Table of contents |

2 As a function on elliptic curves 3 Examples 4 Generalizations 5 References |

- 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*. - 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. - 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.

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

The simplest examples from this point of view are the **Eisenstein series**: For each even integer *k*>2 we define *E*_{k}(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 **R**^{n} 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 SL_{2}(**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 SL_{2}(**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.

- For an elementary introduction to the theory of modular forms, see Chapter VII of Jean-Pierre Serre:
*A Course in Arithmetic*. Graduate Texts in Mathematics 7, Springer-Verlag, New York, 1973. - For a more advanced treatment, see Goro Shimura:
*Introduction to the arithmetic theory of automorphic functions*. Princeton University Press, Princeton, N.J., 1971. - For an introduction to modular forms from the point of view of representation theory, one might consult Stephen Gelbart:
*Automorphic forms on adele groups*. Annals of Mathematics Studies 83, Princeton University Press, Princeton, N.J., 1975.