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Modular form theory is a special case of the more general theory of 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. A '' Modular Function '' is in practical terms a modular form of weight 0; but to be strictly accurate modular functions are Meromorphic Function s rather than analytic. AS A FUNCTION ON LATTICES A modular form can be thought of as a function ''F'' from the set of Lattice s Λ in C to the set of Complex Number s which satisfies certain conditions: :(1) If we consider the lattice Λ = <α, ''z''> 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−''k''''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. When ''k'' = 0, condition 2 says that ''F'' depends only on the 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 polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on ''c'', letting ''F''(''cv'') = ''c''''k''''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 Algebro-geometric answer is that they are ''sections'' of a Sheaf (one could also say a Line Bundle in this case). The situation with modular forms is precisely analogous. AS A FUNCTION ON ELLIPTIC CURVES Every lattice Λ in C 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 α. Modular functions can be thought of as functions on the 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 :$f\backslash left(\{az+b\backslash over\; cz+d\}\; ight)\; =\; (cz+d)^k\; f(z)$ for ''a'', ''b'', ''c'', ''d'' integers with ''ad'' − ''bc'' = 1 (the Modular Group ). For example, :$f(-1/z)\; =\; F(\backslash langle\; 1,-1/z\; angle)\; =\; z^k\; F(\backslash langle\; z,-1\; angle)\; =\; z^k\; F(\backslash langle\; 1,z\; angle)\; =\; z^k\; f(z).$ Functions which satisfy the modular functional equation for all matrices in a finite index subgroup of SL2(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.) GENERAL DEFINITIONS Let $N$ be a positive integer. The Modular Group Γ0(''N'') is defined as :$\backslash Gamma\_0(N)\; =\; \backslash left\backslash \{$ \begin{pmatrix} a & b \ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : c \equiv 0 \pmod{N} ight\} Let $k$ be a positive integer. A modular form of '''weight''' $k$ with '''level''' $N$ (or level group $\backslash Gamma\_0(N)$) is a Holomorphic Function $f$ on the Upper Half-plane such that for any : and any $z$ in the Upper Half-plane , we have :$$ f\left(rac{az+b}{cz+d} ight) = (cz+d)^k f(z) and $f$ is Holomorphic at the Cusp . By "holomorphic at the cusp", it is meant that the modular form is holomorphic as $z\; ightarrow\; i\backslash infty$, or equivalently, has a Fourier Series :$f(z)=\backslash sum\_\{n=0\}^\backslash infty\; c(n)\; \backslash exp(2\backslash pi\; inz)=\; \backslash sum\_\{n=0\}^\backslash infty\; c(n)x^n$ where $x=\backslash exp(2\backslash pi\; iz)$ is the square of the Nome . Such a form, having no pole at ''x''=0, is sometimes called an ''entire modular form''. If ''c''(0)=0, then the form is called a Cusp Form (''Spitzenform'' in German). The smallest ''n'' such that $c(n)$ e 0 is called the ''order of the zero of f at $i\backslash infty$''. More general treatments allow poles at ''x''=0; thus, for example, the J-invariant is a non-entire modular form of weight 0, because it has a simple pole at $i\backslash infty$.

even though the lattices L₈×L₈ and L₁₆
are not similar. John Milnor observed that the 16-dimensional Tori obtained by dividing R¹⁶ by these two lattices are consequently examples of Compact Riemannian Manifold s which are Isospectral but not Isometric (see Hearing The Shape Of A Drum .)

The Dedekind Eta Function is defined as

:$\backslash eta(z)\; =\; q^\{1/24\}\backslash prod\_\{n=1\}^\backslash infty\; (1-q^n),\backslash \; q\; =\; e^\{2\backslash pi\; i\; z\}.$

Then the Modular Discriminant Δ(''z'')=η(''z'')²⁴ is a modular form of weight 12. A celebrated conjecture of Ramanujan asserted that the ''q''^''p'' coefficient for any prime ''p'' has absolute value ≤2''p''^11/2. 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 . 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 Real-analytic Eigenfunction s of the Laplacian but are not Holomorphic . Groups which are not subgroups of SL₂('''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 SL₂('''R'''); in other words, they are related to 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.


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 Modular Forms and Hecke Operators