Timeline for Which Langlands functoriality conjecture implies the original Ramanujan conjecture?
Current License: CC BY-SA 4.0
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| Oct 13, 2023 at 7:38 | history | edited | 2734364041 | CC BY-SA 4.0 |
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| Mar 3, 2023 at 5:03 | history | edited | 2734364041 | CC BY-SA 4.0 |
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| Mar 1, 2023 at 8:38 | history | edited | 2734364041 | CC BY-SA 4.0 |
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| Mar 1, 2023 at 8:20 | comment | added | Monty | @2734364041, Thank you very much for such clear answer. It helped me a lot. | |
| Mar 1, 2023 at 8:20 | comment | added | Monty | @DavidLoeffler, Thank you very much for such wonderful answers. I understand it. | |
| Mar 1, 2023 at 3:17 | comment | added | 2734364041 | To make the comment by @DavidLoeffler concrete (when the central character is trivial and the base field is $\mathbb{Q}$), I note that the factorization that I mention above is $\zeta(s) L(s,\mathrm{Sym}^2 \pi)$ when $k=1$, $\zeta(s)^2 L(s,\mathrm{Sym}^2\pi)^3 L(s,\mathrm{Sym}^4\pi)$ when $k=2$, and $\zeta(s)^5 L(s,\mathrm{Sym}^2\pi)^9 L(s,\mathrm{Sym}^4\pi)^5 L(s,\mathrm{Sym}^6\pi)$ when $k=3$. The formulas get complicated quickly as $k$ grows, but you'll need the first $2k$ symmetric powers to access the $2k$-th moment. | |
| Mar 1, 2023 at 2:57 | comment | added | 2734364041 | @DavidLoeffler For non-dihedral Hecke-Maass forms, we only know the automorphy up to $n=4$. But that is enough to establish the absolute convergence of the Dirichlet series for $L(s,\mathrm{Sym}^n\pi)$ for $\mathrm{Re}(s)>1$ when $n\leq 8$. | |
| Mar 1, 2023 at 2:38 | comment | added | David Loeffler | What you need is that if $\pi$ is a (non-dihedral) cuspidal automorphic representation of $GL_2$, then the $n$-th symmetric power lifting $Sym^n \pi$ exists as a cuspidal automorphic representation of $GL_{n+1}$. This is known for a few small $n$ (maybe up to $n = 7$ or something like that) but absolutely out of reach for arbitrary $n$. | |
| Mar 1, 2023 at 2:18 | vote | accept | Monty | ||
| Mar 1, 2023 at 2:18 | comment | added | Monty | @Thank you very much for such great answer! It seems that the generalized Ramanuzan conjecture follows from the existence of such isobaric sum of cuspidal representation. Is it not proved yet? | |
| Mar 1, 2023 at 1:40 | history | answered | 2734364041 | CC BY-SA 4.0 |