1.1 Motivation

The Bruggeman-Kuznetsov formula, one of the core tools of analytic number theory since the late 1970s, can be stated in its simplest form as follows. Given a test function $h_\infty (t)$ one defines the Kuznetsov transform of it by

(1.1) $$ \begin{align} H_\infty(x) = \frac{i}{2} \int_{-\infty}^\infty \frac{J_{2it}(x)}{\cosh (\pi t)} h_\infty(t) t\,dt. \end{align} $$

For sufficiently well-behaved test functions $h_\infty $ (see (1.5)) and integers $m,n$ with $mn>0$ one has

(1.2) $$ \begin{align} \sum_{u \in \mathcal{B}_0} h_\infty(t_u) a_u(m)&\overline{a_u(n)}+ \frac{1}{4\pi} \int_{-\infty}^\infty h_\infty(t) \frac{\pi}{|\zeta(1+2it)|^2} \lambda_t(m) \overline{\lambda_t(n)}\,dt \\ & = \delta_{m=n} \frac{1}{4\pi} \int_{-\infty}^{\infty} h_{\infty}(t) t \tanh(\pi t) dt + \sum_{c \in \mathbb{N}} \frac{S(m,n;c)}{c} H_{\infty}\Big(\frac{4 \pi \sqrt{|mn|}}{c}\Big) \nonumber \end{align} $$

where $\mathcal {B}_0$ is an orthonormal basis of Hecke-Maass waveforms u on $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathcal {H}$ normalized by $\operatorname {\mathrm {vol}}( \operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathcal {H})=\pi /3$ , $1/4 + t_u^{2}$ is the Laplace eigenvalue of u, $a_u(m)$ are the Fourier coefficients given by

$$ \begin{align*}u(x+iy) = 2 \sqrt{y} \left( \frac{\cosh (\pi t_u)}{\pi}\right)^{1/2} \sum_{n \neq 0} a_u(n) K_{it_u}(2 \pi |n| y)e(nx),\end{align*} $$

the $\lambda _t(n) = \sum _{ab=|n|}(b/a)^{it}$ , and

$$ \begin{align*}S(m,n;c)= \sideset{}{^*}\sum_{x\ \pmod{c}} e\left(\frac{mx + n \overline{x}}{c}\right)\end{align*} $$

is the Kloosterman sum, in which the $*$ indicates that the sum runs over invertible residue classes modulo c and $\overline {x}$ denotes the inverse of x modulo c. There is also an opposite-sign version of (1.2) that holds in the case $mn<0$ with the modification that $H_\infty (x)$ is replaced by a function $H_\infty ^-(x)$ (see (1.23)).

Even more classical is the holomorphic counterpart to (1.2), i.e. the Petersson formula:

(1.3) $$ \begin{align} \sum_{f \in \mathcal{B}_\kappa} a_{f}(m) \overline{a_{f}(n)} = \frac{\kappa-1}{4\pi} \left(\delta_{m=n} + 2 \pi i^{-\kappa} \sum_{c\in \mathbb{N}} \frac{S(m,n;c)}{c} J_{\kappa -1}\Big(\frac{4 \pi \sqrt{mn}}{c}\Big)\right), \end{align} $$

where $\mathcal {B}_\kappa $ is an orthonormal basis of holomorphic cusp forms f for $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ of weight $\kappa $ and Fourier coefficients $a_f(m)$ given by

$$ \begin{align*}f(z) = \left( \frac{(4\pi)^\kappa}{\Gamma(\kappa)}\right)^{1/2} \sum_{n\geq 1} a_f(n) n^{\frac{\kappa-1}{2}}e(nz).\end{align*} $$

In the representation-theoretic framework for automorphic forms, the parity $p(u)$ (see (1.22)) and spectral parameter $t_u$ or weight $\kappa =\kappa _f$ parametrize the possible archimedean local components $\pi _\infty $ of trivial central character cuspidal automorphic representations $\pi $ of $\operatorname {\mathrm {GL}}_2/\mathbb Q$ . Therefore, the above Bruggeman-Kuznetsov (both $mn>0$ and $mn<0$ cases) and Petersson formulas can be combined to give a spectral summation device for automorphic forms on $\operatorname {\mathrm {PGL}}_2/\mathbb Q$ with specified local representation at infinity (and that are unramified at all finite places).

The goal of this paper is to analogously develop Petersson/Bruggeman-Kuznetsov (PBK) formulas at finite places p that allow control on the associated representations of $\operatorname {\mathrm {GL}}_2(\mathbb Q_p)$ at those places (as well as at $\infty $ ). To generate such formulas, we use the adelic relative trace formula approach to the PBK formulas of Jacquet [Reference JacquetJac86] and Zagier [Reference ZagierZag81, Reference JoynerJoy90], as exposited by Knightly and Li [Reference Knightly and LiKL06a, Reference Knightly and LiKL13]. We restrict our attention to automorphic forms over $\mathbb Q$ in this paper, but many of the local aspects of our work should carry over to more general non-archimedean local fields.

In this perspective, one chooses a test function f on the group $\operatorname {\mathrm {GL}}_2(\mathbb {A})$ for the pretrace formula and then integrates along left and right unipotent orbits to obtain the PBK formula. To aid this strategy and to produce a reasonably explicit formula, we introduce two assumptions on the test function f called the geometric and spectral assumptions. The geometric assumptions place a constraint on the support of the local test function $f_p$ on $\operatorname {\mathrm {GL}}_2(\mathbb Q_p)$ and allow us to establish the standard properties of the geometric side of the formula. The spectral assumption puts a strong constraint on the integral operators $\pi (f)$ and allows us to explicate the spectral side of the formula. The result is Theorem 1.8.

As an application of Theorem 1.8, we give a harmonically weighted Weyl-Selberg Law for the family of cusp forms $\mathcal {F}_0(f)$ cut out by our chosen test function f and interpret the leading constant in terms of local Plancherel volumes. For this result, see Corollary 1.14. In a similar context, Palm [Reference PalmPal12, Thm. 3.2.1] gave a Weyl law for cusp forms with specified local components as an application of the Selberg trace formula.

As a second application of Theorem 1.8, we give an axiomatized Large Sieve Inequality for the families $\mathcal {F}_0(f)$ cut out by f. Under additional local hypotheses (stated in Section 1.5) these large sieve inequalities are optimally strong: the estimate is of the shape $\ll (X|\mathcal {F}|)^\varepsilon (X + |\mathcal {F}|)\|\mathbf {a}\|_2^2$ , where X is the length of summation of the sequence $\mathbf {a}$ and $|\mathcal {F}|$ is the cardinality of the family of cusp forms.

Probably the most important part of the paper however are the examples. Most notably, in Section 6.4 we give an elegant expression for the generalized Kloosterman sum that arises from a specified (trivial central character) supercuspidal representation $\sigma $ of $\operatorname {\mathrm {GL}}_2(\mathbb Q_p)$ . See Theorem 6.45 for this formula.

For the specified supercuspidal formula, we set the local test function $f_p$ equal to the diagonal newform matrix coefficient of $\sigma $ restricted to a maximal compact subgroup, building on earlier work of the first author [Reference HuHu24]. This test function $f_p$ generates a generalized PBK formula that selects on the spectral side automorphic forms with local component at p isomorphic to either $\sigma $ or at most two other supercuspidal representations of the same conductor. This is essentially the narrowest possible support on the spectral side under the geometric assumptions. For precise statements, see Theorems 6.20 and 6.29.

In a parallel fashion, given a primitive Dirichlet character $\chi $ modulo a power of p with $\chi ^2 \neq 1$ , we construct local test functions $f_\chi $ in Section 7.2 whose generalized PBK formula selects on the spectral side automorphic forms with local component at p isomorphic to a principal series representations $\pi (\chi |\cdot |_p^{it} ,\chi ^{-1}|\cdot |_p^{-it})$ for some $t\in \mathbb R$ . The generalized Kloosterman sum on the geometric side of the formula (7.16) is in complete analogy with the supercuspidal Kloosterman sum mentioned above. Again, this generalized PBK formula has the narrowest possible support on the spectral side under the geometric assumptions (see Lemma 3.11).

These examples lay the groundwork for future important analytic applications. That we can produce several interesting examples that satisfy both the geometric and spectral hypotheses shows that while these two hypotheses together may appear fairly restrictive, they nonetheless contain the families of greatest interest to us.

An important feature of the Bruggeman-Kuznetsov (BK) formula is that the integral transform (1.1) relating the test function $h_\infty $ on the spectral side to the archimedean test function $H_\infty $ on the geometric side is relatively simple and can often be analyzed effectively using standard techniques such as stationary phase estimates. The situation (at present) with finite places is not quite as clean: the local test function $f_p$ on $\operatorname {\mathrm {GL}}_2(\mathbb Q_p)$ continues to play a strong role in the formula whereas the test function $f_\infty $ on $\operatorname {\mathrm {GL}}_2(\mathbb R)$ can be completely suppressed from the classical BK formula.

Nonetheless, we develop the sequence of transforms

$$ \begin{align*}h_p \to f_p \to H_p\end{align*} $$

to some extent, where $h_p: \pi \mapsto \pi (f_p)$ is an operator-valued function on the unitary dual of $\operatorname {\mathrm {PGL}}_2(\mathbb Q_p)$ (assumed to be projections with finite-dimensional image) and $H_p$ are the generalized (local) Kloosterman sums defined in (3.13). Indeed, Proposition 4.1 gives an expression for $f_p$ as an integral transform (of sorts) of $h_p$ in terms of matrix coefficients over the unitary dual $\operatorname {\mathrm {PGL}}_2(\mathbb Q_p)$ with respect to Plancherel measure. Then $H_p$ is by definition an integral of $f_p$ against additive characters. Furthermore, Section 7 gives a list of transform pairs $h_p \to H_p$ for which one can mostly forget about the function $f_p$ on the group entirely.

As already mentioned, much of this paper builds on previous work of the first author [Reference HuHu24]. To be definite, our paper contains the following new features:

• • an axiomatized treatment of PBK formulas given by adelic test functions (see the geometric and spectral assumptions in Section 1.2),

• • some preliminary applications of the axiomatized PBK formulas (see Section 5),

• • a test function with smallest possible spectral support around a specified supercuspidal representation $\sigma $ with even conductor exponent, i.e. we achieve $l_0=0$ in the sense of [Reference HuHu24, Rem. 1.4], see the discussion around Theorem 6.20 in this paper,

• • a more natural expression for the generalized Kloosterman sum associated to a specified supercuspidal representation (compare [Reference HuHu24, Def. 4.6] and Theorem 6.45), and

• • a detailed treatment of the dihedral $p=2$ case, which is often omitted for convenience but hides many technical complications (see Section 6.3).

1.2 Statement of results

Let $G=\operatorname {\mathrm {GL}}_2$ and $\overline {G} =\operatorname {\mathrm {PGL}}_2$ . Write ${\mathcal {H}}_{\mathrm {fin}} =C_c^\infty (\overline {G}(\mathbb {A}_{\mathrm {fin}}))$ for the non-archimedean Hecke algebra of $\overline {G}$ , that is the space of locally constant functions on $G(\mathbb {A}_{\mathrm {fin}})$ that are invariant by and compactly supported modulo center $Z(\mathbb {A}_{\mathrm {fin}})$ . Define the local Hecke algebra ${\mathcal {H}}_p= C_c^\infty (\overline {G}(\mathbb Q_p))$ similarly.

Let $K_p = G(\mathbb {Z}_p)$ and $ZK_p = Z(\mathbb Q_p)G(\mathbb {Z}_p)$ for $p<\infty $ . We say that a pure tensor $f= \bigotimes _p f_p \in {\mathcal {H}}_{\mathrm {fin}}$ is ramified at a prime p if $f_p$ is not a constant multiple of $1_{ZK_p}$ . Let $K = \prod _p K_p$ be the standard maximal compact subgroup of $G(\mathbb {A}_{\mathrm {fin}})$ and let $K(N)$ be the principal congruence subgroup of K. The minimal $N \in \mathbb {N}$ such that $f \in {\mathcal {H}}_{\mathrm {fin}}$ is bi- $K(N)$ -invariant is called the level of f.

Our generalized Bruggeman-Kuznetsov formula is an equality between a spectral sum of Fourier coefficients/Hecke eigenvalues over a family of automorphic forms and a geometric sum of generalized Kloosterman sums over a set of admissible moduli. In the next several paragraphs, we define these objects.

For an irreducible admissible representation $(\pi , V_\pi )$ with $\pi _{\mathrm {fin}}$ the underlying $\overline {G}(\mathbb {A}_{\mathrm {fin}})$ -representation and $f\in {\mathcal {H}}_{\mathrm {fin}}$ , we write $\pi (f): V_{\pi } \to V_{\pi }$ for the integral operator

(1.4) $$ \begin{align} \pi(f): v \mapsto \int_{\overline{G}(\mathbb{A}_{\mathrm{fin}})} f(g) \pi_{\mathrm{fin}}(g)v\,dg. \end{align} $$

Note that $\pi (f):=\pi _{\mathrm {fin}}(f)$ but we have dropped the subscript to avoid cluttering the notation.

The family $\mathcal {F}_0(f)$ has no restrictions on the archimedean spectral parameters of the representation it contains. Such restrictions will be imposed in our formulas in the standard way: by selecting a test function $h_\infty $ . Note that $\mathcal {F}_0(f)$ is a harmonic family in the sense of [Reference Sarnak, Shin and TemplierSST16], and at least in spirit every harmonic family on $\operatorname {\mathrm {PGL}}_2$ over $\mathbb Q$ arises in this way.

For $\pi $ a cuspidal automorphic representation, write $\mathcal {B}(\pi )$ for an orthonormal basis of $V_\pi $ (with respect to (1.61)). Let $K_\infty = \operatorname {\mathrm {SO}}_2(\mathbb R)$ . The subspace $V_\pi ^{K_\infty \times K(N)}$ of fixed vectors in $\pi $ is finite-dimensional, and for cuspidal $\pi $ let $u=u_\varphi $ be the classical Maass waveform with respect to $\Gamma (N)$ corresponding to $\varphi \in V_\pi ^{K_\infty \times K(N)}$ by $u(x+iy) = \varphi ( \left ( \begin {smallmatrix} y & x \\ & 1\end {smallmatrix}\right ) \times 1_{\mathrm {fin}}).$ Recall from (2.9) the Fourier coefficients $a_u(m)$ for $m\in \frac {1}{N}\mathbb {Z}$ of a Maass form u for $\Gamma (N)$ .

Let $h_\infty (t)$ be a test function as in the classical Kuznetsov formula. Iwaniec and Kowalski [Reference Iwaniec and KowalskiIK04, (15.19)] give the following sufficient conditions: For some $\delta>0$

(1.5) $$ \begin{align} \begin{cases} h_\infty(t) \text{ is holomorphic in } |\mathrm{Im}(t)| \leq 1/2+\delta, \\ h_{\infty}(t)\ll (1+|t|)^{-2-\delta}, \text{ and } \\ h_\infty(t)= h_\infty(-t) \text{ for all } t. \end{cases} \end{align} $$

Let $\psi _p: \mathbb Q_p\to \mathbb {C}^\times $ be the standard additive character $\psi _p(x)= e(\{x\}_p)$ and $\psi _{\mathrm {fin}} : \mathbb {A}_{\mathrm {fin}} \to \mathbb {C}^\times $ be given by $\psi _{\mathrm {fin}} = \prod _p \psi _p$ .

Definition 1.2. For $f \in {\mathcal {H}}_{\mathrm {fin}}$ , $m,n \in \mathbb Q$ and $c \in \mathbb Q_+$ , the generalized Kloosterman sum (attached to f) that appears in this paper is defined as

(1.6) $$ \begin{align} H(m_1,m_2;c) = \iint_{\mathbb{A}_{\mathrm{fin}}^2} f\left( \left( \begin{smallmatrix}1 & -t_1 \\ & 1 \end{smallmatrix}\right) \left( \begin{smallmatrix} & -c^{-2} \\ 1 & \end{smallmatrix}\right) \left( \begin{smallmatrix}1 & t_2 \\ & 1 \end{smallmatrix}\right) \right) \psi_{\mathrm{fin}}(m_1t_1- m_2 t_2)\,dt_1\,dt_2. \end{align} $$

Note that $H(m_1, m_2;c)$ is in fact a finite sum since $f\in {\mathcal {H}}_{\mathrm {fin}}$ . While the sum $H(m,n;c)$ is à priori defined for all $m,n \in \mathbb Q$ , it vanishes unless $m,n \in \frac {1}{N}\mathbb {Z}$ (see Theorem 3.8(1)). We also define local generalized Kloosterman sums $H_p(m,n;c)$ by the same formula (1.6) but where $\mathbb {A}_{\mathrm {fin}}, \psi _{\mathrm {fin}},$ and f are replaced by their local versions $\mathbb Q_p, \psi _p,$ and $f_p$ (to be definite, see (3.13)).

When $f=\bigotimes _p f_p$ is a pure tensor, one has $H(m,n;c) = \prod _p H_p(m,n;c)$ . Of course the generalized Kloosterman sum $H(m,n;c)$ depends on $f \in {\mathcal {H}}_{\mathrm {fin}}$ and the local $H_p(m,n;c)$ depend on $f_p\in {\mathcal {H}}_p$ , but these are suppressed in the notation. Recall, we also defined the transform $H_\infty (x)$ of $h_\infty (t)$ by (1.1), as in the classical Kuznetsov formula.

Next we define the index set of the sum on the geometric side of our formula.

See Lemma 2.5 for a proof. Under the mild hypothesis in the second sentence of Proposition/Definition 1.3, it is not too hard to show that one has an “unrefined” Bruggeman-Kuznetsov formula of the shape

(1.7) $$ \begin{align} \sum_{ \pi \in \mathcal{F}_0(f)} h_\infty(t_\pi) & \sum_{\varphi \in \mathcal{B}(\pi)}a_{u_{\pi(f)\varphi}}(m_1) \overline{a_{u_\varphi}(m_2) } + (\text{cts.}) \\ & = (\text{diag.\ weight})\delta_{m_1=m_2} + \sum_{c \in \mathcal{C}(\mathcal{F}) } \frac{H(m_1,m_2;c)}{c}H_\infty\left(\frac{4 \pi \sqrt{ m_1m_2}}{c}\right).\nonumber \end{align} $$

For more details, see Theorem 2.1. On its own, (1.7) is not very useful without additional information on f.

We next introduce the geometric and spectral assumptions alluded to in Section 1.1, which permit a practically useful refinement of (1.7). Let $a(y)= \left (\begin {smallmatrix} y & 0 \\ 0 & 1 \end {smallmatrix}\right )$ and $A\subset G$ be as in Section 1.7.3.

We say that a rational number $y \in \mathbb Q_+$ for which Geometric Assumption (2) holds “controls the support of f”. Caution: y is not necessarily uniquely determined by f. Note, Geometric Assumption (2) ensures that the hypothesis on the support of f of Theorem 2.1 is satisfied. Another useful fact to keep in mind is that under Geometric Assumption (2), the function f is ramified at p if and only if p divides the level of f, for which see Section 1.3.1.

The geometric assumptions control the set of admissible moduli $\mathcal {C}(\mathcal {F})$ as follows.

For a proof, see Lemmas 3.5 and 3.6. With additional information on the support of f, the geometric conductor $k(\mathcal {F})$ can be determined exactly (see Section 3.2). One also has that $k(\mathcal {F}) = \prod _p p^{k_p}$ for “local geometric conductors” $k_p$ defined analogously by the nonidentical vanishing of $H_p$ , for which see Theorem 3.8(6).

In addition to controlling $\mathcal {C}(\mathcal {F})$ , the geometric assumptions also endow the generalized Kloosterman sums $H(m,n;c)$ with many of the same basic structural properties as the standard Kloosterman sums, as in [Reference IwaniecIwa97, Ch. 4.3]. For a detailed list of these, see Theorem 3.8.

We now move on to the spectral assumption. Let $\overline {G}(\mathbb Q_p)^{\wedge }$ denote the unitary dual of $\overline {G}(\mathbb Q_p)$ , i.e. the space of isomorphism classes of smooth irreducible unitary representations of $\overline {G}(\mathbb Q_p)$ on a complex vector space.

With future and past applications in mind, we also want to allow the classical PBK formula with level structure at finitely many primes (as in [Reference Knightly and LiKL06a, Reference Knightly and LiKL13], recovering the classical formulae). Let $\nu (n) = [\operatorname {\mathrm {SL}}_2(\mathbb {Z}):\Gamma _0(n)] = n \prod _{p \mid n} (1+p^{-1})$ .

Note, when $c=0$ the test function $1_{ZK_p}$ is itself a newform projector, but when $c\geq 1$ the test function $\nu (p^c)1_{ZK_0(p^c)}$ is not.

The main purpose of the spectral assumption is to simplify the left hand (spectral) side of (1.7) (but see also Section 4.3). Indeed, writing $\pi = \bigotimes ^{\prime }_v \pi _v$ and $V = \bigotimes ^{\prime }_v V_v$ , the operator $\pi (f)$ is an orthogonal projection onto the subspace

(1.8) $$ \begin{align}V_f:=V_\infty \otimes \bigotimes_{p: f_p \text{ newform proj.}} \mathbb{C} \varphi_{0,p} \otimes \bigotimes_{\substack{p : f_p=\nu(p^c)1_{ZK_0(p^{c})} \\ \text{ with } c\geq 1}} V_p^{K_0(p^c)}\end{align} $$

of $V_{}^{K(N)}$ , where $\varphi _{0,p}$ is an $L^2$ -normalized newvector in $V_p$ if $\pi _p(f_p) \neq 0$ , and $\varphi _{0,p} =0$ otherwise. For the implementation of this, see Theorem 4.8.

For our intended applications, we need generalized PBK formulas in terms of Hecke eigenvalues in lieu of Fourier coefficients. These are made possible by the spectral assumption. If f is a newform projector, then the space $V_f^{K_\infty }$ is 1-dimensional so that there is essentially only one choice of basis $\mathcal {B}_f(\pi )$ . On the other hand, if f is the classical test function with $c\geq 1$ at some primes, then $\dim V_f^{K_\infty }>1$ and the problem of choosing an orthonormal basis for this space that recovers Hecke eigenvalues from Fourier coefficients has been studied by many authors e.g. [Reference Iwaniec, Luo and SarnakILS00, Reference RouymiRou11, Reference NgNg12, Reference Blomer and MilićevićBM15, BBD+17]. Indeed, following e.g. [Reference PetrowPet18, §7] there exists an orthonormal basis $\mathcal {B}_f(\pi )$ of $V_f^{K_\infty }$ and weights $w(\pi , f) \in \mathbb {C}$ such that for all $m_1,m_2 \in \mathbb {N}$ and $(m_1m_2,N)=1$

(1.9) $$ \begin{align} \sum_{\varphi \in \mathcal{B}_f(\pi)} a_{u_{\varphi}}(m_1) \overline{a_{u_\varphi}(m_2)} = w(\pi,f) \lambda_{\pi}(m_1)\overline{\lambda_\pi(m_2)}, \end{align} $$

where $\lambda _\pi (m)$ are the Hecke eigenvalues of $\pi $ normalized so that the Ramanujan conjecture predicts that $|\lambda _\pi (m)|\leq d(m)$ . Note that the left hand side of (1.9) is independent of the choice of orthonormal basis $\mathcal {B}_f(\pi )$ , and therefore so is $w(\pi ,f)$ .

To continue our discussion, we introduce the “naive Rankin-Selberg L-series”. For $\Pi $ a standard (in the sense of [Reference Michel and VenkateshMV10, §2.2.1]) generic automorphic representation of $\overline {G}$ , let

(1.10) $$ \begin{align} \mathcal{L}_\Pi(s) = \sum_{n \geq 1} \frac{|\lambda_\Pi(n)|^2}{n^s}, \end{align} $$

and following a notation of Michel and Venkatesh, write $\mathcal {L}_\Pi ^*(1)$ for its leading Laurent series coefficient at $s=1$ . For $\pi \in \mathcal {F}_0(f)$ with $q(\pi )$ the (finite) conductor of $\pi $ , write

(1.11) $$ \begin{align} r_\pi(p)^{-1} := \begin{cases} (1+p^{-1}) \sum_{\alpha \geq 0} \frac{\lambda_\pi(p^{2\alpha})}{p^\alpha} & \text{ if } p \nmid q(\pi), \\ (1-p^{-2})^{-1} & \text{ if } p \, \| \, q(\pi), \\ 1 & \text{ if } p^2 \mid q(\pi). \end{cases} \end{align} $$

Then, for f of level N the weights $w(\pi ,f)$ in (1.9) are given by

(1.12) $$ \begin{align} w(\pi,f) = \frac{1}{2 \xi(2) \mathcal{L}_\pi^*(1)} \prod_{\substack{ p^2 \mid N/q(\pi) \\ p \nmid q(\pi)}} (1-p^{-2})^{-1} \prod_{p \mid N/q(\pi)} r_\pi(p)^{-1}=: \frac{1}{2 \xi(2) \mathcal{L}_\pi^*(1)}\frac{1}{\rho_\pi(N/q(\pi))}. \end{align} $$

In (1.12), the weights $\rho _\pi (\ell )$ defined on the right are exactly the same weights $\rho _f(\ell )$ or $\rho _E(\ell )$ defined in [Reference Petrow and YoungPY20, §2.4], with f being the newform in $\pi $ . In particular, we have $w(\pi ,f) = ((1+|t_\pi |)N)^{o(1)}$ by [Reference Goldfeld, Hoffstein and LiemanGHL94, Reference IwaniecIwa90]. Note that the factor $2 \xi (2) = \operatorname {\mathrm {vol}}(\overline {G}(\mathbb Q) \backslash \overline {G}(\mathbb {A}))$ appearing in (1.12) is a global volume factor (see e.g. [Reference Michel and VenkateshMV10, §4.1.2] for a more general statement).

The spectral assumption also allows us to give a motivated expression for the diagonal term constant in the generalized PBK formula in terms of Plancherel volumes.

Definition 1.7 (Local family).

For $f = \bigotimes _p f_p \in {\mathcal {H}}_{\mathrm {fin}}$ , the subspace

(1.13) $$ \begin{align} \mathcal{F}_p(f):= \{\pi \in {\overline{G}(\mathbb Q_p)}^{\wedge}: \pi(f_p) \neq 0\} \end{align} $$

is called the local family of f at p.

Let $\alpha $ be the quasicharacter of $\mathbb Q_p^\times $ defined by $\alpha : x \mapsto |x|_{p}$ . For $\pi $ a smooth irreducible unitary generic representation of $\overline {G}(\mathbb Q_p)$ set

(1.14) $$ \begin{align} \mathcal{L}_\pi(1) = \begin{cases} \frac{(1-p^{-2})}{ ( 1-e^{2 i \theta}p^{-1})(1-p^{-1})(1- e^{-2 i \theta}p^{-1} )} & \text{ if } \pi \simeq \pi(\alpha^{i\theta/\log p}, \alpha^{-i\theta/\log p}), \\ (1+ \frac{1}{p})^{-1} & \text{ if } c(\pi) = 1, \\ (1-\frac{1}{p}) & \text{ if } c(\pi) \geq 2, \end{cases} \end{align} $$

where in the first line either $\theta \in [0, \pi ]$ , or $\theta = i \tau \log p$ or $\pi + i \tau \log p$ with $\tau \in (0,1/2)$ . If $\Pi = \pi _\infty \otimes \bigotimes ^{\prime }_p \pi _p$ is a standard generic automorphic representation of $\overline {G}$ , then the leading Laurent series coefficient $\mathcal {L}_\Pi ^*(1)$ admits the Euler product factorization

(1.15) $$ \begin{align} \mathcal{L}_{\Pi}^*(1) = \prod_p \mathcal{L}_{\pi_p}(1), \end{align} $$

in the regularized sense of [Reference Michel and VenkateshMV10, §4.1.5].

Let $f_\infty $ be the bi- $K_\infty $ -invariant function on $\operatorname {\mathrm {GL}}_2^+(\mathbb R)$ defined by [Reference Knightly and LiKL13, (3.5) and Prop. 3.7] in terms of $h_\infty $ . Then, by the Plancherel theorem (see (3.17) of loc. cit.) we have

(1.16) $$ \begin{align}f_\infty(1) = \frac{1}{4 \pi} \int_{-\infty}^\infty h_\infty(t) \tanh(\pi t) t \,dt. \end{align} $$

Define

(1.17) $$ \begin{align} f_{\mathbb{A}} = f_\infty \cdot f. \end{align} $$

Suppose that f satisfies the spectral assumption. For each place v, define the diagonal weight $\delta _v$ at v as follows. If $v=\infty $ , set $\delta _\infty = f_\infty (1)$ . If $v=p<\infty $ and $f_p$ is a newform projector, set

(1.18) $$ \begin{align} \delta_p = \int_{\mathcal{F}_p(f)} \frac{1}{\mathcal{L}_\pi(1)} \, d \widehat{\mu}(\pi), \end{align} $$

and if $f_p= \nu (p^c)1_{ZK_0(p^c)}$ for some $c\in \mathbb {Z}_{\geq 0}$ , set

(1.19) $$ \begin{align} \delta_p= \int_{\mathcal{F}_p(f)} \dim \pi^{K_0(p^c)}\, d \widehat{\mu}(\pi). \end{align} $$

Note that $\delta _p=1$ for all but finitely many p. Finally, set $\delta _{\mathrm {fin}}= \prod _p \delta _p$ and $\delta = \delta _\infty \delta _{\mathrm {fin}}$ .

Theorem 1.8. Let $f \in {\mathcal {H}}_{\mathrm {fin}}$ be a pure tensor satisfying the geometric and spectral assumptions. For all $m_1,m_2 \in \mathbb {Z}$ with $m_1m_2>0$ and $(m_1m_2,N)=1$ we have

(1.20) $$ \begin{align} \sum_{ \pi \in \mathcal{F}_0(f)} h_\infty(t_\pi) w(\pi, f) \lambda_\pi & (m_1)\overline{\lambda_\pi(m_2)} + (\text{ cts. }) \\ & = \delta_{m_1=m_2}\delta + \sum_{c \equiv 0\ \pmod{k(\mathcal{F})} } \frac{H(m_1,m_2;c)}{c}H_\infty\left(\frac{4 \pi \sqrt{ m_1m_2}}{c}\right),\nonumber \end{align} $$

where $(\text { cts. })$ is a similar continuous spectrum term that we give explicitly in (4.22).

Remark 1.10. Theorem 1.8 also holds for other choices of archimedean test functions. For example, let $\kappa \geq 2$ be even and let $\pi _\kappa $ be the discrete series representation of $\operatorname {\mathrm {GL}}_2(\mathbb R)$ of weight $\kappa $ . Define $\mathcal {F}_\kappa (f)$ as in Definition 1.1 to be the set of cuspidal automorphic representations $\pi $ with $\pi _\infty \simeq \pi _\kappa $ and such that $\pi (f): V_{\pi } \to V_{\pi }$ is not the zero map. Define $V_f^\kappa $ to be the weight $\kappa $ isotypic subspace of $V_f$ . Choose $f_\infty $ to be given by the Bergman kernel (2.23), in particular one has $f_\infty (1) = \frac {\kappa -1}{4 \pi }$ . Then, under the same hypotheses as Theorem 1.8 with $\mathcal {F}_0(f)$ and $V_f^{K_\infty }$ replaced by $\mathcal {F}_\kappa (f)$ and $V_f^\kappa $ , respectively, we have

(1.21) $$ \begin{align} \sum_{ \pi \in \mathcal{F}_\kappa(f)} w(\pi, f) & \lambda_\pi(m_1)\overline{\lambda_\pi(m_2)} \\ & = \delta_{m_1=m_2} \delta + \frac{(\kappa-1)}{2} i^{-\kappa} \sum_{c \equiv 0\ \pmod{k(\mathcal{F})} } \frac{H(m_1,m_2;c)}{c}J_{\kappa-1}\left(\frac{4 \pi \sqrt{ m_1m_2}}{c}\right), \nonumber \end{align} $$

with $w(\pi ,f) = (\kappa N)^{o(1)}$ . For $\kappa \geq 4$ the archimedean aspects of the holomorphic forms variation (1.21) were worked out in [Reference Knightly and LiKL06a] and while a relative trace formula proof of the $\kappa =2$ case strictly speaking has not appeared in the literature, it is expected to follow from a limiting argument and in any case is well-known from the classical Poincaré series approach to the Petersson formula. We give more details on the holomorphic/discrete series case in Sections 2.3 and 4.4.

Similarly, one expects the opposite-sign case of Theorem 1.8 in which $m_1m_2<0$ to hold, but at present there is not a relative trace formula proof for this case. The shape of the formula would be similar but with an additional factor of $p(\pi )$ on the spectral side, where

(1.22) $$ \begin{align} p(\pi) = \text{ parity of } \pi = \begin{cases} \text{ eigenvalue of } u_\varphi, \varphi \in V_\pi \text{ under the involution } x+iy \mapsto -x+iy, \text{ or } \\ (-1)^\epsilon \text{ where } \pi_\infty \simeq \pi(\operatorname{\mathrm{sgn}}^\epsilon |\cdot|^s , \operatorname{\mathrm{sgn}}^\epsilon |\cdot|^{-s}), \end{cases} \end{align} $$

and the factor $H_\infty (x)$ on the geometric side of the formula is replaced with

(1.23) $$ \begin{align} H_\infty^-(x) = \frac{1}{\pi} \int_{-\infty}^\infty K_{2it}(x) \sinh (\pi t) h(t) t \,dt. \end{align} $$

We posit that when $\kappa =2$ the formula (1.21) holds, and when $m_1m_2<0$ the formula (1.20) with modifications (1.22) and (1.23) holds, and assume these to be so in the following discussion. Since this paper concerns non-archimedean aspects, and for the sake of brevity, we do not provide any details for these assertions.

Remark 1.11. Since we have not modified any archimedean aspects of the classical PBK formulas when deducing Theorem 1.8 etc., we also get the “backwards” Kuznetsov formula as in [Reference Iwaniec and KowalskiIK04, §16.4] mutatis mutandis. Indeed, for the rest of this paragraph let $\Phi \in C^2([0,\infty ))$ with

$$ \begin{align*}\Phi(0)=0 \quad \text{ and } \quad \Phi^{(a)}(x) \ll_a (1+x)^{-\alpha}\end{align*} $$

for $a=0,1,2$ and some $\alpha>2$ . Let $\mathcal {M}_\Phi (t)$ be the Hankel transform of $\Phi $ as defined in [Reference Iwaniec and KowalskiIK04, 16.40] and $\mathcal {N}_f(k)$ be the Neumann coefficients of $\Phi $ as defined in [Reference Iwaniec and KowalskiIK04, 16.41]. Let $f\in {\mathcal {H}}_{\mathrm {fin}}$ be as in Theorem 1.8 with associated generalized Kloosterman sum $H(m,n;c)$ . If $m_1m_2>0$ , then

(1.24) $$ \begin{align} & \sum_{c \equiv 0\ \pmod{k(\mathcal{F})} } \frac{H(m_1,m_2;c)}{c}\Phi\left(\frac{4 \pi \sqrt{ m_1m_2}}{c}\right) \\ & \quad= \frac{4}{\pi} \sum_{ \pi \in \mathcal{F}_0(f)} \mathcal{M}_\Phi(t_\pi) \cosh (\pi t_\pi) \sum_{\varphi \in \mathcal{B}_f(\pi)} a_{u_{\varphi}}(m_1) \overline{a_{u_\varphi}(m_2)} + (\text{ cts. }) \nonumber\\ & \quad\quad+ \sum_{\substack{\kappa>0 \\ \kappa \equiv 0\ \pmod 2}} \frac{(4\pi)^\kappa}{\Gamma(\kappa)} \mathcal{N}_\Phi(\kappa) \sum_{ \pi \in \mathcal{F}_\kappa(f)} \sum_{\varphi \in \mathcal{B}_f(\pi)} a_{u_{\varphi}}(m_1) \overline{a_{u_\varphi}(m_2)} ,\nonumber \end{align} $$

where $(\text { cts. })$ is a continuous spectrum term given in (4.19) with $h_\infty (t)$ there replaced by $\frac {4}{\pi } \cosh (\pi t) \mathcal {M}_\Phi (t)$ . See Section 2.3 for definitions of $u_\varphi $ and $a_u$ in the holomorphic / discrete series case. Meanwhile, if $m_1m_2<0$ , then we set $\mathcal {K}_\Phi (t)$ to be the integral transform of $\Phi $ given in [Reference Iwaniec and KowalskiIK04, (16.44)]. In this case, we have

(1.25) $$ \begin{align} \sum_{c \equiv 0\ \pmod{k(\mathcal{F})} } & \frac{H(m_1,m_2;c)}{c}\Phi\left(\frac{4 \pi \sqrt{ |m_1m_2|}}{c}\right) \\ & = \frac{4}{\pi} \sum_{ \pi \in \mathcal{F}_0(f)} \mathcal{K}_\Phi(t_\pi) \cosh (\pi t_\pi) \sum_{\varphi \in \mathcal{B}_f(\pi)} a_{u_{\varphi}}(m_1) \overline{a_{u_\varphi}(m_2)} + (\text{ cts. }),\nonumber \end{align} $$

where similarly $(\text { cts. })$ is a continuous spectrum term given in (4.19) with $h_\infty (t)$ there replaced by $\frac {4}{\pi } p(\pi _{\chi ,\chi ^{-1}}) \cosh (\pi t) \mathcal {K}_\Phi (t)$ .

Examples. Choose a finite set S of primes. Let $f = \bigotimes _p f_p \in {\mathcal {H}}_{\mathrm {fin}}$ be a pure tensor such that if $p \not \in S$ , then $f_p = 1_{ZK_p}$ , and if $p \in S$ , then $f_p \in {\mathcal {H}}_p$ may be chosen from any of the following:

• • for some integer $c\geq 0$ , the classical test function $f_{\leq c}$ defined in (7.1),

• • for a nonquadratic character $\chi $ of $\mathbb {Z}_p^\times $ , the function $f_\chi $ defined in (7.7), or

• • for a quadratic extension E of $\mathbb Q_p$ and a character $\xi $ of $E^\times $ satisfying the hypotheses in the first paragraph of Section 7.3, the function $f_{\xi }$ defined in (7.18),

• • for $(E/\mathbb Q_p,\xi )$ as in the previous point and $1\leq n<c(\xi ')$ with $\xi '$ a twist-minimal character underlying $\xi $ , the function $f_{\xi ,n}$ defined in (7.27), or

• • for some integer $c\geq 3$ , the test function $f_{=c}$ introduced by Nelson, see (7.31).

Then f satisfies the geometric and spectral assumptions (see Sections 7.1.1, 7.2.1, 7.3.1, 7.4, and 7.5).

Let $\mathcal {F}_0(f)$ be the family of cuspidal automorphic representations cut out by f as in Definition 1.1. In particular, $\mathcal {F}_0(f)$ consists of $\operatorname {\mathrm {GL}}_2/\mathbb Q$ cuspidal automorphic representations $\pi $ of trivial central character (spherical at infinity) whose local components $\pi _p$ at finite places are constrained to lie in the local families

$$ \begin{align*}\mathcal{F}_p(f) = \begin{cases} \mathcal{F}_{\leq c} & \text{ if } f_p = f_{\leq c}, \\ \mathcal{F}_\chi & \text{ if } f_p= f_{\chi}, \\ \mathcal{F}_{\xi} & \text{ if } f_p= f_{\xi}, \\ \mathcal{F}_{\xi, n} & \text{ if } f_p= f_{\xi,n}, \text{ and }\\ \mathcal{F}_{=c} & \text{ if } f_p = f_{=c} \end{cases} \end{align*} $$

(see Definition 1.7), where:

• • For $c\geq 0$

  (1.26) $$ \begin{align} \mathcal{F}_{\leq c} = \{ \pi \in {\overline{G}}(\mathbb Q_p)^{\wedge}: c(\pi) \leq c\}. \end{align} $$

• • For $\chi \in {\mathbb {Z}_p^{\times }}^{\wedge }$ not quadratic

  (1.27) $$ \begin{align} \mathcal{F}_\chi = \{ \pi(\mu,\mu^{-1}) \in {\overline{G}}(\mathbb Q_p)^{\wedge}: \mu \vert_{\mathbb{Z}_p^\times} = \chi\}. \end{align} $$

• • For $(E/\mathbb Q_p,\xi )$ as above, let $\sigma = \sigma (\rho )$ be the (trivial central character) supercuspidal representation of $G(\mathbb Q_p)$ corresponding to $\rho = \operatorname {\mathrm {Ind}}_{E}^{\mathbb Q_p} \xi $ under the Local Langlands Correspondence (LLC). Writing $\eta $ for the unramified quadratic character of $\mathbb Q_p^\times $ , we have

  (1.28) $$ \begin{align} \mathcal{F}_{\xi} = \begin{cases} \{\sigma\} & \text{ if } E/\mathbb Q_p \text{ is unramified and } p \neq 2, \\ \{\sigma, \sigma \times \eta\} & \text{ if } E/\mathbb Q_p \text{ is ramified,} \\ \{ \sigma( \operatorname{\mathrm{Ind}}_{E}^{\mathbb Q_p} \xi_1) : c(\xi_1 \xi^{-1})\leq 1, \xi_1\vert_{\mathbb Q_p^\times} = \xi \vert_{\mathbb Q_p^\times} \} & \text{ if } E/\mathbb Q_p \text{ is unramified and } p = 2. \end{cases} \end{align} $$

  The set in the last line consists of 3 supercuspidal representations of the same conductor as $\sigma $ . For interpretation, it may be helpful to recall that when $p \neq 2$ , the extension $E/F$ is unramified if and only if $c(\sigma )$ is even. See Section 6.1.2 for a quick overview of the parametrization of dihedral trivial central character supercuspidal representations in terms of pairs $(E/\mathbb Q_p,\xi )$ .

• • For $1\leq n<c(\xi ')$ , we have

  $$ \begin{align*}\mathcal{F}_{\xi,n}= \{ \sigma( \operatorname{\mathrm{Ind}}_{E}^{\mathbb Q_p} \xi_1) : c(\xi_1 \xi^{-1})\leq n, \xi_1\vert_{\mathbb Q_p^\times} = \xi \vert_{\mathbb Q_p^\times} \}.\end{align*} $$

• • For $c\geq 3$ , we have

  $$ \begin{align*}\mathcal{F}_{=c} = \{\pi \in {\overline{G}}(\mathbb Q_p)^{\wedge} : c(\pi ) = c\}.\end{align*} $$

See Sections 7.1.2, 7.2.2, 7.3.2, 7.4, and 7.5 for more details.

The level N of f satisfies

(1.29) $$ \begin{align} v_p(N) = \begin{cases} c & \text{ if } f_p= f_{\leq c} \text{ or } f_{=c},\\ 2c(\chi) & \text{ if } f_p=f_{\chi}, \\ c(\sigma) & \text{ if } f_p = f_{\xi} \text{ or } f_{\xi,n}. \end{cases} \end{align} $$

On the geometric side of the formula, the diagonal weights $\delta _p$ may be given explicitly by

(1.30) $$ \begin{align} \delta_p = \begin{cases} \nu(p^c) & \text{ if } f_p = f_{\leq c}, \\ \frac{\nu(p^{c(\chi)})}{1-p^{-1}} & \text{ if } f_p = f_\chi, \\ \text{ see (7.21)} & \text{ if } f_p=f_\xi, \\ \text{ see (7.22)} & \text{ if } f_p = f_{\xi,n}, \\ p^c(1-p^{-2}) & \text{ if } f_p =f_{=c}. \end{cases}\end{align} $$

for which see (7.3), (7.13) and Section 7.5. In the supercuspidal cases, we write $d=v_p(\operatorname {\mathrm {disc}} (E/\mathbb Q_p))$ . The geometric conductor $k(\mathcal {F})= \prod _p p^{k_p}$ and the local geometric conductors for the above test functions are given explicitly by

(1.31) $$ \begin{align} k_p = \begin{cases} c & \text{ if } f_p = f_{\leq c}, \\ c(\chi) & \text{ if } f_p = f_\chi, \\ c(\xi) & \text{ if } f_{p} = f_{\xi} \text{ with }d = 0, \\ \frac{c(\xi)}{2} +1 & \text{ if } f_p = f_{\xi} \text{ with } d = 1 \text{ or } 2, \\ \frac{c(\xi)}{2} +2 & \text{ if } f_p = f_{\xi} \text{ with } d = 3, \\ \text{ see (7.30)} & \text{ if } f_p = f_{\xi,n}, \\ c-1 & \text{ if } f_p = f_{=c}, \end{cases}\end{align} $$

for which see Sections 7.1.6, 7.2.6, 7.3.6, 7.4, and 7.5.

Lastly, for $c \equiv 0\ \pmod {k(\mathcal {F})}$ , the generalized Kloosterman sum is given by

$$ \begin{align*}H(m,n;c) = \prod_{p \mid c} H_p(m,n;c),\end{align*} $$

where each local Kloosterman sum $H_p$ can be explicitly described as follows. For each p, let us write $c=c_0p^k$ with $(c_0,p)=1$ , and where we assume that $k\geq k_p$ (otherwise $H(m,n;c)=0$ ). Write $\overline {c_0}$ for the inverse of $c_0$ modulo $p^k$ .

If $ f_p = f_{\leq \mathfrak c}$ (including the case $\mathfrak c=0$ ), then we have

(1.32) $$ \begin{align} H_p(m,n;c) = \delta_p S(\overline{c_0}m, \overline{c_0}n; p^k). \end{align} $$

If $f_p=f_\chi $ with $\chi $ not quadratic, then we have

(1.33) $$ \begin{align} H_p(m,n;c) = \delta_p \sideset{}{^*}\sum_{\substack{x,y\ \pmod{p^k} \\ xy =mn\overline{c_0}^2}} \overline{\chi(x)}\chi(y)e\left( \frac{x+y}{p^k}\right) \end{align} $$

when $(mn,p)=1$ and $H_p(m,n;c) = 0$ otherwise.

If $f_p = f_{\xi }$ for a pair $(E/\mathbb Q_p,\xi )$ satisfying the hypotheses in the first paragraph of Section 7.3, then

(1.34) $$ \begin{align} H_p(m,n;c)=\delta_p \overline{\gamma} p^{-\frac{d}{2}} \sum_{\substack{u \in (\mathcal{O}_E/p^k\mathcal{O}_E)^{\times} \\ \operatorname{\mathrm{Nm}}(u) = mn\overline{c_0}^2}} \xi(u) e\left( -\frac{\operatorname{\mathrm{Tr}} u}{p^{k}} \right) \end{align} $$

when $(mn,p)=1$ and $H_p(m,n;c) = 0$ otherwise. In (1.34), $\operatorname {\mathrm {Nm}}, \operatorname {\mathrm {Tr}}: E \to \mathbb Q_p$ are the field norm and trace, and $\gamma $ is the Langlands constant associated to E and the additive character $\psi _p=e(\{.\}_p)$ of $\mathbb Q_p$ . See Remark 6.46 following Theorem 6.45 for more detailed information on $\gamma $ and Propositions 6.56 and 6.58 for bounds on $H_p(m,n;c)$ .

In [Reference HuHu24, Def. 4.6] the first author gave an alternative formula for $H_p(m,n;c)$ that at first glance looks quite different from (1.34). However, these two formulas are in fact equal (up to leading constants) whenever the former formula is valid, as can be seen by computing the Fourier/Mellin transform of both formulas and applying p-adic stationary phase analysis.

If $f_p = f_{\xi ,n}$ , then $H_p(m,n;c)$ is exactly the same as in (1.4), but with $\delta _p$ and $k_p$ given by (7.28) and (7.30) in lieu of (7.21) and (7.23).

1.3 Relations between parameters

The reader may have already observed that the families of automorphic forms in this paper have several different parameters associated with them. These include:

• • the level N of f,

• • the primes p at which f is ramified,

• • the conductors $q(\pi )$ of representations $\pi \in \mathcal {F}_0(f)$ and the conductor exponents $c(\pi )$ of local representations $\pi \in \mathcal {F}_p(f)$ ,

• • the geometric conductor $k(\mathcal {F})$ and local geometric conductors $k_p$ ,

• • the value $f(1)$ and local values $f_p(1)$ , and

• • the diagonal weight $\delta _{\mathrm {fin}}$ and local diagonal weights $\delta _p$ .

We explicate some of the relations between the above quantities.

1.3.4 Conductors of representations versus $f(1)$

Suppose that $f\neq 0$ satisfies the geometric and spectral assumptions. Let us work locally at p. If $f_p=f_{\leq c}$ is the classical test function it is clear that

$$ \begin{align*}\{c(\pi) : \pi \in \mathcal{F}_p(f) \} = \{0,\ldots, c\},\end{align*} $$

so we henceforth assume that $f_p$ is a newform projector.

Since $f\neq 0$ , then $\mathcal {F}_p(f) \neq 0$ . If $\mathcal {F}_p(f)$ contains an irreducible principal series representation $\pi (\chi ,\chi ^{-1})$ with $\chi \vert _{\mathbb {Z}_p^\times }$ not quadratic, then by Lemma 3.11 it contains $\pi (\chi \alpha ^{it},\chi ^{-1} \alpha ^{-it})$ for all $t \in \mathbb R$ . Suppose that $\mathcal {F}_p(f)$ only contains $\pi (\chi ,\chi ^{-1})$ with $\chi \vert _{\mathbb {Z}_p^\times }$ nontrivial quadratic. Then, by Remark 3.12 it also contains a special representation. Thus, $\mathcal {F}_p(f)$ either contains a square-integrable representation, or for some $\chi $ with $\chi \vert _{\mathbb {Z}_p^\times }$ not quadratic it contains $\mathcal {F}_\chi = \{\pi (\chi \alpha ^{it},\chi ^{-1} \alpha ^{-it}):t \in \mathbb R\} \subseteq \mathcal {F}_p(f)$ , or it contains $\mathcal {F}_p(f)= \{\pi : \pi \text { unramified }\}$ . Thus, since $f_p$ is a newform projector, by the Plancherel formula we have

(1.35) $$ \begin{align} f_p(1) = \widehat{\mu}(\mathcal{F}_p(f)) \geq \begin{cases} \widehat{\mu}(\{ \pi \} ) & \text{ if } \pi \in \mathcal{F}_p(f) \text{ is square integrable, } \\ \widehat{\mu}(\mathcal{F}_\chi) & \text{ if } \pi(\chi,\chi^{-1}) \in \mathcal{F}_p(f), \chi \text{ not quadratic, } \\ 1 & \text{ if } c(\pi)= 0 \text{ for all } \pi \in \mathcal{F}_p(f). \end{cases} \end{align} $$

Let $d_\mu (\pi )$ denote the formal degree of $\pi $ . If $\pi $ is square integrable, then $\widehat {\mu }(\{\pi \}) = d_\mu (\pi ) \gg p^{c(\pi \times \pi )/2}\gg p^{\lceil c(\pi )/2\rceil }$ by e.g. [Reference Ichino, Lapid and MaoILM17, Thm. 2.1], with an absolute implied constant. In the principal series case, one has $\widehat {\mu }(\mathcal {F}_\chi )=\nu (p^{c(\chi )})$ . In all cases, if $ \pi \in \mathcal {F}_p(f)$ with f a newform projector satisfying the geometric assumptions, then $f_p(1) \gg p^{\lceil c(\pi )/2\rceil }$ with an absolute implied constant.

In the other direction, if we set $c_{\mathrm {max}} = \max \{c(\pi ) : \pi \in \mathcal {F}_p(f)\}$ , then

(1.36) $$ \begin{align} f(1) \leq \int_{\mathcal{F}_p(f)} \dim V_\pi^{K_0(p^{c_{\mathrm{max}}})}\,d\widehat{\mu}(\pi) = \nu(p^{c_{\mathrm{max}}}) \leq \tfrac{3}{2}p^{c_{\mathrm{max}}}. \end{align} $$

1.3.5 Level versus $f(1)$

Suppose that f satisfies the spectral assumption. Working locally, suppose $c= \max \{c(\pi ): \pi \in \mathcal {F}_p(f)\}$ . Then, by Proposition 4.1, $f_p$ is bi- $K_0(p^{c})$ -invariant and also bi- $K(N)$ -invariant, so that f is bi- $K_0(N)$ -invariant. If $f_p$ is a newform projector, then by the Plancherel formula and newform theory

$$ \begin{align*}f_p(1) = \widehat{\mu}(\mathcal{F}_p(f)) \leq \widehat{\mu}(\{\pi: c(\pi) \leq v_p(N)\}) = \nu * \mu(N_p) \leq N_p.\end{align*} $$

On the other hand if $f_p$ is the classical test function, then $f_p(1)=\nu (N_p)$ . Therefore globally,

(1.37) $$ \begin{align} f(1) \leq \nu(N). \end{align} $$

In the other direction, we work locally and assume that $f_p$ satisfies the geometric and spectral assumptions. If $f_p$ is the classical test function, then $f_p(1)=\nu (N_p)$ , so suppose $f_p$ is a newform projector. Let $\pi \in \mathcal {F}_p(f)$ be a representation of maximal conductor exponent. Then by Proposition 4.1, $f_p$ is bi- $K_0(p^{c(\pi )})$ -invariant, thus $N_p \mid p^{c(\pi )}$ , and so by Section 1.3.4, we have

(1.38) $$ \begin{align} N_p^{1/2} \leq p^{c(\pi)/2} \leq p^{\lceil c(\pi)/2\rceil} \ll f_p(1). \end{align} $$

1.3.6 Diagonal weight versus $f(1)$

Suppose that f satisfies the spectral assumption. In the first case when $f_p$ is a newform projector, we have by inspecting (1.14) that $\delta _p = (1+O(p^{-1})) f_p(1)$ and moreover $\tfrac {1}{6} f_p(1) \leq \delta _p \leq 2 f_p(1)$ , so that $\delta _p$ is nonvanishing. In the second case that $f_p$ is the classical test function, the situation is even simpler as we have $\delta _p = f_p(1)$ . Then, by (1.37) and (1.38) one has

(1.39) $$ \begin{align} \delta_{\mathrm{fin}} = f(1)N^{o(1)} = f(1)^{1+o(1)}. \end{align} $$

1.4 Weighted Weyl-Selberg Law and equidistribution

In this section and in Section 1.5, we consider families of automorphic representations. That is, we consider sequences of varying test functions f or $f_{\mathbb {A}}$ with some parameter, usually $f(1)$ or $f_{\mathbb {A}}(1)$ , going to infinity. Recall by the Plancherel formula (see e.g. (4.3)), that $f_p(1)$ is equal to an integral over the local family $\mathcal {F}_p(f)$ of representations with respect to Plancherel measure, and that the spectral assumption implies that $f(1)\geq 0$ .

To streamline the exposition in this section and the next, we only present our results in detail for Maass cusp forms, but similar results hold for the holomorphic / discrete series variation. Accordingly, we now choose the archimedean test function $h_\infty $ to be one of either

(1.40) $$ \begin{align} h_\infty(t) = \frac{t^2+\frac14}{T^2} \left[ \frac{1}{\cosh\left( \frac{t-T}{\Delta} \right)} + \frac{1}{\cosh\left( \frac{t+T}{\Delta} \right)} \right], \end{align} $$

where $1 \leq \Delta < T/100$ , to give a smooth approximation to the small window $T -\Delta < \pm t \leq T + \Delta $ , or alternatively

(1.41) $$ \begin{align} h_\infty(t) = \frac{t^2+\frac14}{T^2} \exp\left(-\left(\frac{t}{T}\right)^2\right) \end{align} $$

for a smooth approximation to the large window $|t| \leq T$ . We call $\pm [T-\Delta ,T+\Delta ]$ the effective support of (1.40) and $[-T,T]$ the effective support of (1.41).

With these weights, we have the following crude bound.

Lemma 1.13. Let $h_\infty $ be one of the two test functions given by (1.40) or (1.41). If $f \in {\mathcal {H}}_{\mathrm {fin}}$ and $w(\pi ,f)$ are as in Theorem 1.8, then for all $m_1,m_2 \in \mathbb {Z}$ with $m_1m_2>0$ and $(m_1m_2,N)=1$ we have

(1.42) $$ \begin{align} \sum_{ \pi \in \mathcal{F}_0(f)} h_\infty(t_\pi) w(\pi, f) \lambda_\pi(m_1)\overline{\lambda_\pi(m_2)} + (\text{ cts. }) \delta_{m_1=m_2} \delta + O\left( \frac{f_{\mathbb{A}}(1)m_1m_2}{T^2 k(\mathcal{F})}\right). \end{align} $$

We emphasize that we made no effort for optimality in Lemma 1.13, including at the archimedean place, but are rather just recording a simple bound. As an illustration, we only use the trivial bound on the generalized Kloosterman sums in this proof, and any nontrivial bound on the ramified part of the generalized Kloosterman sums would improve the error term in (1.42). Note that by (1.39) the main term in (1.42) is larger than the error term as soon as there exists $\delta>0$ so that $\frac {T^2 k(\mathcal {F})}{m_1m_2} \gg f_{\mathbb {A}}(1)^\delta $ . Particularly pleasing is the shape of the main term in the following corollary.

Corollary 1.14 (Harmonically weighted Weyl-Selberg Law).

Let $h_\infty $ be one of the two test functions given by (1.40) or (1.41). If f satisfies Geometric Assumption (2) and is a newform projector, then we have

(1.43) $$ \begin{align} \sum_{\pi \in \mathcal{F}_0(f)} \frac{h_\infty(t_\pi)}{\mathcal{L}_\pi^*(1)} + (cts.) = \operatorname{\mathrm{vol}}( \overline{G}(\mathbb Q) \backslash \overline{G}(\mathbb{A})) f_\infty(1) \prod_p \int_{\mathcal{F}_p(f)} \frac{1}{\mathcal{L}_{\pi_p}(1)} \,d\widehat{\mu}(\pi_p) + O\left( \frac{f_{\mathbb{A}}(1)}{T^2 k(\mathcal{F})}\right). \end{align} $$

See Section 5.1 for the proofs of Lemma 1.13 and Corollary 1.14. For the holomorphic/discrete series variation alluded to at the beginning of Section 1.4, note that equation (5.4) used in the proof of Lemma 1.13 may no longer hold due the possible presence of weight $2$ holomorphic forms. In this situation, the trivial bound (5.1) on Kloosterman sums is no longer sufficient to prove the analogue of Lemma 1.13. Instead, we may factor $H(m,n;c)$ by Theorem 3.8(3) and use the classical Weil bound on the unramified factor to conclude the analogues of Lemma 1.13 and Corollary 1.14.

Remark 1.17. Corollary 1.14 can be interpreted as an instance of a general equidistribution statement for cusp forms. Let $\mathcal {A}_0(G/k)$ be the set of all unitary cuspidal automorphic representations of G over a number field k. Drawing inspiration from the work of Brumley and Milićević [Reference Brumley and MilićevićBM24, §1.1, §2], who studied the universal family $\mathcal {A}_0(\operatorname {\mathrm {GL}}_n/k)$ ordered by analytic conductor, one expects that for any sufficiently well-behaved test function h on $\mathcal {A}_0(G/k)$

(1.44) $$ \begin{align} \sum_{\pi \in \mathcal{A}_0(G/k)} h(\pi) \sim \operatorname{\mathrm{vol}}( G(k) \backslash G(\mathbb{A})^1) \int_{\pi \in {G(\mathbb{A})^1}^{\wedge}} h(\pi) \, d \widehat{\mu}(\pi), \end{align} $$

as the average analytic conductor of the effective support of h tends to infinity. Indeed, Brumley and Milićević (Thm. 1.2) prove for $G=\operatorname {\mathrm {GL}}_n$ over a number field that if h is the indicator function of forms having analytic conductor $\leq Q$ that (1.45) holds as $Q \to \infty $ with an explicit effective savings of $(\log Q)^{-1}$ over the main term. To see this, follow the proof of their Theorem 1.2, but instead of the final sentence of loc. cit. Lemma 12.1, use the final displayed equation in loc. cit. Proof of Proposition 6.1 and Corollary 6.2 to express the main term of loc. cit. (12.2) summed over all $\mathfrak {q}$ and $\mathfrak {d} \mid \mathfrak {q}$ as the adelic Plancherel volume of a conductor ball.

Corollary 1.14 is also an instance of (1.45) in the case that $G = \operatorname {\mathrm {PGL}}_2$ and $k=\mathbb Q$ and with h the harmonic weights given by $h(\pi ) = h_\infty (t_\pi )/\mathcal {L}_\pi ^*(1)$ .

1.5 Large sieve inequality

As remarked at the beginning of Section 1.4, here we consider families of automorphic representations, which in practice means that certain implied constants should hold uniformly within a given family. To streamline the exposition we focus on the Maass form / spherical at $\infty $ case, but claim that the results of this section go through in the holomorphic / discrete series variation. See Remark 1.24.

We now propose a framework for optimal large sieve inequalities. Let $\mathcal {F}$ be a finite set of cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2$ over $\mathbb Q$ with trivial central character all having the same (finite) conductor $q=q(\mathcal {F})$ . Suppose that there exists a pure tensor $f \in {\mathcal {H}}_{\mathrm {fin}}$ and an $h_\infty $ as in the PBK formula such that $\mathcal {F} \subseteq \mathcal {F}_0(f)$ and with the effective support (see Section 1.4 for definition) of $h_\infty $ containing the spectral parameters $\{ t_\pi : \pi \in \mathcal {F}\}$ . We will show in Theorem 1.23 that $\mathcal {F}$ satisfies an optimal large sieve inequality if the test function f satisfies the hypotheses introduced next.

Let $T= T(\mathcal {F})$ be the infimum of the $T\geq 0$ such that the set of spectral parameters $\{t_\pi : \pi \in \mathcal {F}\}$ is contained in $[-T,T]$ .

We assume that Hypothesis 1.18 (TF) holds for the remainder of this section. The next hypothesis encodes the assumption that $\mathcal {F}_0(f)$ is not too much larger than $\mathcal {F}$ .

Hypothesis 1.19 ( $\mathcal {F}_0(f)$ not much larger than $\mathcal {F}$ (NmL)).

We suppose that

• • $f \in {\mathcal {H}}_{\mathrm {fin}}$ has $N \mid q^{\infty }$ and $\mathcal {F} \subseteq \mathcal {F}_0(f)$ ,

• • if $T\ll q^\varepsilon $ for all $\varepsilon>0$ , then $h_\infty $ is given by (1.41) with the spectral parameters of $\mathcal {F}$ in the effective support of $h_\infty $ ,

• • if there exists uniform $C, \delta>0$ such that $T> C q^\delta $ , then $h_\infty $ is given by (1.40) with $T^\varepsilon \ll \Delta \ll T^{1-\varepsilon }$ and the spectral parameters of $\mathcal {F}$ in the effective support of $h_\infty $ ,

and f, $h_\infty $ are such that

(1.45) $$ \begin{align} \sum_{\pi \in \mathcal{F}_0(f)} h_\infty(t_\pi) w(\pi, f) + (\text{ cts. })= |\mathcal{F}| (qT)^{o(1)} \end{align} $$

where the weights $w(\pi , f)$ are as in Theorem 1.8.

In practice, the existence of appropriate $h_\infty $ satisfying Hypothesis (1.19) (NmL) can often be achieved after partitioning $\mathcal {F}$ into subsets according to the sizes of the archimedean spectral parameters $t_\pi $ of $\pi \in \mathcal {F}$ relative to q. See the example at the end of this section.

Next, we need a hypothesis asserting some control on the generalized Kloosterman sums $H(m,n;c)$ of f. In fact, we do not need a bound on $H(m,n;c)$ itself, but only on its Fourier/Mellin transform for ramified moduli. Note that $c\in \mathbb {Z}$ for any nonvanishing $H(m,n;c)$ by Hypothesis 1.18 (TF), see Lemma 4.6(4). For a Dirichlet character $\chi\ \pmod {c}$ , let

(1.46) $$ \begin{align} \widehat{H}(\chi) = \frac{1}{\varphi(c)} \sideset{}{^*}\sum_{y\ \pmod{c}} H(y,1;c) \overline{\chi}(y), \end{align} $$

so that Fourier inversion gives

(1.47) $$ \begin{align} H(y,1;c) = \sum_{\chi\ \pmod{c}} \widehat{H}(\chi) \chi(y). \end{align} $$

Hypothesis 1.20 (Fourier transform bound (FTB)).

Suppose that for any $c \mid N^\infty $ and $\chi\ \pmod {c}$ we have

(1.48) $$ \begin{align} \|\widehat{H}\|_{\infty} := \max_{\chi\ \pmod{c}} |\widehat{H}(\chi)| \ll f(1) c^{\varepsilon} \end{align} $$

uniformly in f and for all $\varepsilon>0$ .

Hypothesis FTB reduces to checking local statements at each $p\mid N$ . Indeed, suppose $\chi $ is a Dirichlet character modulo c with factorization $\chi = \prod _{p \mid c} \chi _p$ and for each $p \mid c$ we write $c=c_0p^{v_p(c)}$ . Then, by (3.12), Lemma 3.9 and (3.16) we have

$$ \begin{align*}\widehat{H}(\chi) = \prod_{p \mid c}\overline{\chi}(c_0)^2 \widehat{H}_p(\chi_p,v_p(c)),\end{align*} $$

where for $\alpha $ a Dirichlet character with p-power conductor (equivalently, a character of $\mathbb {Z}_p^\times $ ) and $k\geq 0$ we have set

(1.49) $$ \begin{align} \widehat{H}_p(\alpha,k) = \frac{1}{\varphi(p^k)} \sideset{}{^*}\sum_{y\ \pmod{p^k}} H_p(y,1;p^k) \overline{\alpha(y)}= \zeta_p(1)\int_{\mathbb{Z}_p^\times} H_p(y,1;p^k) \overline{\alpha(y)} \,dy \end{align} $$

with $dy$ the additive Haar measure that gives $\mathbb {Z}_p$ volume 1. Thus, to verify Hypothesis 1.20 (FTB), it suffices to show that for $p \mid N$ and all $\alpha $ with p-power conductor and $k\geq 0$ that

(1.50) $$ \begin{align} \widehat{H}_p(\alpha,k) \ll f_p(1), \end{align} $$

with implicit constants independent of $p, \alpha , k, f_p$ , but possibly depending on the family in which f varies.

Note also that if Hypothesis 1.20 (FTB) holds (with $c|N^{\infty }$ ), then the bound (1.49) holds for any character $\chi $ of any modulus c since at primes away from N the generalized Kloosterman sum $H(m,n;c)$ reduces to the classical Kloosterman sum, and we easily derive the required bounds.

Finally, we state our last hypothesis.

Hypothesis 1.21 (Conductor versus size of family (CvF)).

We suppose that

(1.51) $$ \begin{align} k(\mathcal{F}) \gg f(1)^{1-\varepsilon} \end{align} $$

uniformly in f and for all $\varepsilon>0$ .

Again, note that to verify Hypothesis 1.21 (CvF), it suffices (using (1.38)) to show for $p \mid N$ that

(1.52) $$ \begin{align} p^{k_p} \gg f_p(1) \end{align} $$

with implicit constants independent of $p, f_p$ , but possibly depending on the family in which f varies.

Here is an example application of Hypothesis 1.21 (CvF), which is moreover used in the proof of the following theorem.

Lemma 1.22. Let $h_\infty $ be one of the two test functions given by (1.40) or (1.41). If $f \in {\mathcal {H}}_{\mathrm {fin}}$ satisfies Hypotheses 1.18 (TF) and 1.21 (CvF), and $w(\pi ,f)$ are as in Theorem 1.8, then

(1.53) $$ \begin{align} f_{\mathbb{A}}(1) \ll_\varepsilon f(1)^\varepsilon\Big(\sum_{\pi \in \mathcal{F}_0(f)} h_\infty(t_\pi) w(\pi, f) + (\text{ cts. }) \Big). \end{align} $$

Proof. By Lemma 1.13 with $m_1=m_2=1$ , (1.39), and the definition $\delta _\infty = f_\infty (1)$ , we have

$$ \begin{align*}f_{\mathbb{A}}(1)\Big( f(1)^{o(1)} + O \Big(\frac{1}{T^2 k(\mathcal{F})}\Big)\Big) = \sum_{\pi \in \mathcal{F}_0(f)} h_\infty(t_\pi)w(\pi, f) + (\mathrm{cts.}).\end{align*} $$

By Hypothesis CvF, we have that the sum in parentheses on the left is nonvanishing and $\gg f(1)^{-\varepsilon }$ for $f_{\mathbb {A}}(1)$ sufficiently large.

Recall we write $\lambda _\pi (n)$ for the nth Hecke eigenvalue of $\pi $ , normalized so that the Ramanujan conjecture predicts that $|\lambda _\pi (n)|\leq d(n)$ for all $n \in \mathbb {N}$ .

Theorem 1.23 (Optimal Large Sieve Inequality).

Suppose that $\mathcal {F}$ is a finite set of trivial central character automorphic representations for $\operatorname {\mathrm {GL}}_2$ over $\mathbb Q$ , all with (finite) conductor q and spectral parameters contained in $[-T,T]$ . Suppose that there exists a pure tensor $f \in {\mathcal {H}}_{\mathrm {fin}}$ such that hypotheses TF, NmL, FTB and CvF hold for $f,\mathcal {F}$ . Then for any sequence of complex numbers $(a_n)_{n \in \mathbb {N}}$ we have

(1.54) $$ \begin{align} \sum_{\pi \in \mathcal{F}} \Big| \sum_{n \leq X} a_n \lambda_\pi(n)\Big|^2 \ll_\varepsilon (|\mathcal{F}|+ X) (XqT)^\varepsilon \sum_{n \leq X} |a_n|^2. \end{align} $$

Example. The classical Spectral Large Sieve Inequality is a special case of Theorem 1.23. Indeed, set

$$ \begin{align*}\mathcal{S}_{p^c,T} = \{\pi \in \mathcal{A}_{0}(\operatorname{\mathrm{PGL}}_2/\mathbb Q) : c(\pi_p) = c \text{ and } |t_\pi| \leq T\}\end{align*} $$

with $T^2p^c\to \infty $ .

To see this, first we partition the space $S:=[0,T] \cup (0,i/2)$ of potential spectral parameters into the disjoint union of $S_0:=([0,q^\varepsilon ] \cup (0,i/2))\cap S$ and its complement $(q^\varepsilon ,T]\cap S$ . We further $(1+T^{-\varepsilon })$ -adically partition $(q^\varepsilon ,T]\cap S$ , i.e. decompose it as a disjoint union

$$ \begin{align*}(q^\varepsilon,T] \cap S = \bigcup_{i>0} [\frac{T}{(1+T^{-\varepsilon})^{i-1}}, \frac{T}{(1+T^{-\varepsilon})^i})\cap S =: S_1 \cup S_2 \cup \ldots\end{align*} $$

with $S_i$ empty for i sufficiently large. Let $\mathcal {S}^{(i)}_{p^c,T} := \{\pi \in \mathcal {S}_{p^c,T}: t_\pi \in S_i\}$ , so that $\mathcal {S}_{p^c,T} = \cup _{i\geq 0} \mathcal {S}^{(i)}_{p^c,T}$ is a partition of $\mathcal {S}_{p^c,T}$ into $\ll T^{\varepsilon }$ nonempty subsets. Thus, to prove an optimal LSI for $\mathcal {S}_{p^c,T}$ it suffices to do so for each $\mathcal {S}^{(i)}_{p^c,T}$ . For $i=0$ we choose $h_\infty $ to be the test function in (1.41) of width $\min (T,q^\varepsilon )$ , while for $i>0$ we choose $h_\infty $ to be the test function in (1.40) with $\Delta =T^{1-\varepsilon }$ .

We take f equal to $f_{\leq c}$ at p and unramified elsewhere. Then f satisfies Hypotheses TF, FTB and CvF (since this choice of f satisfies $k_p = c$ in (1.31)).

We check Hypothesis 1.19 (NmL). Since $N=p^c$ , we have $N \mid p^\infty $ and $\mathcal {S}_{p^c,T}^{(i)}\subset \mathcal {F}_0(f)$ . In either case $i=0$ or $i>0$ the hypothesis on $h_\infty $ in the second or third bullet points of Hypothesis 1.19 (NmL) hold by construction. The last statement (1.46) of Hypothesis 1.19 (NmL) is given by Lemma 1.13. The Optimal Large Sieve Inequality (1.55) then holds for $\mathcal {F} = \mathcal {S}_{p^c,T}^{(i)}$ by Theorem 1.23, and for $\mathcal {S}_{p^c,T}$ by trivial summation over $i\geq 0$ .

1.6 Moments of L-functions

Let $\sigma $ be a supercuspidal representation of $\operatorname {\mathrm {GL}}_2(\mathbb Q_p)$ with trivial central character. Let $\mathcal {S}_\sigma $ be the family of automorphic representations

(1.55) $$ \begin{align} \mathcal{S}_\sigma := \{ \pi \in \mathcal{A}_{0}(\operatorname{\mathrm{PGL}}_2/\mathbb Q) : \pi_p \simeq \sigma, \, \pi_\infty \text{ spherical with } |t_\pi| \leq 1000\}. \end{align} $$

Note that we have $\#\mathcal {S}_{\sigma } \asymp p^{\lceil c(\sigma )/2 \rceil }$ by Corollary 1.14, (1.30) and (6.7). It is well-known that a large sieve inequality may be used to estimate certain moments of L-functions; see [Reference Iwaniec and KowalskiIK04, Section 7.9] for the method. As a simple application of Theorem 1.23, we have the following Lindelöf-on-average upper bound.

Corollary 1.26. Let $\sigma $ be a supercuspidal representation of $\operatorname {\mathrm {GL}}_2(\mathbb Q_p)$ with trivial central character. For all $\varepsilon>0$ we have

(1.56) $$ \begin{align} \sum_{\pi \in \mathcal{S}_\sigma} |L(1/2,\pi)|^2 \ll_\varepsilon (p^{\lceil c(\sigma)/2\rceil})^{1+\varepsilon}. \end{align} $$

Corollary 1.26 also holds if in (1.56) we replace the word “spherical” with “discrete series” and take the spectral parameter $t_\pi $ of a weight $\kappa $ discrete series to be $\frac {\kappa -1}{2}$ cf. Remark 1.24.

Let $\chi $ be a character of $\mathbb Q_p^\times $ whose restriction to $\mathbb {Z}_p^\times $ is not quadratic. Theorem 1.23 also gives a Lindelöf-on-average upper bound for the 2nd moment of central values of L-functions over the family

(1.57) $$ \begin{align} \mathcal{S}_\chi := \{ \pi \in \mathcal{A}_{0}(\operatorname{\mathrm{PGL}}_2/\mathbb Q) : \pi_p \simeq \pi(\chi \alpha^{i \theta},\chi^{-1}\alpha^{-i \theta}) \text{ for some } \theta \in \mathbb R \text{ and } |t_\pi| \leq 1000\}. \end{align} $$

However, such a second moment estimate already follows easily from previous cubic moment estimates [Reference Petrow and YoungPY23, Thm. 1.2] by Hölder’s inequality.

Since Corollary 1.26 follows from a large sieve inequality, it cannot give a subconvex bound by general principles. However, when $c(\sigma )$ is even, dropping all but one term recovers the convexity bound. In a forthcoming work, we intend to give a Lindelöf-on-average bound for the cubic moment of central values of L-functions over $S_{\sigma }$ and similar families, which will recover strong subconvex bounds for these L-functions.

1.7 Notation, conventions, and measure normalizations

1.7.2 Additive characters

Outside of Section 6, we take $\psi $ to be the standard additive character $\psi : \mathbb {A}/ \mathbb Q \to \mathbb {C}^\times $ , that is, $\psi = \prod _v \psi _v$ , where

(1.58) $$ \begin{align}\psi_v (x) = \begin{cases} e(-x_\infty) & \text{ if } v = \infty \\ e(\{x_p\}_p) & \text{ if } v=p \end{cases} \qquad (x \in \mathbb{A}),\end{align} $$

where $\{ \cdot \}_p : \mathbb Q_p \to \mathbb Q$ is the fractional part function.

At the outset of Section 6, $\psi $ is an arbitrary additive character of the non-archimedean local field F. We say $\psi \neq 1$ has conductor $c(\psi ) = n$ if ${\mathfrak {p}}^n$ is the largest fractional ideal of F on which $\psi $ is trivial. In Section 6.1.3 only we take $\psi $ to have conductor $1$ to match a convention in the compact induction theory of Bushnell-Henniart-Kutzko. On the other hand, from Remark 6.35 until the end of Section 6, we assume that $\psi $ has conductor 0 (e.g. the one in (1.59)).

If $E/F$ is a field extension, we denote by $\psi _E$ the additive character $\psi \circ \operatorname {\mathrm {Tr}}_{E/F}$ of E.

1.7.3 Groups and subgroups

Let G be the algebraic group $G= \operatorname {\mathrm {GL}}_2$ , Z be the subgroup of diagonal matrices of G, and $\overline {G} = Z \backslash G= \operatorname {\mathrm {PGL}}_2$ .

Let $N\subset B \subset G$ be the standard upper-triangular unipotent and Borel algebraic subgroups of G. Let A be the subgroup of matrices of the form $a(y):= \left (\begin {smallmatrix} y & 0 \\ 0 & 1 \end {smallmatrix}\right )$ for y in any commutative ring R. We have $B= ZAN=ZNA$ . For any $x,t\in R$ let

$$ \begin{align*}n(x) = \left( \begin{smallmatrix} 1 & x\\ 0 & 1 \end{smallmatrix} \right) \quad \text{ and } \quad z(t)= \left( \begin{smallmatrix} t & 0\\ 0 & t \end{smallmatrix} \right).\end{align*} $$

Let $K_p=G(\mathbb {Z}_p)$ be the standard maximal compact subgroup of $G(\mathbb Q_p)$ , and $K_\infty = \operatorname {\mathrm {SO}}_2(\mathbb R)$ . We write $Z = Z(\mathbb Q_p)$ when the prime p is clear from context, e.g. $ZK_p$ denotes $Z(\mathbb Q_p)G(\mathbb {Z}_p)$ .

Let $K= \prod _{p} K_p = G(\widehat {\mathbb {Z}})$ . We also use the subgroups $K(N)\subseteq K_d(N)$ , $K_1(N)\subseteq K_0(N)$ of K given by

$$ \begin{align*}K(N)= \{\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in K : a \equiv d \equiv 1\ \pmod{N}, b \equiv c \equiv 0\ \pmod{N}\},\end{align*} $$

$$ \begin{align*}K_d(N)= \{\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in K : b \equiv c \equiv 0\ \pmod{N}\},\end{align*} $$

$$ \begin{align*}K_1(N)= \{\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in K : d \equiv 1\ \pmod{N}, c \equiv 0\ \pmod{N}\},\end{align*} $$

$$ \begin{align*}K_0(N)= \{\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in K : c \equiv 0\ \pmod{N}\}.\end{align*} $$

We use the same notation for the corresponding subgroups of $K_p$ . For $* = \varnothing , d, 1,$ or $0$ , we set as usual $\Gamma _*(N) = K_*(N) \cap \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ .

For $m+n\geq 0$ let us define $K_0(n,m)\subset G(\mathbb Q_p)$ to be the compact open subgroup

(1.59) $$ \begin{align} K_0(m,n) = \begin{cases} \left\{ \begin{pmatrix} \mathbb{Z}_p^\times & (p^m) \\ (p^n) & \mathbb{Z}_p^\times \end{pmatrix}\right\} & \text{ if } m+n>0, \\ a(p^{-m})K_pa(p^{m}) & \text{ if } m+n=0.\end{cases}\end{align} $$

For an algebraic group H over $\mathbb Q$ , let $[H]$ denote the adelic quotient $[H]:= H(\mathbb Q) \backslash H(\mathbb {A})$ .

1.7.4 Measure normalizations

We choose $dx$ to be Lebesgue measure on $\mathbb R$ and $d^\times x=dx/|x|$ on ${\mathbb {R}}^\times $ . For F a non-archimedean local field, we take $dx$ to be the Haar measure on F that gives the maximal compact subgroup $\mathcal {O}$ measure 1. We set the Haar measure $d^\times x$ on $F^\times $ to be given by $d^\times x = \zeta _F(1)dx/|x|_F$ . Here $\zeta _F(1)=\zeta _{\mathfrak {p}}(1)=(1-\operatorname {\mathrm {Nm}} {\mathfrak {p}}^{-1})^{-1}.$

We let $dk$ be the Haar probability measure on $K_\infty $ . Take the measures on $Z(\mathbb R)$ , $A(\mathbb R)$ and $N(\mathbb R)$ induced by $dx$ and $d^\times x$ . These together determine a Haar measure on $G(\mathbb R)$ by the Iwasawa decomposition. Let $dg$ be the Haar measure on $G(\mathbb Q_p)$ that gives $\operatorname {\mathrm {vol}}(K_p)=1$ .

For H one of the algebraic groups in Section 1.7.3, we give $H(\mathbb {A})$ and $H(\mathbb {A}_{\mathrm {fin}})$ the associated product measures. We give $\overline {G}(\mathbb {A})$ and $ \overline {G}(\mathbb {A}_{\mathrm { fin}})$ the quotient measure. With these choices we have $\operatorname {\mathrm {vol}}(\mathbb Q \backslash \mathbb {A}) = 1$ and $\operatorname {\mathrm {vol}}([\overline {G}]) =2\xi (2)= \pi /3$ .

Each cuspidal automorphic representation $\pi $ (resp. global principal series $\pi _{\chi ,\chi ^{-1}}$ in the induced model) is endowed with the inner product

(1.60) $$ \begin{align}\langle \varphi_1, \varphi_2\rangle = \int_{[\overline{G}]} \varphi_1(g) \overline{\varphi_2(g)}\,dg \quad \left( \text{resp.\ } \langle \phi_1, \phi_2\rangle =\int_{K_\infty \times K} \phi_1(k) \overline{\phi_2(k)}\,dk \right).\end{align} $$

If H is a unimodular p-adic linear algebraic group and $\mu $ is a Haar measure on H, then there exists a unique $\sigma $ -finite measure $\widehat {\mu }$ called the Plancherel measure on the unitary dual $H^\wedge $ such that the Plancherel formula (4.1) holds. In particular, for any locally constant compactly supported function f on H, one has

(1.61) $$ \begin{align} f(1) = \int_{\pi \in H^\wedge} \operatorname{\mathrm{Tr}} \pi(f)\, d\widehat{\mu}(\pi), \end{align} $$

which we also refer to as the Plancherel formula. For more details, see Section 4.1.

1.7.5 Test functions and Hecke algebras

Write ${\mathcal {H}}_{\mathrm {fin}} =C_c^\infty (\overline {G}(\mathbb {A}_{\mathrm {fin}}))$ for the non-archimedean Hecke algebra of $\overline {G}= \operatorname {\mathrm {PGL}}_2$ , that is the space of locally constant functions on $G(\mathbb {A}_{\mathrm {fin}})$ that are invariant by and compactly supported modulo center the $Z(\mathbb {A}_{\mathrm {fin}})$ . Define the local Hecke algebra ${\mathcal {H}}_p= C_c^\infty (\overline {G}(\mathbb Q_p))$ similarly.

Throughout this paper (with the exception of in Section 2.1) we will always assume the $f \in {\mathcal {H}}_{\mathrm {fin}}$ that we use as test functions are nonzero pure tensors, i.e. that f admits a representative $\prod _p f_p$ with $f_p \in {\mathcal {H}}_p :=C_c^\infty ( \overline {G}(\mathbb Q_p))$ for each $p < \infty $ , which we may moreover assume satisfy $f_p(1) =1$ for all but finitely many p. Note, if $f = \bigotimes _p f_p \in {\mathcal {H}}_{\mathrm {fin}}$ is a pure tensor, then $f_p$ is a constant multiple of the indicator function $1_{ZK_p}$ for all but finitely many p (see e.g. [Reference CartierCar79, §1.3]). We say that such an f is ramified at p if $f_p$ is not a constant multiple of $1_{ZK_p}$ .

Let $N \in \mathbb {N}$ be minimal such that $f \in {\mathcal {H}}_{\mathrm {fin}}$ is bi- $K(N)$ -invariant. We call N the level of f and define similarly the local level $N_p$ of $f_p\in {\mathcal {H}}_p$ . If $f= \bigotimes _pf_p$ , then $N_p=p^{v_p(N)}$ .

If $f \in {\mathcal {H}}_{\mathrm {fin}}$ and $(\pi ,V)$ with $\pi = \pi _\infty \otimes \pi _{\mathrm {fin}}$ and $V = V_\infty \otimes V_{\mathrm {fin}}$ is an irreducible admissible representation of $\overline {G}(\mathbb {A})$ , then define $\pi (f) \in \text {End}(V_{\mathrm {fin}})$ to be given by

(1.62) $$ \begin{align} \pi(f): v \mapsto \int_{\overline{G}(\mathbb{A}_{\mathrm{fin}})} f(g) \pi_{\mathrm{ fin}}(g)v\,dg.\end{align} $$

If $f \in {\mathcal {H}}_p$ and $(\pi ,V) $ is an irreducible admissible representation of $\overline {G}(\mathbb Q_p)$ , then define similarly $\pi (f)\in \text {End}(V)$ by (1.63) with $\mathbb Q_p$ in place of $\mathbb {A}_{\mathrm {fin}}$ and $\pi $ in place of $\pi _{\mathrm {fin}}$ . In Section 4.1, where H is a unimodular p-adic linear algebraic group with a Haar measure $dg$ , we define $\pi (f) \in \text {End}(V)$ for $f \in L^1(H)$ and $(\pi ,V)$ an irreducible admissible representation of H by (1.63) with H in place of $\overline {G}(\mathbb {A}_{\mathrm {fin}})$ and $\pi $ in place of $\pi _{\mathrm {fin}}$ .