For what automorphic representations is Ramanujan-Petersson known?

Question

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that

If an automorphic representation on $GL(2)$ is ramified at a finite place, then it is supercuspidal or a twist of a ramified principal series. These two kind of automorphic representations satisfy the Ramanujan-Petersson conjecture.

Is that true or have I misunderstood? I am desperately searching for a clear reference of this fact. More generally, in what cases do we know that the Ramanujan-Petersson conjecture is true?

Answer 1

Denote by $\mathbb{A}$ the adeles over a number field $F$. An cuspidal automorphic representation $\pi$ of $\text{GL}_2(\mathbb{A})$ now factors by the tensor product theorem as $\pi\cong \prod_v\pi_v$, where $\pi_v$ are smooth irreducible unitary representations of $\text{GL}_2(F_v)$.

The Ramanujan-Petersson conjecture now states that the representations $\pi_v$ are tempered for all finite places $v$. (One could include the archimedean places, which then amounts to a generalisation of Selbergs Eigenvalue conjecture.) As far as I know this conjeture is only known for Hilbert modular forms with some mild technical conditions on the weight (see for example Don Blasius: Hilbert Modular Forms and the Ramanujan Conjecture). In general there are only results towards the Ramanujan-Petersson conjecture available. See Blomer, Brumley: On the Ramanujan Conjecture over number fields.

What you might have heard is the following. Say $\pi$ is ramified at a finite place $v$, then $\pi_v$ is either supercuspidal, a (unitary) twist of Steinberg or a unitary principal series representation. It is now true that supercuspidal representations and (unitary) twists of Steinberg are tempered by nature. The same is true for most unitary principal series representations with the following exception. Given a non-tempered unramified representation $\sigma_v$ of $\text{GL}_2(F_v)$. Then a unitary twist $\chi_v\otimes \sigma_v$ is also non-tempered. Also the latter example can theoretically occur in nature as long as the Ramanujan-Petersson conjecture is not known. (Take a cuspidal automorphic representation $\pi$ violating the it at a unramified finite place $v$ and twist it by a suitable Hecke-character.)

To summarise: If $\pi$ ramifies at a finite place $v$, then $\pi_v$ is tempered as long as it is not a twist of an unramified not-tempered representation. One might abuse language and say $\pi$ satisfies the Ramujan-Petersson conjecture at $v$. Howeever this is far from knowing the full conjecture for $\pi$ globally.

Answer 2

From the way you phrase your question, I suspect that you misunderstand something. Being supercuspidal is a local condition. If $\pi$ is an automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$, then $\pi_p$ might be supercuspidal for some ramified primes $p$, and non-supercuspidal for some other ramified primes $p$. The Ramanujan conjecture is a global statement, but it can be phrased as: $\pi_p$ is tempered for any prime $p$.

I will try to summarize for you what is known from local theory. For more details see Schmidt: Some remarks on local newforms for GL(2). Let $\omega$ be the central character of $\pi$, and assume that it is unitary (this is a simple normalization condition). For any prime $p$, there are five possibilities:

• $\pi_p$ is induced from two unramified characters of $\mathbb{Q}_p^\times$;
• $\pi_p$ is induced from one ramified and one unramified character of $\mathbb{Q}_p^\times$;
• $\pi_p$ is induced from two ramified characters of $\mathbb{Q}_p^\times$;
• $\pi_p$ is a Steinberg representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ twisted by a character of $\mathbb{Q}_p^\times$;
• $\pi_p$ is a supercuspidal representation of $\mathrm{GL}_2(\mathbb{Q}_p)$.

In the first case, $\pi_p$ is unramified, and $\pi_p$ is tempered if and only if the inducing characters of $\mathbb{Q}_p^\times$ are unitary. This is known when $\pi_\infty$ belongs to the discrete series (Deligne's famous theorem), but unknown otherwise.

In the second case, $\omega_p$ is ramified (hence $\pi_p$ is also ramified), and $\pi_p$ is tempered if and only if the inducing unramified character of $\mathbb{Q}_p^\times$ is unitary. Again, this is only known in special cases.

In the remaining three cases, $\pi_p$ is tempered.