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Given an automorphic representation, I would like to bound $\alpha_1^\nu(p) + \alpha_2^\nu(p)$ where the $\alpha_i$ are the Satake parameters of an automorphic form $f$ of, say, $GL_2$. So that $\alpha_1+\alpha_2$ is the Hecke eigenvalue $\lambda_f$.

In terms of coefficients or Hecke eigenvalues for "forms", I know that the Ramanujan conjecture holds by Deligne, so that those are bounded (by $1$ each) for holomorphic cusp forms. For Maass forms, Kim and Sarnak showed that $|\alpha_1(p) + \alpha_2(p)| \leqslant p^{7/64}$.

However, for an automorphic cuspidal representation of $GL_2$, what can be said about $\alpha_1^\nu(p) + \alpha_2^\nu(p)$? Do the holomorphic cusp forms and the Maass forms cover all the "cases" that can arise in such a representation? Is there another formulation of the result (or the conjecture) in case of representations?

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  • $\begingroup$ Actually, what Kim and Sarnak showed is that $|\alpha_1(p) + \alpha_2(p)| \leq p^{7/64} + p^{-7/64}$, or rather that $|\alpha_1(p)| \leq p^{7/64}$ and $|\alpha_2(p)| = |\alpha_1(p)|^{-1}$ (assuming that $p$ does not divide the level; the bounds are better when $p$ does divide the level). In any case, there is a bijective correspondence between the set of all holomorphic newforms and Maass newforms with the set of all cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A_Q})$. $\endgroup$
    – Peter Humphries
    Commented Oct 30, 2017 at 9:41
  • $\begingroup$ So one can bound the Satake parameters of any cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A_Q})$: $\alpha_1^{\nu}(p) + \alpha_2^{\nu}(p)$ is bounded by $p^{7\nu/64} + p^{-7\nu/64}$. Note that the "classical" version of this statement is about newforms, not just any cusp form. $\endgroup$
    – Peter Humphries
    Commented Oct 30, 2017 at 9:43
  • $\begingroup$ @PeterHumphries Oh, thank you for the correction and most of all for positively answering to this equivalence representations/form. Is it trivial that this correspondence holds? Is there an "explicit" version of the correspondence (from a given representation $\pi$, how can I find the/one form corresponding to it?) $\endgroup$
    – Gory
    Commented Oct 30, 2017 at 9:51
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    $\begingroup$ The correspondence is nontrivial. Basically, it comes down to showing that for every cuspidal automorphic representation $\pi = \pi_{\infty} \otimes \bigotimes_p \pi_p$ of $\mathrm{GL}_2(\mathbb{A_Q})$, there is a distinguished vector $\phi_p \in \pi_p$ for every prime (and similarly a choice of $\phi_{\infty}$) such that $\phi_{\infty} \otimes \bigotimes_p \phi_p$ is the adèlic lift of a classical newform. Conversely, a classical newform gives rise to a cuspidal automorphic representation in the same way. $\endgroup$
    – Peter Humphries
    Commented Oct 30, 2017 at 10:50
  • $\begingroup$ This is essentially a result of Casselman, "On Some Results of Atkin and Lehner", and is explained in more detail in Gelbart's book, section 5.C, and the second volume of Goldfeld and Hundley's book, section 13.8. $\endgroup$
    – Peter Humphries
    Commented Oct 30, 2017 at 10:52

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