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:
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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.
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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!)
Best Answer
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).