0
$\begingroup$

If we consider the set of all unramified Satake parameters $S$ of all automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ as $n$ varies, do we know absolute (that is, independent of $n$) lower and upper bounds for the norm of elements of $S$?

Motivation: denoting by $\mathcal{M}_{a,b}$ the set of such automorphic representations whose set $S_{a,b}$ of Satake parameters at unramified primes $p$ fulfill $a\leq\vert\alpha_{p,j}\vert\leq b$, a proof that all the relevant Satake parameters are bounded above by a quantity independent of $n$ (and $p$ as well) would imply that if $\mathcal{M}_{a,b}$ is closed under the Rankin-Selberg convolution, then $a=b=1$.

| cite | improve this question | | | | |
$\endgroup$
  • $\begingroup$ The archimedean Satake parameters of a classical Maass form of Laplace eigenvalue $1/4+t^2$ are $\pm it$. The eigenvalues are unbounded, so the Satake parameters are also unbounded. So it is not clear what you are after. $\endgroup$ – GH from MO Mar 8 at 18:22
  • $\begingroup$ I'm interested in the Satake parameters whose norm equals $1$ assuming Ramanujan conjecture. $\endgroup$ – Sylvain JULIEN Mar 8 at 20:21
  • 1
    $\begingroup$ We don't know any upper bound for $|\alpha_{p,j}|$ that is independent of $p$. $\endgroup$ – GH from MO Mar 8 at 20:37
  • $\begingroup$ Thank you for your comment. Feel free to turn it into an answer so that I can accept it and the question get closed. $\endgroup$ – Sylvain JULIEN Mar 8 at 20:48
  • 1
    $\begingroup$ If $\alpha_{p,j}:=p^{\mu_{p,j}}$ are bounded independent of $p$ then $\Re(\mu_{p,j})\le 0$ for all $j$. At least for $PGL(n)$ this implies that $\Re(\mu_{p,j})=0$ for all $j,p$, i.e. Ramanujan. $\endgroup$ – Subhajit Jana Mar 9 at 9:27
1
$\begingroup$

We don't know any upper bound for $|\alpha_{p,j}|$ that is independent of $p$. On the other hand, we do know that each $|\alpha_{p,j}|$ is bounded by $p^{1/2}$, hence if $\pi$ is an automorphic representation whose Rankin-Selberg powers $\pi\otimes\dots\otimes\pi$ are all automorphic, then the Satake parameters of $\pi$ at every unramified prime lie on the unit circle. This simple observation is one of the motivations for the Langlands program.

| cite | improve this answer | | | | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.