A **Hecke operator** is a certain kind of linear transformation on the space of modular forms or cusp forms (see also Modular Forms) of a certain fixed weight . They were originally used (and now named after) Erich Hecke, who used them to study L-functions (see also Zeta Functions and L-Functions) and in particular to determine the conditions for whether an L-series has an Euler product. Together with the meromorphic continuation and the functional equation, these are the important properties of the Riemann zeta function, which L-functions are supposed to be generalizations of. Hecke’s study was inspired by the work of Bernhard Riemann on the zeta function.

An example of a Hecke operator is the one commonly denoted , for a prime number. To understand it conceptually, we must take the view of modular forms as functions on lattices. This is equivalent to the definition of modular forms as functions on the upper half-plane, if we recall that a lattice can also be expressed as where is a complex number in the upper half-plane (see also The Moduli Space of Elliptic Curves).

In this view a modular form is a function on the space of lattices on such that

- is holomorphic as a function on the upper half-plane
- is bounded as goes to
- for some nonzero complex number , and is the
**weight**of the modular form

Now we define the Hecke operator by what it does to a modular form of weight as follows:

where runs over the sublattices of of index . In other words, applying to a modular form gives back a modular form whose value on a lattice is the sum of the values of the original modular form on the sublattices of of index , times some factor that depends on the Hecke operator and the weight of the modular form.

Hecke operators are also often defined via their effect on the Fourier expansion of a modular form. Let be a modular form of weight whose Fourier expansion is given by , where we have adopted the convention which is common in the theory of modular forms (hence this Fourier expansion is also known as a **-expansion**). Then the effect of a Hecke operator is as follows:

where when does not divide . To see why this follows from our first definition of the Hecke operator, we note that if our lattice is given by , there are sublattices of index : There are of these sublattices given by for ranging from to , and another one given by . Let us split up the Hecke operators as follows:

where and . Let us focus on the former first. We have

But applying the third property of modular forms above, namely that with , we have

Now our argument inside the modular forms being summed are in the usual way we write them, except that instead of we have , so we expand them as a Fourier series

We can switch the summations since one of them is finite

The inner sum over is zero unless divides , in which case the sum is equal to . This gives us

where again . Now consider . We have

Expanding the right hand side into a Fourier series, we have

Reindexing, we have

and adding together and gives us our result.

The Hecke operators can be defined not only for prime numbers, but for all natural numbers, and any two Hecke operators and commute with each other. They preserve the weight of a modular form, and take cusp forms to cusp forms (this can be seen via their effect on the Fourier series). We can also define Hecke operators for modular forms with **level structure**, but it is more complicated and has some subtleties when for the Hecke operator we have sharing a common factor with the level.

If a cusp form is an eigenvector for a Hecke operator , and it is **normalized**, i.e. its Fourier coefficient is equal to , then the corresponding eigenvalue of the Hecke operator on is precisely the Fourier coefficient .

Now the Hecke operators satisfy the following multiplicativity properties:

- for and mutually prime
- for prime

Suppose we have an L-series . This L-series will have an Euler product if and only if the coefficients satisfy the following:

- for and mutually prime
- for prime

Given that the Fourier coefficients of a normalized **Hecke eigenform** (a normalized cusp form that is a simultaneous eigenvector for *all* the Hecke operators) are the eigenvalues of the Hecke operators, we see that the L-series of a normalized Hecke eigenform has an Euler product.

In addition to the Hecke operators , there are also other closely related operators such as the **diamond operator** and another operator denoted . These and more on Hecke operators, such as other ways to define them with **double coset operators** or **Hecke correspondences** will hopefully be discussed in future posts.

References:

Hecke Operator on Wikipedia

Modular Forms by Andrew Snowden

Congruences between Modular Forms by Frank Calegari

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman