# Hecke Operators

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 $k$. 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 $\sum_{n=1}^{\infty}a_{n}n^{-s}$ 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 $T_{p}$, for $p$ 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 $\Lambda$ can also be expressed as $\mathbb{Z}+\tau\mathbb{Z}$ where $\tau$ 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 $\mathbb{C}$ such that

• $f(\mathbb{Z}+\tau\mathbb{Z})$ is holomorphic as a function on the upper half-plane
• $f(\mathbb{Z}+\tau\mathbb{Z})$ is bounded as $\tau$ goes to $i\infty$
• $f(\mu\Lambda)=\mu^{-k}f(\Lambda)$ for some nonzero complex number $\mu$, and $k$ is the weight of the modular form

Now we define the Hecke operator $T_{p}$ by what it does to a modular form $f(\Lambda)$ of weight $k$ as follows:

$\displaystyle T_{p}f(\Lambda)=p^{k-1}\sum_{\Lambda'\subset \Lambda}f(\Lambda')$

where $\Lambda'$ runs over the sublattices of $\Lambda$ of index $p$. In other words, applying $T_{p}$ to a modular form gives back a modular form whose value on a lattice $\Lambda$ is the sum of the values of the original modular form on the sublattices of $\Lambda$  of index $p$, 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 $f(\tau)$ be a modular form of weight $k$ whose Fourier expansion is given by $\sum_{n=0}^{\infty}a_{i}q^{i}$, where we have adopted the convention $q=e^{2\pi i \tau}$ which is common in the theory of modular forms (hence this Fourier expansion is also known as a $q$-expansion). Then the effect of a Hecke operator $T_{p}$ is as follows:

$\displaystyle T_{p}f(\tau)=\sum_{n=0}^{\infty}(a_{pn}+p^{k-1}a_{n/p})q^{n}$

where $a_{n/p}=0$ when $p$ does not divide $n$. To see why this follows from our first definition of the Hecke operator, we note that if our lattice is given by $\mathbb{Z}+\tau\mathbb{Z}$, there are $p+1$ sublattices of index $p$: There are $p$ of these sublattices given by $p\mathbb{Z}+(j+\tau)\mathbb{Z}$ for $j$ ranging from $0$ to $p-1$, and another one given by $\mathbb{Z}+(p\tau)\mathbb{Z}$. Let us split up the Hecke operators as follows:

$\displaystyle T_{p}f(\mathbb{Z}+\tau\mathbb{Z})=p^{k-1}\sum_{j=0}^{p-1}f(p\mathbb{Z}+(j+\tau)\mathbb{Z})+p^{k-1}f(\mathbb{Z}+p\tau\mathbb{Z})=\Sigma_{1}+\Sigma_{2}$

where $\Sigma_{1}=p^{k-1}\sum_{j=0}^{p-1}f(p\mathbb{Z}+(j+\tau)\mathbb{Z})$ and $\Sigma_{2}=p^{k-1}f(\mathbb{Z}+p\tau\mathbb{Z})$. Let us focus on the former first. We have

$\displaystyle \Sigma_{1}=p^{k-1}\sum_{j=0}^{p-1}f(p\mathbb{Z}+(j+\tau)\mathbb{Z})$

But applying the third property of modular forms above, namely that $f(\mu\Lambda)=\mu^{-k}f(\Lambda)$ with $\mu=p$, we have

$\displaystyle \Sigma_{1}=p^{-1}\sum_{j=0}^{p-1}f(\mathbb{Z}+((j+\tau)/p)\mathbb{Z})$

Now our argument inside the modular forms being summed are in the usual way we write them, except that instead of $\tau$ we have $((j+\tau)/p)$, so we expand them as a Fourier series

$\displaystyle \Sigma_{1}=p^{-1}\sum_{j=0}^{p-1}\sum_{n=0}^{\infty}a_{n}e^{2\pi i n((j+\tau)/p)}$

We can switch the summations since one of them is finite

$\displaystyle \Sigma_{1}=p^{-1}\sum_{n=0}^{\infty}\sum_{j=0}^{p-1}a_{n}e^{2\pi i n((j+\tau)/p)}$

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

$\displaystyle \Sigma_{1}=p^{-1}\sum_{n=0}^{\infty}a_{pn}q^{n}$

where again $q=e^{2\pi i \tau}$. Now consider $\Sigma_{2}$. We have

$\displaystyle \Sigma_{2}=p^{k-1}f(\mathbb{Z}+p\tau\mathbb{Z})$

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

$\displaystyle \Sigma_{2}=p^{k-1}\sum_{n}a_{n}e^{2\pi i n p\tau}$

Reindexing, we have

$\displaystyle \Sigma_{2}=p^{k-1}\sum_{n}a_{n/p}q^{n}$

and adding together $\Sigma_{1}$ and $\Sigma_{2}$ 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 $T_{m}$ and $T_{n}$ 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 $T_{n}$ we have $n$ sharing a common factor with the level.

If a cusp form $f$ is an eigenvector for a Hecke operator $T_{n}$, and it is normalized, i.e. its Fourier coefficient $a_{1}$ is equal to $1$, then the corresponding eigenvalue of the Hecke operator $T_{n}$ on $f$ is precisely the Fourier coefficient $a_{n}$.

Now the Hecke operators satisfy the following multiplicativity properties:

• $T_{m}T_{n}=T_{mn}$ for $m$ and $n$ mutually prime
• $T_{p^{n}}T_{p}=T_{p^{n+1}}+p^{k-1}T_{p}$ for $p$ prime

Suppose we have an L-series $\sum_{n}a_{n}n^{-s}$. This L-series will have an Euler product if and only if the coefficients $a_{n}$ satisfy the following:

• $a_{m}a_{n}=a_{mn}$ for $m$ and $n$ mutually prime
• $a_{p^{n}}T_{p}=a_{p^{n+1}}+p^{k-1}a_{p}$ for $p$ 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 $T_{n}$, there are also other closely related operators such as the diamond operator $\langle n\rangle$ and another operator denoted $U_{p}$. 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