The GL $(n)$ case
For $G=\mathrm{GL}(n)$, the relations are pretty well understood:
- For a cuspidal newform $\varpi \in V_{\pi}$, the Fourier coefficients $A_{\varpi}(1, \cdots, 1, m)$ is proportional to Hecke eigenvalues $\lambda_{\pi}(m)$.
- We know well about the Mellin transform for $W_{\mu}$.
- We have some non-trivial bounds for $\mathrm{GL}(n)$ Kloosterman sums.
The Kuznetsov formula
The Kuznetsov formula is obtained by computing the inner product of Poincaré series in two different ways.
Very roughly, the formula has the following shape:
$$
\int_{(q)}\left|A_{\varpi}(M)\right|^{2}\left|\left\langle W_{\mu_{\pi}^{\top}}, E\right\rangle\right|^{2} d \varpi ” \approx ” 1+\sum_{\mathrm{id} \neq w \in W} \sum_{c=\left(c_{1}, \cdots, c_{n}\right)} \frac{\mathrm{KI}{q, w}(c, M, N)}{c{1} \cdots c_{n}}
$$
for some test function $E$.
Very general!
In principle, this contains information on
$\mu_{\pi}(p)$, through the Fourier coefficients $A_{\varpi}(M) ;$ and
- $\mu_{\pi}(\infty)$, through the Whittaker function $W_{\mu_{\pi}}$.
The formula says that it suffices to understand the Kloosterman sums instead!
But these “principles” are not well understood in general. Neither are Kloosterman sums.
Density Hypothesis
Let $\mathcal{F}$ be a finite family of automorphic representations of $G$. For $\sigma \geq 0$, we define
$$
N_{v}(\sigma, \mathcal{F})=\left|\left{\pi \in \mathcal{F} \mid \sigma_{\pi}(v) \geq \sigma\right}\right|
$$
Trivially, we have $N_{v}(0, \mathcal{F})=|\mathcal{F}| .$ On the other hand, if $\mathcal{F}$ contains the trivial representation, then $N_{v}\left(\sigma_{\text {triv }}(v), \mathcal{F}\right) \geq 1$.
The density hypothesis says maybe we can extrapolate linearly between these two extremes.
A non-GL( $n$ ) case?
What do we know when $G \neq \mathrm{GL}(n) ?$ Let $G=\mathrm{Sp}(4)$. The naive generalisation of Ramanujan conjecture claims that cuspidal automorphic representations of $G$ is tempered. But this is false! Saito-Kurokawa lifts are not tempered. But they are also not generic, i.e. they don’t have a Whittaker model. So we can modify the conjecture a little bit.
Conjecture Generic cuspidal automorphic representations of $\mathrm{Sp}(4)$ are tempered.
The Kuznetsov formula only deals with the generic spectrum.
Let $G=\operatorname{Sp}(4), q$ a prime, and
$$
\Gamma_{0}(q)=\left{\left(\begin{array}{ll}
A & B \
C & D
\end{array}\right) \in \operatorname{Sp}(4, \mathbb{Z}) \mid C \equiv 0 \quad(\bmod q)\right} \subseteq \operatorname{Sp}(4, \mathbb{Z})
$$
be the Siegel congruence subgroup of level $q$. For $M=\left(M_{1}, M_{2}\right) \in \mathbb{Z}^{2}$ we define characters
$$
\theta_{M}: U(\mathbb{R}) \rightarrow \mathbb{C}^{\times}, \quad \theta_{M}\left(\left(\begin{array}{cccc}
1 & x_{12} & x_{13} & x_{14} \
& 1 & x_{23} & x_{24} \
& & 1 & \
& & -x_{12} & 1
\end{array}\right)\right)=e\left(M_{1} x_{12}+M_{2} x_{24}\right)
$$
Let $\mathcal{F}=\mathcal{F}_{I}(q)$ be the family of generic cuspidal automorphic representations for
sufficiently large.
Hecke eigenvalues
Consider the Hecke algebra $\mathscr{H}$ of $\operatorname{GSp}(4, \mathbb{Q})^{+} .$ We define Hecke operators $T_{a, b}^{(r)}(p)$ corresponding to the double coset $\Gamma \operatorname{diag}\left(p^{a}, p^{b}, p^{r-a}, p^{r-b}\right) \Gamma$. For an
irreducible cuspidal representation $\pi$ we denote by $\lambda_{a, b}^{(r)}(p, \pi)$ the eigenvalue under $T_{a, b}^{(r)}(p) .$ The local Hecke algebra is generated by $T(p)=T_{0,0}^{(1)}(p)$ and $T_{0,1}^{(2)}(p)$, along with the identity.
So the Hecke operators $T_{a, b}^{(r)}$ relates the Fourier coefficients $A_{\varpi}\left(p^{k_{1}}, p^{k_{2}}\right) .$
Kloosterman sums
We give a definition of Kloosterman sums in the context of $\mathrm{Sp}(4)$.
Let $\gamma \in \Gamma_{0}(q)$. Bruhăt decomposition says we can write $\gamma=x w c^{} x^{\prime}$, where $x, x^{\prime} \in U(\mathbb{Q}), w \in W$, and $c^{} \in T$ the diagonal torus. Here we embed $c=\left(c_{1}, c_{2}\right) \mapsto c^{}=\operatorname{diag}\left(1 / c_{1}, c_{1} / c_{2}, c_{1}, c_{2} / c_{1}\right) .$ Classifying elements in $\Gamma_{0}(q)$ by the Weyl element, we have a decomposition $\Gamma_{0}(q)=\coprod_{w \in W} G_{w}$. Now we define the Kloosterman sum to be $$ \mathrm{KI}{q, w}(c, M, N)=\sum{x w c^{} x^{\prime} \in U(\mathbb{Z}) \backslash G_{w} / U_{w}(\mathbb{Z})} \theta_{M}(x) \theta_{N}\left(x^{\prime}\right)
$$
if it is well-defined. Otherwise we set $\mathrm{KI}{q, w}(c, M, N)=0$. If $\theta{M}, \theta_{N}$ are non-degenerate, then the Kloosterman sums are well-defined only for
$$
w=\text { id, } s_{\alpha} s_{\beta} s_{\alpha}=\left(\begin{array}{ccc}
& & -1 & \
1 & & & \
& & 1
\end{array}\right), s_{\beta} s_{\alpha} s_{\beta}=\left(\begin{array}{ccc}
& & -1 \
-1 & 1 & 1 & \
-1 & &
\end{array}\right), w_{0}=\left(\begin{array}{ccc}
& & -1 & \
1 & & & -1 \
& 1 & &
\end{array}\right) .
$$
Meanwhile, Kloosterman sums enjoy certain multiplicativity properties. If entries of $M, N$ are coprime to $q$, and $c=\left(q^{a} c_{1}^{\prime}, q^{b} c_{2}^{\prime}\right)$ with $\left(q, c_{1}^{\prime} c_{2}^{\prime}\right)=1$, then
$$
\mathrm{KI}{q, w}(c, M, N)=\mathrm{KI}{q, w}\left(\left(q^{a}, q^{b}\right), M^{\prime}, N^{\prime}\right) \mathrm{K} I_{1, w}\left(\left(c_{1}^{\prime}, c_{2}^{\prime}\right), M^{\prime \prime}, N^{\prime \prime}\right),
$$
for some $M^{\prime}, N^{\prime}, M^{\prime \prime}, N^{\prime \prime} .$ In particular, entries of $M^{\prime}, N^{\prime}$ are also coprime to $q$. We also have a trivial bound
$$
\mathrm{KI}{1, w}\left(\left(c{1}^{\prime}, c_{2}^{\prime}\right), M^{\prime \prime}, N^{\prime \prime}\right) \leq c_{1}^{\prime} c_{2}^{\prime}
$$
So it remains to compute Kloosterman sums of the form $\mathrm{KI}_{q, w}\left(\left(q^{a}, q^{b}\right), M, N\right)$.
The density theorem
So the contribution from $w=s_{\beta} s_{\alpha} s_{\beta}$ on the right hand side is given by
$$
\begin{array}{l}
\sum_{c_{1}^{\prime} \ll m Z / q} \frac{\mathrm{Kl}{q, w}\left(\left(q c{1}^{\prime},\left(q c_{1}^{\prime}\right)^{2}\right), M, M\right)}{c_{1} c_{2}} \int E(\cdots) \
\ll \sum_{c_{1}^{\prime} \ll m Z / q} q^{-1+\varepsilon} \ll q^{\varepsilon}
\end{array}
$$
So the proposition is proved.
Now we are ready to prove the density theorem.
Assume the settings above. Then
$$
N_{v}\left(\sigma, \mathcal{F}{l}(q)\right) \ll{I, v, \varepsilon} q^{3-4 \sigma+\varepsilon}
$$
This says $a=3 / 4$. This is also halfway towards Ramanujan conjecture!
Let $v=p \neq q$ be a finite place. Choose $\nu_{0}$ maximal such thtat $p^{\nu_{0}} \ll q^{2}$ with an implied constant admissible to the proposition. From a previous estimate, there is $\nu_{0}-5 \leq \nu_{\pi} \leq \nu_{0}$ such that
$$
\left|A_{\varpi}\left(1, p^{\nu_{\pi}}\right)\right|^{2} \gg q^{-3-\varepsilon} p^{2 \nu_{\pi} \sigma_{\pi}(p)}
$$
Applying the proposition with $m=p^{\nu_{\pi}}, Z=1$ gives
$$
\begin{aligned}
N_{p}\left(\sigma, \mathcal{F}{l}(q)\right) \leq \sum{\pi \in \mathcal{F}{1}(q)} \frac{p^{2 \nu{\pi} \sigma_{\pi}(p)}}{p^{2 \nu_{\pi} \sigma}} & \ll q^{3-4 \sigma+\varepsilon} \int_{(q)} \sum_{\nu_{0}-5 \leq \nu \leq \nu_{0}}\left|A_{w}\left(1, p^{\nu}\right)\right|^{2} \delta_{\lambda_{\infty} \in I} \
& \ll_{I, \varepsilon} q^{3-4 \sigma+\varepsilon} .
\end{aligned}
$$
For $v=\infty$, applying the proposition with $m=1, Z \ll q^{2}$ gives
$$
N_{\infty}\left(\sigma, \mathcal{F}{I}(q)\right) \leq \sum{\pi \in \mathcal{F}{1}(q)} Z^{2 \sigma{\pi}(\infty)-2 \sigma} \ll q^{3-4 \sigma+\varepsilon} \int_{(q)}\left|A_{\varpi}(1,1)\right|^{2} Z^{2 \sigma_{\pi}(\infty)}
$$
$$
\ll_{I, \varepsilon} q^{3-4 \sigma+\varepsilon}
$$
