Ramanujan-Petersson Conjecture–Part 1

1. Preliminaries

Let $G$ be a reductive group over a local field $F$. Note that here $F$ can be archimedean or non-archimedean. Let $(\pi ,V)$ be a Hilbert space representation of $G(F)$.

Definition 1.1.

A matrix coefficient of $\pi$ is a function of the form

$m:G(F)$ $\to ℂ$

$g$ $\mapsto (\pi (g){\phi }_1,{\phi }_2)$

for some ${\phi }_1,{\phi }_2\in V$.

Definition 1.2.

An irreducible representation $(\pi ,v)$ of $G(F)$ is square-integrable (resp. tempered) if its matrix coefficients lie in $L^2(G(F))$ (resp. $L^{2+\epsilon }(G(F))$ for any $ϵ>0)$. Square-integrable representations are also known as discrete series representations.

Definition 1.3.

Assume $F$ is non-archimedean. A supercuspidal representation of $G(F)$ is a representation whose matrix coefficients are compactly supported modulo the center.

Supercuspidal representations are clearly essentially square-integrable. Also recall supercuspidality can be equivalently characterized by the vanishing of all Jacquet modules ([2] Theorem 8.3.3).

Assume $F$ is non-archimedean. Let $\sigma$ be a smooth representation of a Levi subgroup $M$ of $G(F)$. Let $I(\sigma )$ be the parabolic induction of $\sigma$ to $G(F)$.

Proposition 1.4.

If $\sigma$ is unitary, then $I(\sigma )$ is unitarizable.

Proof.

[2] Proposition 8.2.2. ∎

Still assume $F$ is non-archimedean. We also recall the classification of representations of ${\mathrm{GL}}_2(F)$. It is an easy consequence of the Bernstein-Zelevinsky classification.

Theorem 1.5.

Let $\pi$ be an irreducible admissible representation of $G={\mathrm{GL}}_2(F)$. Let $\chi ,{\chi }_1$, and ${\chi }_2$ be characters of ${𝐆}_m(F)=F^{\times }$. Then $\pi$ is one of the following disjoint types:

1. irreducible principal series $\pi ({\chi }_1,{\chi }_2)$, where ${\chi }_1{\chi }_2^{-1}\ne |\cdot {|}_p^{\pm }$. We have $\pi ({\chi }_1,{\chi }_2)\sim \pi ({\chi }_2,{\chi }_1)$ and no other equivalences.

2. a special representation, i.e., a twisted Steinberg representation; it has the form

   $$\pi (\chi |\cdot {|}_p^{1/2},\chi |\cdot {|}_p^{-1/2})=\mathrm{St}\otimes \chi .$$

   Two such representations $\mathrm{St}\otimes \chi ,\mathrm{St}\otimes {\chi }^{\prime }$ are isomorphic if and only if $\chi ={\chi }^{\prime }$.

3. a supercuspidal representation.

4. $1$-dimensional. It is of the form $\chi \circ det$.

For more discussions on the Bernstein-Zelevinsky classification and therefore a proof of this theorem, see [2]. Representations of type (1), (2), and (4) in the theorem are often together called the principal sereis representations. Their common characterization is that they all appear as quotients of parabolically induced representations from the standard Levi of ${\mathrm{GL}}_2$.

Corollary 1.6.

The irreducible admissible representations of ${\mathrm{PGL}}_2(K)$ are of one of the following types:

1. irreducible principal series $\pi (\chi ,{\chi }^{-1})$ where $\chi \ne |\cdot {|}_p^{\pm 1/2}$ is any character of $K^{\times }$.

2. a quadratic twist of Steinberg: $\mathrm{St}\otimes \chi$, where ${\chi }^2=1$.

3. a supercuspidal representation of ${\mathrm{GL}}_2(K)$ with trivial central character. Such a representation is viewed as a ${\mathrm{PGL}}_2(K)$-representation via the canonical quotient map ${\mathrm{GL}}_2(K)\twoheadrightarrow {\mathrm{PGL}}_2(K)$.

4. $1$-dimensional, which is of the form $\chi \circ det$, where ${\chi }^2=1$.

Proof.

Denote by $Z=Z({\mathrm{GL}}_2)$ the center of ${\mathrm{GL}}_2$, which consists of the scalar matrices in ${\mathrm{GL}}_2$. A representation $(\pi ,V)$ of ${\mathrm{PGL}}_2(K)$ can be lifted to a representation $(\tilde{\pi },\tilde{V})$ of ${\mathrm{GL}}_2(K)$ via the quotient map $\mathrm{GL}(K)\twoheadrightarrow {\mathrm{PGL}}_2(K)$. Notice here we have used the fact

$${\mathrm{GL}}_2(K)/Z(K)\cong ({\mathrm{GL}}_2/Z)(K)={\mathrm{PGL}}_2(K)$$

where the first isomorphism holds because $K$ is henselian. The representation $(\tilde{\pi },\tilde{V})$ has trivial central character, i.e., ${\tilde{\pi }|}_{Z(K)}$ is trivial. If $(\tilde{\pi },\tilde{V})\cong \pi ({\chi }_1,{\chi }_2)$ is a principal series representation if and only if ${\chi }_1{\chi }_2=𝟏$ and ${\chi }_1{\chi }_2^{-1}={\chi }_1^2\ne |\cdot {|}_p^{\pm }$. The last two conditions are equivalent to ${\chi }_1\ne |\cdot {|}_p^{\pm 1/2}$. Suppose $(\tilde{\pi },\tilde{V})\cong \pi (\chi |\cdot {|}_p^{1/2},\chi |\cdot {|}_p^{-1/2})=\mathrm{St}\otimes \chi$ is a twisted Steinberg. Then one must have ${\chi }^2=𝟏$. Now suppose $(\tilde{\pi },\tilde{V})$ is $1$-dimensional of the form $\chi \otimes det$. Then for the restriction ${\chi \otimes det|}_{Z(K)}$, we have

Thus, ${\chi }^2=𝟏$. The only other possibility is therefore that $(\tilde{\pi },\tilde{V})$ is a supercuspidal representation with trivial central character. ∎

The classifications theorems above may have some refinements. To serve the interest of this note better, we want to understand when the representations may have (natural) Hilbert space strucutres.

Corollary 1.7.

An irreducible admissible representation $\pi$ of ${\mathrm{GL}}_2(F)$ is unitary if it belongs to one of the following types:

1. irreducible principal series $\pi ({\chi }_1,{\chi }_2)$ such that ${\chi }_1,{\chi }_2$ are unitary.

2. $\mathrm{St}\otimes \chi$ such that $\chi$ is unitary.

3. $\chi \circ det$ such that $\chi$ is unitary.

A similar result holds for ${\mathrm{PGL}}_2(F)$.

Proof.

A direct consequence of Proposition 1.4. ∎

The representations in the corollary form the tempered principal series for ${\mathrm{GL}}_2(F)$ (resp. ${\mathrm{PGL}}_2(F)$). There are other principal series representations of G which are unitary but not tempered. These are exactly the $\pi ({\chi }_1,{\chi }_2)$, where ${\chi }_2={\overline{{\chi }_1}}^{-1}$ and ${\chi }_1{\chi }_2^{-1}(x)={|x|}^{\sigma }$, for . Such representations form the complementary series. The construction of the invariant Hermitian forms on the complementary series representations is nontrivial. See [4] 1.58 Theorem 12 for more details. We will show shortly after that the unramified tempered principal series representations are in fact tempered in the sense of Definition 1.2.

The next ingredient we need is the Satake isomorphism, which classifies unramified representations based on the so-called Satake parameters. Let $G$ be unramified split reductive group over a non-archimedean local field $F$ with ring of integers ${𝒪}_F$. Let $K\le G(F)$ be a hyperspecial subgroup. Recall this means $K=𝒢({𝒪}_F)$ for some model $𝒢$ of $G$ over ${𝒪}_F$ with reductive special fiber. If $G$ is unramified, then hyperspecial subgroups always exists. Hyperspecial subgroups are maximal open compact, but the converse may not be true. Let $\widehat{G}$ be the complex dual reductive group of $G$, and let $\widehat{T}\le \widehat{G}$ be a maximal torus.

Theorem 1.8 (Satake).

There is an isomorphism of algebras

$$𝒮:C_c^{\mathrm{\infty }}(G(F)⫽K)\to 𝒪{(\widehat{T})}^{W(\widehat{G},\widehat{T})}$$

where $W(\widehat{G},\widehat{T})$ is the Weyl group with respect to $\widehat{T}$ viewed as a group scheme.

Here $\widehat{T}$ is viewed as an (affine) $ℂ$-scheme, and $𝒪(\widehat{T})$ is the coordinate ring of $\widehat{T}$, which is a $ℂ$-algebra. The group scheme $W(\widehat{G},\widehat{T})$ acts on $\widehat{T}$ and hence on $𝒪(\widehat{T})$.

Definition 1.9.

Let $G$ be an unramified reductive group over $F$ and let $K\le G(F)$ be a hyperspecial subgroup. An irreducible smooth representation $(\pi ,V)$ of $G(F)$ is $K$-unramified if $V^K\ne 0$.

Proposition 1.10.

Let $K\le G(F)$ be a compact open subgroup (not necessarily hyperspecial). If $({\pi }_1,V_1)$ and $({\pi }_2,V_2)$ are irreducible representations of $G(F)$ such that $V_1^K$ and $V_2^K$ are nonzero and isomorphic as $C_c^{\mathrm{\infty }}(G(F)⫽K)$-modules, then $({\pi }_1,V_1)$ is isomorphic to $({\pi }_2,V_2)$ as a smooth representation. In particular, an unramified representation is determined up to isomorphism by its Hecke character.

Proof.

[2] Proposition 7.1.1. ∎

By the Chevalley restriction theorem ([8], §6), the inclusion $\widehat{T}\hookrightarrow \widehat{G}$ induces an isomorphism

$$\mathrm{Spec}\left(𝒪{(\widehat{T})}^{W(\widehat{G},\widehat{T})}\right)=\widehat{T}/W(\widehat{G},\widehat{T})\to \widehat{G}/\mathrm{conj}=\mathrm{Spec}\left(𝒪{(\widehat{G})}^{\widehat{G}}\right)$$

In particular, $(\widehat{G}/\mathrm{conj})(ℂ)$ is just the set of closed conjugacy classes in $\widehat{G}(ℂ)$. The closed conjugacy classes are precisely the conjugacy classes of semisimple elements; a nice reference for this is [8] Corollary 6.13. We therefore have a sequence of bijections

$$\mathrm{Hom}(C_c^{\mathrm{\infty }}(G(F)⫽K),ℂ)\stackrel{{({𝒮}^{-1})}^{\ast }}{\to }\mathrm{Hom}(𝒪{(\widehat{T})}^{W(\widehat{G},\widehat{T})},ℂ)\to (\widehat{G}/\mathrm{conj})(ℂ).$$

Recall Proposition 1.10 says unramified representations are in bijection with Hecke characters, which are precisely elements of $\mathrm{Hom}(C_c^{\mathrm{\infty }}(G(F)⫽K),ℂ)$. We have therefore proven the following corollary of the Satake isomorphism.

Corollary 1.11.

Assume that G is split. The composite isomorphism above induces a bijection between semisimple conjugacy classes in $\widehat{G}(ℂ)$ and isomorphism classes of irreducible unramified representations of $G(F)$.

Proof.

∎

Semisimple conjugacy classes in $\widehat{G}(ℂ)$ are often called the Satake parameters of unramified representations of $G(F)$. In the special case $G={\mathrm{GL}}_n$, $\widehat{G}={\mathrm{GL}}_{n,ℂ}$, and each semisimple conjugacy class in ${\mathrm{GL}}_n(ℂ)$ is determined by its eigenvalues $({\lambda }_1,\mathrm{\cdots },{\lambda }_n)$. We also call the set of eigenvalues $({\lambda }_1,\mathrm{\cdots },{\lambda }_n)$ the Satake parameters for ${\mathrm{GL}}_n$.

Proposition 1.12.

Let $\pi ({\chi }_1,{\chi }_2)$ be an irreducible principal series representation of ${\mathrm{GL}}_2(F)$. If $\pi ({\chi }_1,{\chi }_2)$ is unramified, then its Satake parameter is $({\chi }_1(\varpi ),{\chi }_2(\varpi ))$, where $\varpi$ is a uniformizer of $F$.

Proof.

This follows directly from the Satake isomorphism. ∎

Next we give a nice criterion for temperedness in terms of Satake parameters.

Proposition 1.13.

An unramified irreducible admissible representation of ${\mathrm{GL}}_2(F)$ is tempered if and only if its Satake parameter have complex norm $1$.

Proof 1.

By [2] Theorem 8.4.5, An admissible representation $(\pi ,V)$ of ${\mathrm{GL}}_2(F)$ is tempered and irreducible if and only if it belongs to either of the following two classes:

1. $\pi =\pi ({\chi }_1,{\chi }_2)$ such that $\sigma =({\chi }_1,{\chi }_2)$ is square-integrable as a representation of the standard Levi $M(F)$.

2. $\pi$ is a squre-integrable supercuspidal representation of ${\mathrm{GL}}_2(F)$.

If we require $\pi$ is also unramified, then only the first case can occur. From now on, assume $\pi$ is irreducible and unramified. In this case, $\pi$ is tempered if and only if $\sigma =({\chi }_1,{\chi }_2)$ is square-integrable, which is further equivalent to that both ${\chi }_1$ and ${\chi }_2$ are unitary. Since $({\chi }_1(\varpi ),{\chi }_2(\varpi ))$ is the Satake parameter of $\pi$, the proposition follows. ∎

Proof 2.

One can also show this proposition using Langlands correspondence. Of course this does not require the full correspondence; only the unramified case established already by the Satake isomorphism is needed. Under the Langlands correspondence, an unramified irreducible tempered admissible representation $(\pi ,V)$ of ${\mathrm{GL}}_2(F)$ corresponds to an unramified Galois representation

$$\rho :W_F\to {\mathrm{GL}}_2(ℂ)$$

with bounded image, where, $W_F$ is the Weil group of $F$. This means any power ${\mathrm{Frob}}_F^n$ of the Frobenius element $\mathrm{Frob}$ has bounded eigenvalues, which clearly shows the eigenvalues have complex norm $1$. These eigenvalues are by definition the Satake parameters of $(\pi ,V)$. ∎

Corollary 1.14.

An unramified irreducible admissible representation of ${\mathrm{PGL}}_2(F)$ is tempered if and only if its Satake parameter have complex norm $1$.

Proof.

An unramified irreducible admissible representation $(\pi ,V)$ of ${\mathrm{PGL}}_2(F)$ can be lifted to an unramified irreducible admissible representation $({\pi }^{\prime },V^{\prime })$ of ${\mathrm{GL}}_2(F)$ with trivial central character. If $(\alpha ,\beta )$ is the Satake parameter of an unramified representation of such $({\pi }^{\prime },V^{\prime })$, then $\alpha$ is complex-conjugate to $\beta$, and $\alpha$ is the Satake parameter for $(\pi ,V)$. Now the corollary follows from the previous proposition. ∎

Corollary 1.15.

A tempered principal series representation that is unramified is tempered.

In [7], Satake uses a slightly different way to parametrize unramified representations. Let $K\le G(F)$ be a hyperspecial subgroup. Let $\varphi \in \mathscr{H}(G(F))$ be a Hecke function, and let $\omega \in C_c^{\mathrm{\infty }}(G(F)⫽K)$ be a spherical Hecke function. Then it is an basic fact that $\omega$ is an eigenfunction for the involution operator defined by $\varphi$, i.e.,

$$\varphi \ast \omega ={\lambda }_{\varphi }\omega$$

for some ${\lambda }_{\varphi }\in ℂ$. Define the Hecke character $\widehat{\omega }\in \mathrm{Hom}(C_c^{\mathrm{\infty }}(G(F)⫽K),ℂ)$ associated to $\omega$ by

$$\widehat{\omega }(\varphi )={\int }_{G(F)}\varphi (g)\omega (g^{-1})𝑑g.$$

Write $\mathrm{\Omega }=C_c^{\mathrm{\infty }}(G(F)⫽K)$ and denote by ${\mathrm{\Omega }}^{+}$ the subset of positive-definite spherical functions, i.e.,

$${\int }_G{\int }_G\omega (g_1^{-1}{g}_2)\overline{\varphi (g_1)}\varphi (g_2)𝑑g_1𝑑g_2\ge 0$$

for all functions $\varphi$ of compact support on $G$.

Theorem 1.16 ([3]).

${\mathrm{\Omega }}^{+}$ is in bijection with the set of irreducible unitary unramified representations of $G(F)$.

Now we can state Satake’s formulation of the Ramanujan conjecture for ${\mathrm{PGL}}_2$ in [7].

Conjecture 1.17 (Automorphic Ramanujan Conjecture).

Let $\pi$ be a cuspidal irreducible representation of ${\mathrm{PGL}}_2({𝐀}_{\mathbb{Q}})$, and write ${\pi }_p$ for the local component of $\pi$ at a finite prime $p$. Then ${\pi }_p$ is tempered.

2. $\mathrm{\Delta }$-function

Recall the Ramanujan $\mathrm{\Delta }$-function $\mathrm{\Delta }(z)$ is the unique normalized holomorphic cuspidal eigenform for ${\mathrm{SL}}_2(\mathbb{Z})$ of level $1$ and weight $12$; that is, the complex vector space ${𝒮}_{12}({\mathrm{SL}}_2(\mathbb{Z}))$ has dimension $1$ and is spanned by $\mathrm{\Delta }(z)$. The $\tau$-function $\tau :𝐍\to \mathbb{Z}$ is defined to be the Fourier coefficients of $\mathrm{\Delta }(z)$, i.e.,

$$\mathrm{\Delta }(z)=\sum _{n=1}^{\mathrm{\infty }}\tau (n)e^{2\pi inz}=\sum _{n=1}^{\mathrm{\infty }}{a}_n{q}^n.$$

The following is (a part of) the original version of the Ramanujan conjecture.

Conjecture 2.1 (Classical Ramanujan Conjecture).

For all prime $p$, $|\tau (p)|\le 2p^{11/2}$.

We recall how to associate an automorphic form, and thus an automorphic representation, to a classical modular form.

Denote by ${\mathrm{GL}}_2^{+}(\mathbb{Q})$ the subgroup of ${\mathrm{GL}}_2(\mathbb{Q})$ consisting of matrices with positive discriminants. Let $W_k$ be the space of holomorphic functions on the upper half-plane $\mathscr{H}$ with the property that, for all $\gamma \in {\mathrm{GL}}_2^{+}(\mathbb{Q})$, ${f|}_{\gamma ,k}$ is bounded on the region $\mathrm{\Im }z>y_0$ for every $y_0>0$. The weight $k$ is going to be fixed in the following discussion, so we will just write ${f|}_{\gamma }$ for ${f|}_{\gamma ,k}$. The space $W_k$ has a left action $r_k:{\mathrm{GL}}_2^{+}(\mathbb{Q})\to \mathrm{GL}(W_k)$ by

$$r_k(\gamma )f={f|}_{{\gamma }^{-1}}.$$

Now define a smooth representation ${\rho }_k:{\mathrm{GL}}_2({𝐀}_{\mathbb{Q}}^f)\to \mathrm{GL}(M_k)$ by

$$M_k=\mathrm{sm}\text{-}{\mathrm{Ind}}_{{\mathrm{GL}}_2^{+}(\mathbb{Q})}^{{\mathrm{GL}}_2({𝐀}_{\mathbb{Q}}^f)}{W}_k$$

Let $K$ be one of $K_0(N),K_1(N),K_{(}N)$, and let $\mathrm{\Gamma }=K\cap {\mathrm{GL}}_2^{+}(\mathbb{Q})$. Given $g\in {\mathrm{GL}}_2({𝐀}_{\mathbb{Q}}^f)$, by the strong approximation (see, for example, [6]), we can find $\gamma \in {\mathrm{GL}}_2^{+}(\mathbb{Q})$ and $h\in K$ with $g=\gamma h$. Then define an element ${\varphi }_f\in M_k$ by

$${\varphi }_f(g)={f|}_{{\gamma }^{-1}}.$$

This is well-defined. Also notice ${\varphi }_f$ is smooth since it is $K$-invariant.

Exercise 2.2.

Show this is well-defined.

Therefore there is an injective map

$M_k(\mathrm{\Gamma })$ $\to M_k$

$f$ $\mapsto {\varphi }_f$

for every $\mathrm{\Gamma }$. The maps for different $\mathrm{\Gamma }$ are compatible in the obvious sense.

We can also define the cuspidal subspace ${𝒮}_k$ of $M_k$ by redefining the space $W_k$ so that every ${f|}_{\gamma }(x+iy)$ is not only bounded as $y\to \mathrm{\infty }$ but also approaches $0$ uniformly in $x$.

So in particular, we can associate an automorphic form ${\varphi }_{\mathrm{\Delta }}$ to $\mathrm{\Delta }$. Let ${\pi }_{\mathrm{\Delta }}\subseteq M_{12}$ be the smallest ${\mathrm{GL}}_2({𝐀}_{\mathbb{Q}}^f)$-stable subspace of ${𝒮}_{12}$ containing ${\varphi }_{\mathrm{\Delta }}$; this is the automorphic representation associated to $\mathrm{\Delta }$.

Proposition 2.3.

The automorphic representation ${\pi }_{\mathrm{\Delta }}$ induced by $\mathrm{\Delta }$ is unramified. Its local components ${\pi }_{\mathrm{\Delta },p}$ are all unramified irreducible admissible for all finite primes $p$.

Proof.

This is a rather simple fact. One only needs to follow the definitions to show the proposition. ∎

Corollary 2.4.

If Satake’s Ramanujan-Petersson conjecture holds for ${\pi }_{\mathrm{\Delta }}$, then the classical Ramanujan conjecture holds for $\mathrm{\Delta }$.

Proof.

Suppose Satake’s Ramanujan-Petersson conjecture holds for ${\pi }_{\mathrm{\Delta }}$, i.e., ${\pi }_{\mathrm{\Delta },p}$ is tempered for any finite prime $p$. Then the Satake parameter ${\lambda }_p$ of ${\pi }_{\mathrm{\Delta },p}$ has complex norm $1$. By the description of the Satake parameters of ${\mathrm{PGL}}_2(F)$ in [7], ${\pi }_{\mathrm{\Delta },p}$ must be of principal series. The classication of unramified representations of ${\mathrm{PGL}}_2({\mathbb{Q}}_p)$ in [7] then shows $|a_p|\le 2p^{11/2}$, i.e., the classical Ramanujan conjecture holds for $\mathrm{\Delta }$. ∎