Modular form 286000 221701720 2008-06-25T18:19:17Z Dratman 44578 Added a wiki link In [[mathematics]], a '''modular form''' is a (complex) [[analytic function]] on the [[upper half-plane]] satisfying a certain kind of [[functional equation]] and 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 of weight 0: it is ''invariant'' under the [[modular group]], instead of transforming in a prescribed way, and is thus a function on the modular region (rather than a section of a [[line bundle]]). 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. ==As a function on lattices== A modular form can be thought of as a function ''F'' from the set of [[period lattice|lattice]]s &Lambda; 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 &alpha; and a variable ''z'', then ''F''(&Lambda;) is an [[analytic function]] of ''z''. :(2) If &alpha; is a non-zero [[complex number]] and &alpha;&Lambda; is the lattice obtained by multiplying each element of &Lambda; by &alpha;, then ''F''(&alpha;&Lambda;) = &alpha;<sup>&minus;''k''</sup>''F''(&Lambda;) where ''k'' is a constant (typically a positive integer) called the '''weight''' of the form. :(3) The [[absolute value]] of ''F''(&Lambda;) remains bounded above as long as the absolute value of the smallest non-zero element in &Lambda; is bounded away from 0. When ''k'' = 0, condition 2 says that ''F'' depends only on the [[similarity (mathematics)|similarity]] class of the lattice. This is a very important special case, but the only modular forms of weight 0 are the constants. If we eliminate condition 3 and allow the function to have poles, then weight 0 examples exist: they are called ''modular functions''. 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''&ne; 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. ==As a function on elliptic curves== Every lattice &Lambda; in '''C''' determines an [[elliptic curve]] '''C'''/&Lambda; over '''C'''; two lattices determine [[isomorphic]] elliptic curves if and only if one is obtained from the other by multiplying by some &alpha;. Modular functions can be thought of as functions on the [[moduli problem|moduli space]] of isomorphism classes of complex elliptic curves. For example, the [[j-invariant]] of an elliptic curve, regarded as a function on the set of all elliptic curves, is modular. Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves. To convert a modular form ''F'' into a function of a single complex variable is easy. Let ''z'' = ''x'' + ''iy'', where ''y'' > 0, and let ''f''(''z'') = ''F''(<1, ''z''>). (We cannot allow ''y'' = 0 because then 1 and ''z'' will not generate a lattice, so we restrict attention to the case that ''y'' is positive.) Condition 2 on ''F'' now becomes the [[functional equation]] :<math>f\left({az+b\over cz+d}\right) = (cz+d)^k f(z)</math> for ''a'', ''b'', ''c'', ''d'' integers with ''ad'' &minus; ''bc'' = 1 (the [[modular group Gamma|modular group]]). For example, :<math>f(-1/z) = F(\langle 1,-1/z\rangle) = z^k F(\langle z,-1\rangle) = z^k F(\langle 1,z\rangle) = z^k f(z).</math> Functions which satisfy the modular functional equation for all matrices in a finite index subgroup of SL<sub>2</sub>('''Z''') are also counted as modular, usually with a qualifier indicating the group. Thus modular forms of ''level N'' (see below) satisfy the functional equation for matrices congruent to the identity matrix modulo ''N'' (often in fact for a larger group given by (mod ''N'') conditions on the matrix entries.) ==Modular functions== In [[mathematics]], '''modular functions''' are certain kinds of [[function (mathematics)|mathematical function]]s mapping [[complex number]]s to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details. Formally, a function ''f'' is called '''modular''' or a '''modular function''' [[iff]] it satisfies the following properties: # ''f'' is [[meromorphic function|meromorphic]] in the open [[upper half-plane]] ''H''. # For every [[matrix (mathematics)|matrix]] ''M'' in the [[modular group Gamma|modular group &Gamma;]], ''f''(''M''&tau;) = ''f''(&tau;). # The [[Fourier series]] of ''f'' has the form ::<math>f(\tau) = \sum_{n=-m}^\infty a(n) e^{2i\pi n\tau}.</math> It is bounded below; it is a [[Laurent polynomial]] in <math>e^{2i\pi \tau}</math>, so it is meromorphic at the cusp. It can be shown that every modular function can be expressed as a [[rational function]] of [[Klein's absolute invariant]] ''j''(&tau;), and that every rational function of ''j''(&tau;) is a modular function; furthermore, all [[analytic function|analytic]] modular functions are [[modular form]]s, although the converse does not hold. 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>&Gamma;</sub>. === Other uses === 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 &Delta;(''g'') determined by the conjugation action. ==General definitions== Let <math>N</math> be a positive integer. The [[congruence subgroup|modular group]] &Gamma;<sub>0</sub>(''N'') is defined as :<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> Let <math>k</math> be a positive integer. An '''modular form''' of '''weight''' <math>k</math> with '''level''' <math>N</math> (or level group <math>\Gamma_0(N)</math>) is a [[holomorphic function]] <math>f</math> 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 <math>z</math> in the [[upper half-plane]], we have :<math> f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z) </math> and <math>f</math> is [[meromorphic]] at the [[cusp]]. By "meromorphic at the cusp", it is meant that the modular form is meromorphic as <math>z\rightarrow i\infty</math>. Note that <math>f\left(z+1\right) = f(z)</math>, so modular forms are periodic, with period 1, and thus have a Fourier series. ===''q''-expansion=== The '''''q''-expansion'''<ref>[http://www.msri.org/about/computing/docs/magma/html/text600.htm Elliptic and Modular Functions<!-- Bot generated title -->]</ref> of a modular form is the Laurent series at the cusp. Equivalently, the [[Fourier series]], written as a [[Laurent series]] in terms of <math>q=\exp(2\pi iz)</math> (the square of the [[nome (mathematics)|nome]]). Since <math>\exp</math> is non-vanishing, <math>q \neq 0</math> on the complex plane, but in the limit, <math>\exp(w) \to 0</math> as <math>w \to -\infty</math> (along the negative real axis), so <math>q \to 0</math> as <math>2\pi iz \to -\infty</math>, so as <math>z \to i\infty</math> (along the positive imaginary axis) &mdash; thus the ''q''-expansion is the Laurent series expansion at the cusp. "Meromorphic at the cusp" means that only finitely many negative Fourier coefficients are non-zero, so the ''q''-expansion is bounded below, and meromorphic at <math>q=0</math>: :<math>f(z)=\sum_{n=-m}^\infty c_n \exp(2\pi inz) = \sum_{n=-m}^\infty c_n q^n</math> The coefficients <math>c_n</math> are the Fourier coefficients of <math>f</math>, and the number ''m'' is the '''order of the pole of ''f'' at <math>i\infty</math>'''. ===Entire forms, cusp forms=== If <math>f</math> is [[holomorphic]] at the cusp (has no pole at <math>q=0</math>), it is called an '''entire modular form'''. If <math>f</math> is meromorphic but not holomorphic at the cusp, it is called '''non-entire modular form'''. For example, the [[j-invariant]] is a non-entire modular form of weight 0, and has a simple pole at <math>i\infty</math>. If <math>f</math> is entire and vanishes at <math>q=0</math> (so <math>c_0=0</math>), the form is called a [[cusp form]] (''Spitzenform'' in German). The smallest ''n'' such that <math>c_n \ne 0</math> is the '''order of the zero of ''f'' at <math>i\infty</math>'''. ===Automorphic factors and other generalizations=== Other common generalizations allow the weight ''k'' to not be an integer, and allow a multiplier <math>\epsilon(a,b,c,d)</math> with <math>\left|\epsilon(a,b,c,d)\right|=1</math> to appear in the transformation, so that :<math> f\left(\frac{az+b}{cz+d}\right) = \epsilon(a,b,c,d) (cz+d)^k f(z). </math> Functions of the form <math>\epsilon(a,b,c,d) (cz+d)^k</math> are known as [[automorphic factor]]s. By allowing automorphic factors, functions such as the [[Dedekind eta function]] may be encompassed by the theory, being a modular form of weight 1/2. Thus, for example, let <math>\chi</math> be a [[Dirichlet character]] mod <math>N</math>. A modular form of weight <math>k</math>, level <math>N</math> (or level group <math>\Gamma_0(N)</math>) with '''nebentypus''' <math>\chi</math> is a [[holomorphic function]] <math>f</math> 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 <math>z</math> 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 <math>f</math> is [[holomorphic]] at the [[cusp]]. Sometimes the convention :<math>\chi^{-1}(d) (cz+d)^k f(z)</math> is used for the [[right hand side]] of the above equation. == 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>(&Lambda;) to be the sum of &lambda;<sup>&minus;''k''</sup> over all non-zero vectors &lambda; of &Lambda;: :<math>E_k(\Lambda) = \sum_{\lambda\in\Lambda-0}\lambda^{-k}.</math> The condition ''k'' > 2 is needed for convergence; the condition that ''k'' is even prevents &lambda;<sup>&minus;''k''</sup> from cancelling with (&minus;&lambda;)<sup>&minus;''k''</sup>. 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>&times;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]] [[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]] &Delta;(''z'')=&eta;(''z'')<sup>24</sup> is a modular form of weight 12. A celebrated conjecture of [[Ramanujan]] asserted that the ''q''<sup>''p''</sup> coefficient for any prime ''p'' has absolute value &le;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 various notions of modular form more general than the one discussed above. The assumption of complex analyticity can be dropped; '''Maass forms''' are [[Analytic function|real-analytic]] [[eigenfunction]]s of the [[Laplacian]] but are not [[holomorphic]]. Groups which are not subgroups of SL<sub>2</sub>('''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&times;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<sub>2</sub>('''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. [[Automorphic form]]s extend the notion of modular forms to general [[Lie group]]s. ==History== It was developed, historically speaking, in three or four periods of development: in connection with the theory of [[elliptic function]]s, in the first part of the [[nineteenth century]]; by [[Felix Klein]] and others towards the end of the nineteenth century, as the automorphic form concept was understood (for one variable); by [[Erich Hecke]] from about 1925; and 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. Curiously, [[G. H. Hardy]] is said to have banned it in his circle of students; for example, the deep studies made on the particular [[cusp form]] highlighted by [[Srinivasa Ramanujan]] often do not use the modern term. == 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] [[Category:Modular forms|*]] [[Category:Analytic number theory]] [[Category:Moduli theory]] [[Category:Special functions]] [[de:Modulform]] [[fr:Forme modulaire]] [[it:Forme modulari]] [[he:תבנית מודולרית]] [[ru:Модулярная функция]] [[fi:Modulaarinen muoto]] [[zh:模形式]]