|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], a '''modular form''' is a (complex) [[analytic function]] on the [[upper half-plane]] satisfying a certain kind of [[functional equation]] with respect to the [[group action]] of the [[modular group]], and also satisfying a growth condition. The theory of modular forms therefore belongs to [[complex analysis]] but the main importance of the theory has traditionally been in its connections with [[number theory]]. Modular forms appear in other areas, such as [[algebraic topology]] and [[string theory]].
| |
|
| |
|
| A '''modular function''' is a modular form invariant with respect to the modular group but without the condition that f(z) be [[Holomorphic function|holomorphic]] at infinity. Instead, modular functions are [[Meromorphic function|meromorphic]] at infinity.
| |
|
| |
|
| Modular form theory is a special case of the more general theory of [[automorphic form]]s, and therefore can now be seen as just the most concrete part of a rich theory of [[discrete group]]s.
| | May get download from the below hyperlink, if you're seeking clash of families no charge gems, elixir and gold. You'll get the greatest secret write down to get accessibility concerning assets and endless rocks by downloading from adhering to links.<br><br>amend delivers a bunch of notable enhancements, arc of which could quite possibly be the new Dynasty Conflict Manner. In this mode, you can said combating dynasties and remove utter rewards aloft personal beat.<br><br>clash of clans is a ideal game, which usually requires someone to build your personal village, discover warriors, raid tools and build your hold clan and so up. there is a lot a lot additionally to this video game and for every these you require jewels in play, as you desire. If you have any kind of concerns relating to where and how to utilize [http://prometeu.net hack clash of Clans ifile], you could call us at our own web-page. Clash of Clans hack allows you to obtain as many jewels as you want. There is an unlimited regarding gems you could establish with all the Collide of Clans cheats there online, however you want to be specific about the web page link you are using for the reason some of them only waste materials your serious amounts of also dont get you anything more.<br><br>In case that you are searching with a particular game so that it will buy but want to purchase it at you see, the best price possible, utilize the "shopping" tab that are available on many search sites. This will allow you to immediately equate the prices of our own game at all each major retailers online. You can also catch a glimpse of ratings for the insurer in question, helping someone determine who you should always buy the game because of.<br><br>Supply the in-online game songs possibility. If, nonetheless, you might be annoyed by using this tool soon after one moment approximately, don't be scared to mute the television set or personal computer and as a consequence play some audio of the very own. You will discover a far more enjoyment game playing experience performing this and therefore are a lot of unlikely to get a frustration from actively game play.<br><br>It looks like computer games are just about everywhere these times. You could play them on their telephone, boot a unit in the home and even see them through social media on your [http://Www.Thefreedictionary.com/personal personal] computer system. It helps to comprehend this associated with amusement to help you benefit from the dozens of offers which are out there.<br><br>So your village grows, possess to explore uncharted areas for Gold and Woodgrain effect which are the regarding key resources you is able to expect to require here in start of the game ( addititionally there is Stone resource, that you discover later inside i would say the game ). Through your exploration, you in many cases can expect to stumble located on many islands whereby that villages happen to wind up being held captive under BlackGuard slavery and you profit from free Gold sources if they are vacant. |
| | |
| ==Modular forms for SL<sub>2</sub>(Z)==
| |
| A modular form of weight ''k'' for the ''[[modular group]]''
| |
| :<math>SL(2, \mathbf Z) = \left \{ \left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )| a, b, c, d \in \mathbf Z, ad-bc = 1 \right \}</math>
| |
| is a [[complex numbers|complex-valued]] function ''f'' on the [[upper half-plane]] {{nowrap|'''H''' {{=}} {''z'' ∈ '''C''', [[imaginary part|Im]](''z'') > 0}, }} satisfying the following three conditions: firstly, ''f'' is a [[holomorphic function]] on '''H'''. Secondly, for any ''z'' in '''H''' and any matrix in SL(2,'''Z''') as above, the equation
| |
| | |
| :<math> f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z)</math>
| |
| | |
| is required to hold. Thirdly, ''f'' is required to be holomorphic as {{math|''z'' → [[Imaginary unit|''i'']]∞}}. The latter condition is also phrased by saying that ''f'' is "holomorphic at the cusp", a terminology that is explained below. The weight ''k'' is typically a positive integer.
| |
| | |
| The second condition, with the matrices <math>S = \left ( \begin{array}{cc}0 & -1 \\ 1 & 0 \end{array} \right )</math> and <math>T = \left ( \begin{array}{cc}1 & 1 \\ 0 & 1 \end{array} \right )</math> reads
| |
| :<math>f(-1/z) = z^k f(z)\,</math>
| |
| and
| |
| :<math>f(z+1) = f(z)\,</math>
| |
| respectively. Since ''S'' and ''T'' [[generating set of a group|generate]] the modular group SL(2,'''Z'''), the second condition above is equivalent to these two equations. Note that since
| |
| :<math>f(z+1) = f(z)</math>,
| |
| | |
| modular forms are [[periodic function]]s, with period 1, and thus have a [[Fourier series]].
| |
| | |
| Note that for odd ''k'', only the zero function can satisfy the second condition.
| |
| | |
| ===Definition in terms of lattices or elliptic curves===
| |
| A modular form can equivalently be defined as a function ''F'' from the set of [[period lattice|lattice]]s in '''C''' to the set of [[complex number]]s which satisfies certain conditions:
| |
| | |
| :(1) If we consider the lattice <math>\Lambda = \langle \alpha, z\rangle</math> generated by a constant α and a variable ''z'', then ''F''(Λ) is an [[analytic function]] of ''z''.
| |
| | |
| :(2) If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of Λ by α, then ''F''(αΛ) = α<sup>−''k''</sup>''F''(Λ) where ''k'' is a constant (typically a positive integer) called the '''weight''' of the form.
| |
| | |
| :(3) The [[absolute value]] of ''F''(Λ) remains bounded above as long as the absolute value of the smallest non-zero element in Λ is bounded away from 0.
| |
| | |
| The key idea in proving the equivalence of the two definitions is that such a function ''F'' is determined, because of the first property, by its values on lattices of the form <math>\langle 1, \omega \rangle</math>, where ω ∈ '''H'''.
| |
| | |
| ===Modular functions===
| |
| When the weight ''k'' is zero, the only modular forms are constant functions, as can be shown. However, relaxing the requirement that ''f'' be holomorphic leads to the notion of ''modular functions''. A function ''f'' : '''H''' → '''C''' is called modular [[iff]] it satisfies the following properties:
| |
| | |
| # ''f'' is [[meromorphic function|meromorphic]] in the open [[upper half-plane]] ''H''.
| |
| # For every [[matrix (mathematics)|matrix]] <math>\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )</math> in the [[modular group Gamma|modular group Γ]], <math> f\left(\frac{az+b}{cz+d}\right) = f(z)</math>.
| |
| # As pointed out above, the second condition implies that ''f'' is periodic, and therefore has a [[Fourier series]]. The third condition is that this series is of the form <math>f(z) = \sum_{n=-m}^\infty a_n e^{2i\pi nz}.</math> It is often written in terms of as <math>q=\exp(2\pi i z)</math> (the square of the [[nome (mathematics)|nome]]),
| |
| ::<math>f(z)=\sum_{n=-m}^\infty a_n q^n.</math>
| |
| This is also referred to as the ''q''-expansion<ref>[http://www.msri.org/about/computing/docs/magma/html/text600.htm Elliptic and Modular Functions<!-- Bot generated title -->]</ref> of ''f''. The coefficients <math>a_n</math> are known as the Fourier coefficients of ''f'', and the number ''m'' is called the order of the pole of ''f'' at i∞.
| |
| This condition is called "meromorphic at the cusp", meaning that only finitely many negative-''n'' coefficients are non-zero, so the ''q''-expansion is bounded below, guaranteeing that it is meromorphic at ''q''=0. <ref>A [[meromorphic]] 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 [[Pole_(complex_analysis)|pole]] at q=0, not an [[essential singularity]] as exp(1/''q'') has.</ref>
| |
| | |
| Another way to phrase the definition of modular functions is to use [[elliptic curve]]s: every lattice Λ determines an [[elliptic curve]] '''C'''/Λ over '''C'''; two lattices determine [[isomorphic]] elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number α. Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the [[j-invariant]] ''j''(''z'') 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 [[moduli problem|moduli space]] of isomorphism classes of complex elliptic curves.
| |
| | |
| A modular form ''f'' that vanishes at ''q'' = 0 (equivalently, ''a''<sub>0</sub> = 0, also paraphrased as ''z'' = i∞) is called a ''[[cusp form]]'' (''Spitzenform'' in [[German language|German]]). The smallest ''n'' such that ''a''<sub>''n''</sub> ≠ 0 is the order of the zero of ''f'' at i∞.
| |
| | |
| A ''[[modular unit]]'' is a modular function whose poles and zeroes are confined to the cusps.<ref>{{Citation | last1=Kubert | first1=Daniel S. | author1-link=Daniel Kubert | last2=Lang | first2=Serge | author2-link=Serge Lang | title=Modular units | url=http://books.google.com/books?id=BwwzmZjjVdgC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science] | isbn=978-0-387-90517-4 | id={{MR|648603}} | year=1981 | volume=244 | zbl=0492.12002 | page=24 }}</ref>
| |
| | |
| ==Modular forms for more general groups==
| |
| The functional equation, i.e., the behavior of ''f'' with respect to {{nowrap|''z'' ↦ (''az'' + ''b'')/(''cz'' + ''d'')}} can be relaxed by requiring it only for matrices in smaller groups.
| |
| | |
| ===The Riemann surface ''G''\H<sup>∗</sup>===
| |
| Let ''G'' be a subgroup of SL(2,'''Z''') that is of finite [[Index of a subgroup|index]]. Such a group ''G'' [[group action|acts]] on '''H''' in the same way as SL(2,'''Z'''). The [[quotient topological space]] ''G''\'''H''' can be shown to be a [[Hausdorff space]]. Typically it is not compact, but can be compactified by adding a finite number of points called ''cusps''. These are points at the boundary of '''H''', i.e., either in '''Q''', the rationals, or ∞, such that there is a parabolic element of ''G'' (a matrix with [[trace of a matrix|trace]] ±2) fixing the point. Here, a matrix <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math> sends ∞ to ''a''/''c''. This yields a compact topological space ''G''\'''H'''<sup>∗</sup>. What is more, it can be endowed with the structure of a [[Riemann surface]], which allows to speak of holo- and meromorphic functions.
| |
| | |
| Important examples are, for any positive integer ''N'', either one of the [[congruence subgroup]]s
| |
| :<math>\Gamma_0(N) = \left\{
| |
| \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) :
| |
| c \equiv 0 \pmod{N} \right\}</math>
| |
| and
| |
| :<math>\Gamma(N) = \left\{
| |
| \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) :
| |
| c \equiv b \equiv 0, a \equiv d \equiv 1 \pmod{N} \right\}.</math>
| |
| For ''G'' = Γ0(''N'') or Γ(''N''), the spaces ''G''\'''H''' and ''G''\'''H'''<sup>∗</sup> are denoted ''Y''<sub>0</sub>(''N'') and ''X''<sub>0</sub>(''N'') and ''Y''(''N''), ''X''(''N''), respectively.
| |
| | |
| The geometry of ''G''\'''H'''<sup>∗</sup> can be understood by studying [[fundamental domain]]s for ''G'', i.e. subsets ''D'' ⊂ '''H''' such that ''D'' intersects each orbit of the ''G''-action on '''H''' exactly once and such that the closure of ''D'' meets all orbits. For example, the [[genus]] of ''G''\'''H'''<sup>∗</sup> can be computed.<ref>{{Citation | last1=Gunning | first1=Robert C. | title=Lectures on modular forms | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | year=1962 | volume=48}}, p. 13</ref>
| |
| | |
| ===Definition===
| |
| A modular form for ''G'' of weight ''k'' is a function on '''H''' satisfying the above functional equation for all matrices in ''G'', that is holomorphic on '''H''' and at all cusps of ''G''. Again, modular forms that vanish at all cusps are called cusp forms for ''G''. The '''C'''-vector spaces of modular and cusp forms of weight ''k'' are denoted ''M''<sub>''k''</sub>(''G'') and ''S''<sub>''k''</sub>(''G''), respectively. Similarly, a meromorphic function on ''G''\'''H'''<sup>∗</sup> is called a modular function for ''G''. In case ''G'' = Γ<sub>0</sub>(''N''), they are also referred to as modular/cusp forms and functions of ''level'' ''N''. For ''G'' = Γ(1) = SL<sub>2</sub>('''Z'''), this gives back the afore-mentioned definitions.
| |
| | |
| ===Consequences===
| |
| The theory of Riemann surfaces can be applied to ''G''\'''H'''<sup>∗</sup> to obtain further information about modular forms and functions. For example, the spaces ''M''<sub>''k''</sub>(''G'') and ''S''<sub>''k''</sub>(''G'') are finite-dimensional, and their dimensions can be computed thanks to the [[Riemann-Roch theorem]] in terms of the geometry of the ''G''-action on '''H'''.<ref>{{Citation | last1=Shimura | first1=Goro | title=Introduction to the arithmetic theory of automorphic functions | publisher=Iwanami Shoten | location=Tokyo | series=Publications of the Mathematical Society of Japan | year=1971 | volume=11}}, Theorem 2.33, Proposition 2.26</ref> For example,
| |
| :<math>\text{dim}_{\mathbf C} M_k(SL(2,\mathbf Z)) =
| |
| \left \{ \begin{array}{ll} \lfloor k/12 \rfloor & k \equiv 2 \pmod{12} \\
| |
| \lfloor k/12 \rfloor + 1 & \text{else}
| |
| \end{array} \right.</math>
| |
| where <math>\lfloor - \rfloor</math> denotes the [[floor function]].
| |
| | |
| The modular functions constitute the [[function field of an algebraic variety|field of functions]] of the Riemann surface, and hence form a field of [[transcendence degree]] one (over '''C'''). If a modular function ''f'' is not identically 0, then it can be shown that the number of zeroes of ''f'' is equal to the number of [[pole (complex analysis)|pole]]s of ''f'' in the [[closure (mathematics)|closure]] of the [[fundamental region]] ''R''<sub>Γ</sub>.It can be shown that the field of modular function of level ''N'' (''N'' ≥ 1) is generated by the functions ''j''(''z'') and ''j''(''Nz'').<ref>{{Citation|author=Milne|first=James|title=Modular Functions and Modular Forms|url=http://www.jmilne.org/math/CourseNotes/MF.pdf|year=2010}}, Theorem 6.1.</ref>
| |
| | |
| ===Line bundles===
| |
| The situation can be profitably compared to that which arises in the search for functions on the [[projective space]] P(''V''): in that setting, one would ideally like functions ''F'' on the vector space ''V'' which are polynomial in the coordinates of ''v'' ≠ 0 in ''V'' and satisfy the equation ''F''(''cv'') = ''F''(''v'') for all non-zero ''c''. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let ''F'' be the ratio of two [[homogeneous function|homogeneous]] polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on ''c'', letting ''F''(''cv'') = ''c''<sup>''k''</sup>''F''(''v''). The solutions are then the homogeneous polynomials of degree ''k''. On the one hand, these form a finite dimensional vector space for each ''k'', and on the other, if we let ''k'' vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(''V'').
| |
| | |
| One might ask, since the homogeneous polynomials are not really functions on P(''V''), what are they, geometrically speaking? The [[algebraic geometry|algebro-geometric]] answer is that they are ''sections'' of a [[sheaf (mathematics)|sheaf]] (one could also say a [[vector bundle|line bundle]] in this case). The situation with modular forms is precisely analogous.
| |
| | |
| Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
| |
| | |
| ==Miscellaneous==
| |
| ===Entire forms===
| |
| If ''f'' is [[holomorphic]] at the cusp (has no pole at ''q'' = 0), it is called an '''entire modular form'''.
| |
| | |
| If ''f'' is meromorphic but not holomorphic at the cusp, it is called a '''non-entire modular form'''. For example, the [[j-invariant]] is a non-entire modular form of weight 0, and has a simple pole at i∞.
| |
| | |
| ===Automorphic factors and other generalizations===
| |
| Other common generalizations allow the weight ''k'' to not be an integer, and allow a multiplier <math>\varepsilon(a,b,c,d)</math> with <math>\left|\varepsilon(a,b,c,d)\right|=1</math> to appear in the transformation, so that
| |
| | |
| :<math>f\left(\frac{az+b}{cz+d}\right) = \varepsilon(a,b,c,d) (cz+d)^k f(z).</math>
| |
| | |
| Functions of the form <math>\varepsilon(a,b,c,d) (cz+d)^k</math> are known as [[automorphic factor]]s.
| |
| | |
| Functions such as the [[Dedekind eta function]], a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors. Thus, for example, let χ be a [[Dirichlet character]] mod ''N''. A modular form of weight ''k'', level ''N'' (or level group <math>\Gamma_0(N)</math>) with '''nebentypus''' χ is a [[holomorphic function]] ''f'' on the [[upper half-plane]] such that for any
| |
| | |
| :<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)</math>
| |
| | |
| and any ''z'' in the upper half-plane, we have | |
| | |
| :<math>f\left(\frac{az+b}{cz+d}\right) = \chi(d)(cz+d)^k f(z)</math>
| |
| | |
| and ''f'' is [[holomorphic]] at all the [[cusp form|cusps]]; when the form vanishes at all cusps, it is called a cusp form.
| |
| | |
| == Examples ==
| |
| The simplest examples from this point of view are the [[Eisenstein series]]. For each even integer ''k'' > 2, we define ''E''<sub>''k''</sub>(Λ) to be the sum of ''λ''<sup>−''k''</sup> over all non-zero vectors ''λ'' of Λ:
| |
| | |
| :<math>E_k(\Lambda) = \sum_{\lambda\in\Lambda-0}\lambda^{-k}.</math>
| |
| | |
| The condition ''k'' > 2 is needed for [[absolute convergence|convergence]]; for odd ''k'' there is cancellation between λ<sup>−''k''</sup> and (−λ)<sup>−''k''</sup>, so that such series are identically zero.
| |
| | |
| An [[unimodular lattice|even unimodular lattice]] ''L'' in '''R'''<sup>''n''</sup> is a lattice generated by ''n'' vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in ''L'' is an even integer. As a consequence of the [[Poisson summation formula]], the [[theta function]]
| |
| | |
| :<math>\vartheta_L(z) = \sum_{\lambda\in L}e^{\pi i \Vert\lambda\Vert^2 z} </math> | |
| | |
| is a modular form of weight ''n''/2. It is not so easy to construct even unimodular lattices, but here is one way: Let ''n'' be an integer divisible by 8 and consider all vectors ''v'' in '''R'''<sup>''n''</sup> such that 2''v'' has integer coordinates, either all even or all odd, and such that the sum of the coordinates of ''v'' is an even integer. We call this lattice ''L''<sub>''n''</sub>. When ''n'' = 8, this is the lattice generated by the roots in the [[root system]] called [[E8 (mathematics)|E<sub>8</sub>]]. Because there is only one modular form of weight 8 up to scalar multiplication,
| |
| | |
| :<math>\vartheta_{L_8\times L_8}(z) = \vartheta_{L_{16}}(z),</math>
| |
| | |
| even though the lattices L<sub>8</sub>×L<sub>8</sub> and L<sub>16</sub> are not similar. [[John Milnor]] observed that the 16-dimensional [[torus|tori]] obtained by dividing '''R'''<sup>16</sup> by these two lattices are consequently examples of [[Compact space|compact]] [[Riemannian manifold]]s which are [[isospectral]] but not [[Isometry|isometric]] (see [[Hearing the shape of a drum]].)
| |
| | |
| The [[Dedekind eta function]] is defined as
| |
| | |
| :<math>\eta(z) = q^{1/24}\prod_{n=1}^\infty (1-q^n),\ q = e^{2\pi i z}.</math>
| |
| | |
| Then the [[modular discriminant]] Δ(''z'') = η(''z'')<sup>24</sup> is a modular form of weight 12. The presence of [[24 (number)|24]] can be connected to the [[Leech lattice]], which has 24 dimensions. [[Ramanujan conjecture|A celebrated conjecture]] of [[Ramanujan]] asserted that the ''q''<sup>''p''</sup> coefficient for any prime ''p'' has absolute value ≤2''p''<sup>11/2</sup>. This was settled by [[Pierre Deligne]] as a result of his work on the [[Weil conjectures]].
| |
| | |
| 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 [[quadratic form]]s and the [[Partition function (number theory)|partition function]]. The crucial conceptual link between modular forms and number theory are furnished by the
| |
| theory of [[Hecke operator]]s, which also gives the link between the theory of modular forms and [[representation theory]].
| |
| | |
| == Generalizations ==
| |
| There are a number of other usages of the term '''''modular function''''', apart from this classical one; for example, in the theory of [[Haar measure]]s, it is a function Δ(''g'') determined by the conjugation action.
| |
| | |
| '''[[Maass forms]]''' are [[Analytic function|real-analytic]] [[eigenfunction]]s of the [[Laplacian]] but need not be [[holomorphic]]. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's [[mock theta function]]s. Groups which are not subgroups of SL(2,'''Z''') can be considered.
| |
| | |
| '''[[Hilbert modular form]]s''' are functions in ''n'' variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a [[totally real number field]].
| |
| | |
| '''[[Siegel modular form]]s''' are associated to larger [[symplectic group]]s in the same way in which the forms we have discussed are associated to SL(2,'''R'''); in other words, they are related to [[abelian variety|abelian varieties]] in the same sense that our forms (which are sometimes called ''elliptic modular forms'' to emphasize the point) are related to elliptic curves.
| |
| | |
| '''[[Jacobi form]]s''' 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 analogous to the usual theory of modular forms.
| |
| | |
| '''[[Automorphic form]]s''' extend the notion of modular forms to general [[Lie group]]s.
| |
| | |
| ==History==
| |
| The theory of modular forms was developed in three or four periods: first in connection with the theory of [[elliptic function]]s, in the first part of the nineteenth century; then by [[Felix Klein]] and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable); then by [[Erich Hecke]] from about 1925; and then in the 1960s, as the needs of number theory and the formulation of the [[modularity theorem]] in particular made it clear that modular forms are deeply implicated.
| |
| | |
| The term '''''modular form''''', as a systematic description, is usually attributed to Hecke.
| |
| | |
| == Notes ==
| |
| <references/>
| |
| | |
| == References ==
| |
| * [[Jean-Pierre Serre]], ''A Course in Arithmetic''. Graduate Texts in Mathematics 7, Springer-Verlag, New York, 1973. ''Chapter VII provides an elementary introduction to the theory of modular forms''.
| |
| * [[Tom M. Apostol]], ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0
| |
| * [[Goro Shimura]], ''Introduction to the arithmetic theory of automorphic functions''. Princeton University Press, Princeton, N.J., 1971. ''Provides a more advanced treatment.''
| |
| * Stephen Gelbart, ''Automorphic forms on adele groups''. [[Annals of Mathematics]] Studies 83, Princeton University Press, Princeton, N.J., 1975. ''Provides an introduction to modular forms from the point of view of representation theory''.
| |
| * Robert A. Rankin, ''Modular forms and functions'', (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X
| |
| * Stein's notes on Ribet's course [http://modular.fas.harvard.edu/MF.html Modular Forms and Hecke Operators]
| |
| * [[Erich Hecke]], ''Mathematische Werke'', Goettingen, Vandenhoeck & Ruprecht, 1970.
| |
| * N.P. Skoruppa, [[Don Zagier|D. Zagier]], ''Jacobi forms and a certain space of modular forms'', [[Inventiones Mathematicae]], 1988, Springer
| |
| | |
| [[Category:Modular forms|*]]
| |
| [[Category:Analytic number theory]]
| |
| [[Category:Moduli theory]]
| |
| [[Category:Special functions]]
| |