4645:)
4643:N
4637:N
4635:(
4633:0
4629:G
4608:.
4604:}
4599:)
4596:N
4589:(
4584:1
4578:d
4572:a
4569:,
4566:0
4560:b
4554:c
4551::
4548:)
4544:Z
4540:,
4537:2
4534:(
4521:)
4515:d
4510:c
4503:b
4498:a
4492:(
4486:{
4482:=
4475:)
4472:N
4469:(
4458:}
4453:)
4450:N
4443:(
4438:0
4432:c
4429::
4426:)
4422:Z
4418:,
4415:2
4412:(
4399:)
4393:d
4388:c
4381:b
4376:a
4370:(
4364:{
4360:=
4353:)
4350:N
4347:(
4342:0
4317:N
4306:H
4304:\
4302:G
4294:G
4289:Q
4284:H
4272:H
4270:\
4268:G
4261:)
4259:Z
4253:H
4246:G
4238:)
4236:Z
4230:G
4222:G
4201:d
4198:+
4195:z
4192:c
4187:b
4184:+
4181:z
4178:a
4169:z
4159:f
4138:∞
4136:i
4131:f
4124:n
4122:a
4117:n
4107:(
4099:∞
4097:i
4093:z
4086:0
4083:a
4076:q
4071:f
4060:z
4058:(
4056:j
4049:α
4041:C
4037:C
4023:f
4019:f
4012:q
4008:q
4004:n
4000:f
3996:m
3992:f
3976:n
3972:a
3959:(
3957:f
3953:q
3934:.
3929:n
3925:q
3919:n
3915:a
3904:m
3898:=
3895:n
3887:=
3884:)
3881:z
3878:(
3875:f
3846:)
3843:z
3840:i
3834:2
3831:(
3822:=
3819:q
3794:.
3789:z
3786:n
3780:i
3777:2
3773:e
3767:n
3763:a
3752:m
3746:=
3743:n
3735:=
3732:)
3729:z
3726:(
3723:f
3703:f
3698:.
3686:)
3683:z
3680:(
3677:f
3674:=
3670:)
3664:d
3661:+
3658:z
3655:c
3650:b
3647:+
3644:z
3641:a
3635:(
3631:f
3620:Γ
3602:)
3596:d
3591:c
3584:b
3579:a
3573:(
3556:H
3545:f
3538:C
3534:H
3530:f
3522:f
3514:k
3458:p
3452:p
3448:q
3444:)
3442:z
3424:)
3422:z
3420:(
3418:η
3414:z
3401:q
3384:.
3379:z
3376:i
3370:2
3366:e
3362:=
3359:q
3355:,
3352:)
3347:n
3343:q
3336:1
3333:(
3323:1
3320:=
3317:n
3303:/
3299:1
3295:q
3291:=
3288:)
3285:z
3282:(
3228:R
3211:L
3205:8
3202:L
3198:8
3195:L
3177:,
3174:)
3171:z
3168:(
3157:L
3148:=
3145:)
3142:z
3139:(
3132:8
3128:L
3119:8
3115:L
3095:8
3093:E
3083:n
3077:n
3075:L
3071:v
3066:v
3064:2
3059:R
3054:v
3050:n
3044:n
3020:z
3015:2
3001:i
2994:e
2988:L
2974:=
2971:)
2968:z
2965:(
2960:L
2939:L
2935:n
2930:R
2925:L
2910:)
2908:λ
2901:λ
2896:k
2886:k
2864:.
2861:)
2855:(
2850:k
2846:G
2842:=
2835:)
2832:1
2829:+
2823:(
2818:k
2814:G
2806:,
2803:)
2797:(
2792:k
2788:G
2782:k
2774:=
2766:)
2757:1
2748:(
2742:k
2738:G
2707:,
2699:k
2695:)
2688:n
2685:+
2682:m
2679:(
2675:1
2666:2
2661:Z
2653:)
2650:n
2647:,
2644:m
2641:(
2635:)
2632:0
2629:,
2626:0
2623:(
2615:=
2612:)
2606:(
2601:k
2597:G
2593:=
2590:)
2584:(
2579:k
2575:G
2560:τ
2557:Z
2553:Z
2547:k
2542:k
2540:G
2523:.
2518:k
2493:0
2485:=
2482:)
2476:(
2471:k
2467:G
2453:Λ
2449:λ
2444:λ
2436:k
2434:G
2427:k
2404:H
2400:τ
2394:τ
2391:Z
2387:Z
2382:F
2374:Λ
2368:F
2352:k
2346:F
2343:α
2339:α
2337:(
2335:F
2330:α
2326:Λ
2322:Λ
2320:α
2315:α
2310:.
2308:z
2298:F
2293:z
2289:α
2284:z
2281:Z
2277:α
2274:Z
2259:C
2250:F
2239:.
2233:1
2225:)
2223:z
2219:f
2215:z
2211:f
2200:)
2198:Z
2189:T
2185:S
2167:)
2164:z
2161:(
2158:f
2155:=
2152:)
2149:1
2146:+
2143:z
2140:(
2137:f
2133:,
2130:)
2127:z
2124:(
2121:f
2116:k
2112:z
2108:=
2104:)
2098:z
2095:1
2086:(
2082:f
2054:)
2048:1
2043:0
2036:1
2031:1
2025:(
2020:=
2017:T
2013:,
2008:)
2002:0
1997:1
1990:1
1982:0
1976:(
1971:=
1968:S
1939:f
1919:D
1912:|
1908:)
1905:z
1902:(
1899:f
1895:|
1886:M
1880:)
1877:z
1874:(
1848:0
1842:D
1839:,
1836:M
1824:f
1815:k
1808:k
1798:.
1796:∞
1793:i
1788:z
1781:f
1762:)
1759:z
1756:(
1753:f
1748:k
1744:)
1740:d
1737:+
1734:z
1731:c
1728:(
1725:=
1721:)
1715:d
1712:+
1709:z
1706:c
1701:b
1698:+
1695:z
1692:a
1686:(
1682:f
1670:)
1668:Z
1661:H
1657:z
1651:.
1648:H
1637:f
1626:z
1624:(
1618:C
1614:z
1610:H
1600:f
1576:}
1572:1
1569:=
1566:c
1563:b
1557:d
1554:a
1548:,
1544:Z
1537:d
1534:,
1531:c
1528:,
1525:b
1522:,
1519:a
1515:|
1510:)
1504:d
1499:c
1492:b
1487:a
1481:(
1472:{
1468:=
1465:)
1461:Z
1457:,
1454:2
1451:(
1430:k
1405:)
1403:z
1399:f
1372:)
1368:Z
1364:(
1359:2
1349:=
1319:)
1316:)
1312:Q
1308:(
1303:1
1298:P
1288:H
1283:(
1274:=
1265:X
1215:)
1209:(
1204:k
1200:M
1196:=
1193:)
1188:k
1177:,
1168:X
1164:(
1159:0
1155:H
1148:f
1125:k
1085:)
1081:Z
1077:(
1072:2
1017:)
1014:z
1011:(
986:0
980:)
977:)
974:z
971:(
965:(
962:f
957:k
950:)
946:d
943:+
940:z
937:c
934:(
914:)
910:Z
906:(
901:2
859:.
856:)
852:Z
848:(
843:2
828:)
822:d
817:c
810:b
805:a
799:(
794:=
751:)
748:d
745:+
742:z
739:c
736:(
732:/
728:)
725:b
722:+
719:z
716:a
713:(
710:=
707:)
704:z
701:(
669:)
666:z
663:(
638:)
635:)
632:z
629:(
623:(
620:f
615:k
608:)
604:d
601:+
598:z
595:c
592:(
572:)
568:Z
564:(
559:2
524:)
521:z
518:(
515:f
510:k
506:)
502:d
499:+
496:z
493:c
490:(
487:=
484:)
481:)
478:z
475:(
469:(
466:f
411:C
402:H
397::
394:f
370:k
320:)
316:Z
312:(
307:2
252:)
248:R
244:(
239:2
234:L
231:S
223:)
219:Z
215:(
210:2
205:L
202:S
150:,
119:H
83:)
77:(
72:)
68:(
54:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.