I have a question regarding a result stated without proof by Henryk Iwaniec in his "Topics in Classical Automorphic Forms" Chapter 12, pp.222.

Specifically, given a real quadratic field $\mathbb{Q}(\sqrt{D})$, the result characterizes the eigenspaces of the Atkin-Lehner-Li involution $W_M$ for positive $M|D$ (modulo the Hecke congruence group of level $D$, i.e., $\Gamma_0(D)$) acting on the $\chi_D$ eigenspace of modular forms of weight $k$ w.r.t. $\Gamma_0(D)$ via the slash operator (defined on pp.41 of the same book). Here, $\chi_D$ is a Dirichlet character modulo $D$.The characterization goes something like this:

If $\delta_p$ is an eigenvalue of the aforementioned operator for some modular form $f \in \mathcal{M}_k(\Gamma_0(D), \chi_D)$, then its eigenspace is

$\mathcal{M}_k^{(\delta)}$ = { $f \in \mathcal{M}_k : f(z) = \Sigma a(n)e(nz)$ with $a(n) = 0$ if $\left(\frac{P}{n}\right) = -\delta_p$ for all $p|D$ }

Here, $\left(\frac{P}{n}\right)$ is the Kronecker symbol associated to the prime divisor $p$ of $D$ and $P$ is the corresponding prime discriminant.

My question is, as expected, how can we prove this? The natural thing to do would be to look at how the operator above acts on the Fourier expansion of $f$, and then, in an ideal world, this would allow us to characterize the eigenspaces. However, this is easier said than done since the action of the involution $W_M$ on $f$ is not pretty to look at, which is why I think, people sometimes study it in conjunction with some other operator, so as to potentially simplify things.

To this end, I looked around in the literature and the closest thing I could find were results of a similar nature proved by W. Kohnen in his papers "Newforms of Half-Integral Weight" pp. 40 -41 and "Fourier Coefficients of Modular Forms of Half-Integral Weight" pp.243. The problem is that, in the first paper, probably since he is working with modular forms of half-integral weight, Kohnen defines his involution differently. Basically, his involution is a product of two matrices with one matrix corresponding to the Hecke operator $T_p$ when $p$ divides the level $D$ and the other matrix, more or less, corresponding to what Iwaniec has defined in his book. Also, the order in which the two matrices are being multiplied is important since they do not commute. So, after trying to mimic his calculations, I reached a dead-end.

His second paper gives another proof, which seems to work, but it requires some rather heavy technique and the eigenspace result is a mere by-product after some huffing and puffing.

Does anyone know of any other way to prove the above result?

Thanks in advance.

P.S: Sorry for the long-windedness. I just didn't know how much detail to leave in. Plus, this is my first time asking a question on MO.

  • 1
    $\begingroup$ Could you clarify the notation slightly? E.g. what is $p$ in the subscript $\delta_p$, and what is its relationship to $M$? Also, is $\chi_D$ the primitive quadratic character of conductor $D$, or something more general? And what is the role of $\mathbb Q(\sqrt{D})$? $\endgroup$
    – Emerton
    Feb 22 '11 at 3:47
  • $\begingroup$ Sorry about the confusion. I'll try to be more precise. For every prime $p$ dividing $D$, we define the Atkin-Lehner involution $W_p$ as $f|\frac{1}{\sqrt{p}} (p \,a;D \,pb)$ with the matrix entries satisfying $p^2b-Da = p$. So, then $\delta_p$ is the eigenvalue associated to the action of $W_p$. From what I've read in Iwaniec, it looks like $\chi_D$ is just a character mod $D$. So, basically, the result states that the $\pm$-eigenspace of $W_p$ consists of functions whose n$^{th}$ Fourier coefficient vanishes for $\left(\frac{P}{n}\right) = \mp 1$ for all $p|D$. $\endgroup$
    – mpt-cloud
    Feb 22 '11 at 4:59
  • $\begingroup$ So, $M|D$ doesn't enter into the picture. It was just used by Iwaniec for defining the involution. And as for $\mathbb{Q}(\sqrt{D})$, Iwaniec later on in the page, uses this result to establish a connection with some genus theory of $\mathbb{Q}(\sqrt{D})$. It's just that we are defining the involution w.r.t the divisors of $D$. $\endgroup$
    – mpt-cloud
    Feb 22 '11 at 5:03

Iwaniec says on page 221: "Much of the material presented below can be found in [Asa], [He2] and [Miy]." The first reference is available for free online and seems to contain what you need. See (12) here.


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