$\DeclareMathOperator\sgn{sgn}$For modular forms for $\mathrm{SL}_2(\mathbb{Z})$ there is an easy argument why the (essentially only) Eisenstein series has to be an eigenform of the Hecke operators. What I am now trying to see is that the same assertion is true for Hecke operators acting on modular forms for $\Gamma(N)$ and the Eisenstein series

$$E^{(0, 1)} (\tau) = \frac{1}{2} \sum_{\gcd(c, d)=1 ~\text{and}~ (c,d) \equiv (0,1) \mod N} {(c \tau + d)^{-k}}$$

(see for example Diamond/Shurman, A First course in Modular Forms, p.111).

So the question is: is this function an eigenform for the Hecke operators? (I have the feeling that this might only be true for some of the Hecke operators (see below))

Further remarks:

One can show that $E^{(0, 1)}$ is a modular form for $\Gamma(N)$ and there are the Hecke operators

$$T_p(f) = \sum_{j=0}^{p-1} f\big|_{\begin{pmatrix} 1 & jN \\ 0 & p \end{pmatrix}} + f\big|_{R_p \begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}}$$

(up to a constant) where $R_p$ is an arbitrary matrix in $\mathrm{SL}_2(\mathbb{Z})$ such that $R_p \equiv \begin{pmatrix} p^{-1} & 0 \\ 0 & p \end{pmatrix} \mod N$ (we always assume that $p$ is a prime with $\gcd(p,N)=1$).

One can show that the Fourier expansion of $E^{(0, 1)}$ is given by

$a_n(E^{(0, 1)}) = \sum_{a \in \mathbb{Z}_N^\times} \zeta^{a^{-1}} \sum_{m \in \mathbb{Z},~ m|n,~ n/m \equiv 0 \mod N} \sgn(m) m^{k-1} e(am)$

where $e(z) = e^{2 \pi i z}$, and $\zeta^{a} = \sum_{m \in \mathbb{Z}, m \equiv a \mod N} {\frac{\mu(m)}{m^k}}$ is the restricted zeta function associated to the Moebius function $\mu$. Let us assume for a moment that $p$ has a root $x$ mod N. Let us further assume that $\gcd(n,p)=1$, then the effect on the Fourier coefficients of the $p$-th Hecke operator is

$$a_n(T_p E^{(0,1)}) = a_{np}(E^{(0,1)}) = \sum_{a \in \mathbb{Z}_N^\times} \zeta^{a^{-1}} \sum_{m|np} (...)$$ Now we separate into cases $m|n$ and $m = pm'$ for some $m'|n$ giving $$\sum_{a \in \mathbb{Z}_N^\times} \zeta^{a^{-1}} \sum_{m|n} (\sgn(m)m^{k-1} e(am)) + \sum_{a \in \mathbb{Z}_N^\times} \zeta^{a^{-1}} \sum_{m|n} (\sgn(m)(mp)^{k-1} e(amp))$$

We substitute $a \mapsto ax$ in the first sum and $a\mapsto a/x$ in the second sum yielding

$$\sum_{a \in \mathbb{Z}_N^\times} (\zeta^{a^{-1}x^{-1}} + p^{k-1} \zeta^{a^{-1} x}) \sum_{m|n} (\sgn(m)m^{k-1} e(am))$$

Let us assume that $x \equiv 1 \mod N$ (i.e. also $p \equiv 1 \mod N$). Since the zeta function does only depend on the value modulo $N$, we have $\zeta^{a^{-1}x^{-1}} = \zeta^{a^{-1} x} = \zeta^{a^{-1}}$ so that the above is nothing else than $(1 + p^{k-1}) a_n(E^{(0,1)})$. Since the case $\gcd(n,p)\neq 1$ works similarly, this really proves that $E^{(0,1)}$ is an eigenform whenever $p \equiv 1 \mod N$. In the case I am interested in, $p$ still has a root $x$ modulo $N$ but it is not congruent to one. Does anybody see/know how to handle this case? (Is it even true?)


2 Answers 2


Yes, at primes not dividing the level $N$, Eisenstein series are Hecke eigenfunctions. A proof can be arranged in fairly elementary way, using just Sun-Ze's theorem about solving simultaneous congruences.

At some point, one may find it interesting to see that rewriting things to make Eisenstein series be functions on adele groups, which makes the independence of phenomena at different primes very clear. (One write-up of this is at http://www.math.umn.edu/~garrett/m/mfms/notes_c/ex_eis_transition.pdf)

[Edit: earlier error in URL]

It turns out to be simpler than looking at Fourier expansions.

The subtler questions about Atkin-Lehner Hecke operators at primes dividing the level were treated classically by Atkin-Lehner, and reconsidered from the viewpoint of repn theory by Casselman. It turns out that square-free level is still essentially elementary, even for cuspforms. For general level, Eisenstein series for $GL(2)$ are attached in a transparent way to principal series repns locally everywhere, but at primes $p$ with $p^2$ dividing the level, the local repn attached to a cuspform can include much-less-explicit repns, "supercuspidal" repns, only understood relatively recently, in work of Kutzko and others.

Edit: What I meant by the "essentially elementary approach" is just to prove existence of matrices with determinant $p$ for the $p$th Hecke operator, congruent to $\pmatrix{* & * \cr c_o & d_o}$ for whatever $c_o,d_o$ you want, modulo $N$ not divisible by $p$. That is, one can find representatives for the "usual" Hecke operators at $p$ meeting whatever congruence condition one wants at other primes. Thus, Hecke operators at $p$ are provably disconnected from congruence conditions (e.g., on Eisenstein series) at other primes than $p$.

Indeed, executing such an argument is a form of proving the sort of ("Strong") "Approximation" assertion that systematizes the separation of primes, and effectively turns the whole business into automorphic forms on adele groups, to our benefit.

There is extra interest in the Eisenstein series case, again, because the "local ingredients" in an Eisenstein series are all elementary, while sometimes the local ingredients for cuspforms are not: there can be a supercuspidal repn. Further, the "local parameters" specifying the local data for Eisenstein series (for $GL(2)$, especially) always fit into a simple pattern, while for (typical) cuspforms we do not know any simpler description of the pattern of Hecke eigenvalues than the cuspform itself.

Another Edit: Yes, "elementary" means "on an adele group" (which notion is 50-60+ years old).

Yes, that PDF linked-to above only explicitly mentions level "1", but once the various primes are "separated", one can immediately observe that imposition of arbitrary congruence conditions at one prime has no impact whatsoever on what happens at another.

  • $\begingroup$ Thanks for your quick response but the link does not seem to work. Could you check? Regards. $\endgroup$
    – user25160
    Jul 16, 2012 at 22:36
  • $\begingroup$ Sorry, found it. You did mean math.umn.edu/~garrett/m/mfms/notes_c/ex_eis_transition.pdf right? $\endgroup$
    – user25160
    Jul 16, 2012 at 22:38
  • $\begingroup$ Ok, i will read... it will take some time to understand what is going on... meanwhile; could you sketch what you mean by the "direct"/elementary approach (i.e. i do not see how CRT comes in and/or which congruences to consider)? Thanks. $\endgroup$
    – user25160
    Jul 16, 2012 at 22:44
  • $\begingroup$ Just two more question: I assume, "elemntary" still means "elementary in the adelic setting" so its not as elementary as "study the effect of T(p) on the Fourier coefficients and then apply CRT somehow"? Secondly: I just wanted to make sure: In the pdf-file you published, everything is done for $SL_2(\mathbb{Z})$. Do the arguments still hold in a somewhat similar way for $\Gamma(N)$? $\endgroup$
    – user25160
    Jul 17, 2012 at 22:03

What you can do to get a reasonable theory of Hecke operators for Eisenstein series is given as follows. I usually work in an adelic setting, but I have tried to translate everything to the classical setting.

  1. Work with $\Gamma_1(N)$ instead of $\Gamma(N)$. The Eisenstein theory is essentially the same. I can only explain this on the level of adeles though, and this is connected to Paul Garrett's comment "Eisenstein series factor into principal series representations."

  2. Induce by steps to $\Gamma_0(N)$, and you get a bunch of one-dimensional representations of the abelian quotient group $\Gamma_0(N) / \Gamma_1(N)$.

  3. By induction, you only need to consider the characters, which have conductor $N$ (New-forms).

  4. The pre-Hecke operator at $p^n$ is given then as the convolution by the function $\phi \in GL_2(\mathbb{Q})$ supported on $\Gamma_0(p^k) \begin{pmatrix} p^n & 0 \newline 0 & 1 \end{pmatrix} \Gamma_0(p^k)$ with $p^k || N$ twisted by a character $\mu$ of conductor $N$, i.e., $$\phi \left( \begin{pmatrix} a & b \newline c & d \end{pmatrix} \begin{pmatrix} p^n & 0 \newline 0 & 1 \end{pmatrix} \begin{pmatrix} a' & b' \newline c' & d' \end{pmatrix} \right) = \mu(aa').$$ I have twisted here implicitly by a one-dimensional character.

  5. The Hecke operator is given as convolution with the function $g \mapsto \sum\limits_{\gamma \in GL_2(\mathbb{Z}) / \Gamma(N) } \phi( \gamma g \gamma^{-1}).$ These are commutative. The eigenvalue is a constant multiple of $q^{-s} + q^{s}$ for $\Re s =0$ for Eisenstein series. For cuspforms, you may have additionally the value $0$ (if square integrable at $p$) and $-1/2 < s <1/2$ for $\mu=1$ (impossible if the Ramanujan conjecture is true).


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