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2 As a function on elliptic curves 3 Examples 4 Generalizations 5 References |
At the simplest level, a modular form can be thought of as a function F from the set of lattices Λ in C to the set of complex numbers which satisfies the following conditions:
As a function on lattices
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 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 polynomials of the same degree. Or we can stick with polynomials and loosen the dependence on c, letting F(cv) = ckF(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.
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
As a function on elliptic curves
for a, b, c, d integers with ad-bc=1 (the modular group). For example,
The simplest examples from this point of view are the Eisenstein series: For each even integer k>2 we define Ek(Λ) to be the sum of λ-k over all non-zero vectors λ of Λ (the condition k>2 is needed for convergence and the condition k is even to prevent λ-k from cancelling with (-λ)-k and producing the 0 form.)
A even unimodular lattice L in Rn 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
Let
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 forms and the partition function. The crucial conceptual link between modular forms and number theory are furnished by the
theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.
There are various notions of modular form more general than the one discussed above. The assumption of analyticity can be dropped; Maass forms are eigenfunctions of the Laplacian but are not analytic. Groups which are not subgroups of SL2(Z) can be considered. Hilbert modular forms 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 forms are associated to larger symplectic groups 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.
Generalizations
References