Suppose I have $f\in S_2(\Gamma_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as defined here.

I know the Atkin-Lehner operator commutes with the Hecke operators away from the level, but it seems like there should be some reason to care about it beyond just that it's something else one can do once one has an eigenform. I would guess that it relates in some way to the ramification at $p$ of the Galois representation associated to $f$, but I've had a look through some literature and found nothing beyond the definition of the operator. I'm sure it wouldn't have as much time dedicated to it if it weren't in some way important! Just what that importance is, though, is unclear to me.

I would also be interested in what the interpretation is for Bianchi and Hilbert modular forms, although I know this will probably be harder to specify due to the technical complications there are in these settings. Thanks!

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    $\begingroup$ The associated Galois representation into $GL_2$ will have inertia acting as a unipotent matrix $\begin{pmatrix} 1 &1 \\ 0 & 1\end{pmatrix}$ and Frobenius (well-defined modulo inertia) acting as $\begin{pmatrix} a & * \\ 0 & ap \end{pmatrix}$ where $a^2 =1$ and $a$ is the Atkin-Lehner eigenvalue. (I may be off by a sign or something here, but that's the basic idea). So it's not (just) inertia, but Frobenius, that determines this eigenvalue. $\endgroup$
    – Will Sawin
    Mar 31 at 18:07
  • $\begingroup$ This is interesting... do you have a reference for it? I'd wondered for a while whether there was some alternate definition of Atkin-Lehner for mod 2 representations, as the eigenvalue is always 1, but this makes it sound as though there is not? $\endgroup$
    – ribbcastle
    Mar 31 at 18:25
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    $\begingroup$ It tells you what the local root number is and what $a_p$ is. As Will says, this is more than you get from the local inertial type. If $p$ sharply divides $N$, it tells you exactly what the local representation is. $\endgroup$
    – Kimball
    Mar 31 at 20:15

Write the Fourier expansion of the newform $f\in S_2(\Gamma_0(N))$ as

$\displaystyle\sum_{n=1}^{\infty}\lambda_f(n)n^{1/2}e^{2\pi inz}$

so that the Deligne bound is $|\lambda_f(n)|\leq d(n)$, where $d(n)$ is the divisor function. If for a prime $p|N$ we let $\lambda_p$ denote the eigenvalue of the Atkin-Lehner operator, then

$\displaystyle\lambda_f(p) = \begin{cases} -\lambda_p p^{-1/2}&\mbox{if $p^2\nmid N$,}\\ 0&\mbox{otherwise.} \end{cases}$

If you let $f\in S_k(\Gamma_0(N))$ be a newform of even integral weight $k\geq 2$, then all of this carries through, except that we replace $n^{1/2}$ with $n^{(k-1)/2}$ in the Fourier expansion. This can be found in Chapter 2, Section 5 of Ken Ono's book "The web of modularity."

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    $\begingroup$ The statement that $\lambda_{f}(p) = -\lambda_{p} p^{-1/2}$ is only true if $p^{2} \nmid N$. If $p^{2} \mid N$, then $\lambda_{f}(p) = 0$. $\endgroup$ Mar 31 at 18:17
  • $\begingroup$ @JeremyRouse Edited accordingly. Thanks! $\endgroup$
    – 2734364041
    Mar 31 at 18:23

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