Timeline for Ramanujan conjecture in terms of representations
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2017 at 15:01 | comment | added | Gory | @PeterHumphries Many thanks for the references and providing a quick sketch! | |
| Oct 30, 2017 at 10:52 | comment | added | Peter Humphries | 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. | |
| Oct 30, 2017 at 10:50 | comment | added | Peter Humphries | 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. | |
| Oct 30, 2017 at 9:51 | comment | added | Gory | @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?) | |
| Oct 30, 2017 at 9:43 | comment | added | Peter Humphries | 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. | |
| Oct 30, 2017 at 9:41 | comment | added | Peter Humphries | 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 $\mathrm{GL}_2(\mathbb{A_Q})$. | |
| Oct 30, 2017 at 8:40 | history | asked | Gory | CC BY-SA 3.0 |