# Potential modularity and the Ramanujan conjecture

A little background: Let $f(z)=\sum_{n=1}^{\infty} a(n) e^{2\pi i nz}$ be a classical holomorphic cuspidal eigenform on $\Gamma_1(N)$, of weight $k \geq 2$ normalized with $a(1)=1$. The Ramanujan conjecture is the assertion that $|a(n)| \leq n^{\frac{k-1}{2}}d(n)$. This is a theorem, due to several people, but the main steps in its proof are the following two:

1. Show that $f$ can be "realized" in the etale cohomology of a suitable variety. In fact $f$ can be found in $H^{k-1}$ of a $k-2$-fold self-product of the universal elliptic curve over $X_1(N)$. This step is due to Eichler, Igusa, Kuga/Sato, and Deligne (who showed how to desingularize the variety). This reduces the Ramanujan conjecture to the Weil conjectures.

2. Prove the Weil conjectures. This is of course due originally to Deligne, although if I understand correctly, there are now several other proofs (a p-adic proof by Kedlaya, etc.)

Now, there is another approach to the Ramanujan conjecture, essentially through Langlands functoriality. In particular, if we knew for all $n$ that the L-functions $L(s,\mathrm{sym}^n f)$ were holomorphic and nonvanishing in the halfplane $Re(s)\geq 1$, the Ramanujan conjecture would follow. This observation is, I believe, due to Serre Langlands.

Nowadays these analytic properties are known, by the potential modularity and modularity lifting results of Barnet-Lamb/Clozel/Gee/Geraghty/Harris/Shephard-Barron/Taylor. However, the proofs of potential modularity and modularity lifting seem to utilize the Ramanujan conjecture. Hence my question:

Can the recent proofs of potential modularity for symmetric powers of $GL2$ modular forms be modified so they do not assume the Ramanujan conjecture, hence giving a new proof of the Ramanujan conjecture?

(If I got any of the history or attributions wrong, please correct me!)

• I think that Langlands claims the deduction of Ramanujan from functoriality for symmetric powers to be his own. One could also note that the same argument appears in Deligne's proof of the Riemann hypothesis. May 10, 2010 at 15:48
• "...another approach to the Ramanujan conjecture, essentially through Langlands functoriality. ... if we knew for all $n$ that the $L$-functions $L(sym^n f)$ were holomorphic and nonvanishing in the halfplane $Re(s)\ge 1$, the Ramanujan conjecture [follows]..." I don't know about functorality - a variant of the weaker claim is in a paper of Ogg springerlink.com/content/k24u6q718u3v770w Ogg, A. P. A remark on the Sato-Tate conjecture. Invent. Math. 9 1969/1970 198--200. Hs shows a holomorphic continuation past 1/2-line (follows from functorality?) implies no zeros on 1-line. May 11, 2010 at 2:42
• Actually, I am now confused by your normalization of the half-line. Are the coefficients bounded by 1, so that the symmetry line is $Re(s)=1/2$, and the edge is $Re(s)=1$. This is normalization that Ogg uses. May 11, 2010 at 2:48
• Junkie, yes, that is the normalization I am using. May 11, 2010 at 13:06
• ...was I wrong for thinking that a new, global, proof of the Ramanujan conjecture would be interesting?! May 27, 2010 at 17:04

The claimed analytic properties of $L(s,sym^n f)$ surely imply Sato-Tate, and this is due to Serre, but I don't see how they imply Ramanujan. What Langlands observed is that the automorphy of $L(s,sym^n f)$ on GL(n+1) (assumed for all $n$) implies Ramanujan, but of course analytically this is a much stronger assumption (as far as we know), e.g. all these $L$-functions and most of their twists should be entire with a functional equation (cf. the converse theorems of Cogdell and Piatetski-Shapiro).

• On the other hand, the following is clear: if in a fixed right half-plane each $L(s,sym^n f)$ is given by an absolutely convergent Euler-product, then Ramanujan holds. This is because the assumption implies that all the Euler factors are holomorphic in a fixed right half-plane which is equivalent to Ramanujan. It is also worth mentioning that an elaboration of this idea (pole of an Euler factor or gamma factor forces a zero of the remaining partial $L$-function), originally due to Serre, lead to nontrivial bounds for cusp forms on GL(n), see the work of Luo-Rudnick-Sarnak. Jan 2, 2011 at 12:55
• To deduce Ramanujan: You need to write $L(s,\mathrm{sym}^n(f)\otimes \mathrm{sym}^n(f))$ as a product of $L(s,\mathrm{sym}^j(f))$'s for various j's (use Clebsch-Gordon) and then appeal to the fact that a Dirichlet series with nonnegative coefficients converges absolutely up to its first pole. Jan 2, 2011 at 22:12
• That's a very good point, thanks! Does it also work when the nebentypus of $f$ is nontrivial? Jan 2, 2011 at 23:30
• Sure, just argue with $L(s,\mathrm{sym}^n(f) \otimes \mathrm{sym}^n(\overline{f}))$ instead, and use the fact that $\overline{f}=f\otimes \overline{\chi}$ where $\chi$ is the nebentypus. Jan 17, 2011 at 22:18
• OK, but then we need the relevant properties for $L(s,\text{sym}^n f\otimes\chi^m)$ for all $n$ and many $m$'s, right? I am not familiar with Clebsch-Gordon, and I don't know if the recent work on potential modularity covers these character twists as well. Jan 17, 2011 at 22:33

Leaving aside the original question, it seems that this new paper https://arxiv.org/abs/1812.09999 might be relevant. They claim to prove the Ramanujan conjecture for a class of automorphic forms as a consequence of automorphy results for symmetric powers and not (directly) by using Deligne's theorem, since the corresponding Galois representations in this case apparently don't come in any obvious way from geometry.

• From the abstract, it appears that only weight zero case of the Ramanujan conjecture is treated. Dec 27, 2018 at 21:34
• @VictorProtsak, I did say <i>for a class of automorphic forms</i>. Did I say something misleading? I confess your comment has gone over my head. Dec 27, 2018 at 23:58
• Well, the correspodence with the original question is far from obvious. The setting of this paper, involving ${\rm GL}_n(F)$ and a related group $\widetilde{G}$ for an imaginary CM field $F$, is rather technical. But if I understand correctly, the "weight zero" condition for $n=2$ and $F$ totally real would correspond to Hilbert modular forms of weight 2 at the archimedean places. Ramanujan conjecture in this context asserts that all non-archimedean local components are tempered representations of the corresponding ${\rm GL}_2(F_v)$. Dec 28, 2018 at 5:17
• @VictorProtsak, I am still baffled by your comments. I'm sure the OP knows what the Ramanujan conjecture for GL(2) is. Given the question is about deducing Ramanujan from automorphy of symmetric powers and this is exactly what the linked paper claims to do for the CM analog of weight two forms, I find your comments perplexing. But I think we are talking at cross purposes. It feels like you are complaining about something (I'm not sure), but perhaps you are just trying to add clarifying remarks. Dec 28, 2018 at 6:11
• I'm sorry you feel that way. But given the (honestly confusing) level of hostility in all of your comments, maybe it's for the best. Dec 28, 2018 at 16:16