1 Introduction↩︎

1.1 Background and motivation↩︎

Automorphic \(L\) functions play a central role in the classical Langlands program over global fields. It is known that the meromorphic continuation and functional equations of general automorphic \(L\)-functions would imply Langlands’ transfer conjecture when combined with a suitable converse theorem, see [1] for more detailed explanations.

Automorphic \(L\)-functions admit several complementary constructions. A basic prototype is the work of Godement and Jacquet [2], where the standard \(L\)-function on \(\mathrm{GL}_n\) is realized by zeta integrals together with its functional equation. For the symmetric square of \(\mathrm{GL}_2\), the analytic theory is classically provided by the lift of Gelbart and Jacquet [3], with further developments on automorphy and cuspidality in work such as [4]. From the trace-formula side, Jacquet and Zagier [5] gave a different model for extracting \(L\)-functions from spectral and geometric expansions. There is also a proposal by Kazhdan-Braverman, aiming to prove the meromorphic continuation and functional equation of general automorphic \(L\)-function along the lines of Godement-Jacquet (see [1]).

Langlands’ beyond endoscopy proposal [6] asks whether one can isolate functorial transfers directly from trace formulas by inserting suitable \(L\)-functions on the spectral side and then analyzing the resulting geometric expressions. In the case of \(\mathrm{GL}_2\), this point of view was developed by Altuğ in a series of works beginning with [7] and culminating, for the standard representation, in [8]. A central feature there is the combination of approximate functional equations, Poisson summation, and the smoothing of orbital-integral singularities. Most recently the work [9] has extended the work of Altuğ to the case of \(\mathrm{GL}_3(\mathbb{Q})\). Related perspectives include Sakellaridis’ work on beyond endoscopy for relative trace formulas [10], the extension of Altuğ’s smoothing method to \(\mathrm{GL}_n\) by [11]. We point out that in the work [12], the Kuznetsov Trace Formula is used, instead of the Arthur-Selberg Trace formula, to carry out the ideas put forward by Langlands.

The aim of this note is modest and explicit. We study the symmetric-square \(L\)-function attached to level-one holomorphic cusp forms and even Hecke–Maass cusp forms on \(\mathrm{GL}_2\), and we recover its functional equation from trace-formula identities of Petersson/Kuznetsov type. The key point is an explicit archimedean transform \(\mathcal{T}_\mu\) whose Mellin transform produces the gamma factor of \(\operatorname{Sym}^2\), which is introduced for Voronoi summation formulas for \(\mathrm{GL}_3\) in [13]. After Poisson summation in a quadratic variable and a Weber–Schafheitlin calculation, the geometric side reorganizes into a Voronoi-type reciprocity formula, from which the functional equation is deduced. To our knowledge, this is the first derivation of the symmetric-square functional equation directly from Petersson/Kuznetsov trace formulas without invoking the Gelbart–Jacquet lift. For the standard \(L\) function, similar procedure was first developped in [14] which is corrected in [15]. Our work shows that the beyond endoscopy approach to \(L\)-functions can go beyond the standard case.

1.2 Main results↩︎

1.2.1 Holomorphic cusp forms↩︎

Let \(k\geqslant 2\) be an integer. Let \(S_{2k}\) be the space of holomorphic cusp forms of weight \(2k\) and level \(1\), and let \(B_{2k}\) be an orthogonal basis of Hecke eigenforms with Fourier expansions \[f(z)=\sum_{n\geqslant 1} a_f(n) n^{\frac{2k-1}{2}} e^{2\pi i nz},\qquad a_f(1)=1.\] For such an \(f\) define the symmetric-square coefficients \[A_f(n)=\sum_{d^2\mid n} a_f\bigl((n/d^2)^2\bigr).\] For any cusp form \(f\), the coefficients \(a_f(n)\) satisfy \(a_f(n)\ll n^{\frac{1}{2}}\), so \(A_f(n)\ll n\).

Let \(g\in\mathcal{S}\) be a Schwartz function on \((0,\infty)\) (see Definition 1) and let \(\ell\geqslant1\) be an integer. Set \[I_\ell(g)=\sum_{f\in B_{2k}} \frac{\Gamma(2k-1)}{(4\pi)^{2k-1}\|f\|^2}a_f(\ell)\sum_{n=1}^\infty A_f(n)g(n)=\sideset{}{^h}\sum_{f\in B_{2k}} a_f(\ell)\sum_{n=1}^\infty A_f(n)g(n),\] where \(\sideset{}{^h}\sum\) indicates that the sum is weighted by the reciprocal of the Petersson norm. The first main result is the following average Voronoi formula.

From this we obtain the functional equation and entireness of the symmetric-square \(L\)-functions.

1.2.2 Even Hecke–Maass cusp forms↩︎

Let \(\mathcal{B}\) be an orthonormal basis of even Hecke–Maass cusp forms of level \(1\) and trivial central character. For \(u\in\mathcal{B}\) with spectral parameter \(t_u\) (\(t_u\in [0,\infty)\cup i(0,\frac{1}{2})\) so that \(\Delta u=(\frac{1}{4}+t_u^2)u\)) write the Fourier expansion as \[u(z)=\sqrt{y}\sum_{n\neq0}\rho_u(n)K_{it_u}(2\pi|n|y)e(nx),\qquad \rho_u(n)=\rho_u(1)\lambda_u(n), \lambda_u(1)=1,\] and set \(\omega_u=4\pi|\rho_u(1)|^2/\cosh(\pi t_u)\). Here \(K_{it_u}(y)\) is the Bessel function of the second kind. The coefficients \(\lambda_u(n)\) satisfy \(\lambda_u(n)\ll n^{\frac{1}{2}}\).

Define the symmetric-square coefficients by \[A_u(n)=\sum_{d^2\mid n}\lambda_u\bigl((n/d^2)^2\bigr).\] Then \(A_u(n)\ll n\). For the continuous spectrum we likewise define \(\tau_t(n)=\sum_{ab=n}(a/b)^{it}\) and \[A_t(n)=\sum_{d^2\mid n}\tau_t\bigl((n/d^2)^2\bigr).\] The parameters \(t_u\) satisfy the Weyl Law: \[\#\{u\in\mathcal{B}: |t_u|\leqslant T\}\sim \frac{T^2}{12},\quad \text{as}\quad T\to\infty.\]

Let \(h\) be an even function of complex variable. It is called Kuznetsov-admissible if it is holomorphic in \(|\mathrm{Im}(t)|<1+\varepsilon\) for some \(\varepsilon>0\) and satisfies \(h(s)\ll |s|^{-A}\) for every \(A>0\) as \(|s|\to\infty\). In our paper, we also acquire \(h(\pm i/2)=0\).

Theorem 3 gives the following Maass‑form analogue of Theorem 1.

From this we deduce the functional equation and entireness of the associated symmetric-square \(L\)-functions.

1.3 Outline of the method↩︎

Our strategy follows the “spectral reciprocity” approach developed in [16] for the standard \(L\)-function of holomorphic cusp forms. Here we adapt it to the symmetric-square setting, replacing the standard gamma factor by \(\mathcal{G}_\mu\) and handling the quadratic Dirichlet series \(\mathscr{L}_{n^2-4\ell}(s)\) that arises from Poisson summation in a quadratic variable.

More precisely, we prove averaged Voronoi identities for symmetric-square coefficients via the following path. (For simplicity, we only outline the holomorphic case.)

1. Trace formula expansion. Express \(I_\ell(g)\) using the Petersson trace formula, decomposing it into a diagonal term \(\Delta_\ell(g)\) and a Kloosterman sum term \(\mathcal{G}_\ell(g)\).

2. Poisson summation. Apply Poisson summation to \(\mathcal{G}_\ell(g)\). This yields \[\mathcal{G}_\ell(g)=\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)} \widetilde{g}(1-z) \sum_{n\in\mathbb{Z}} \mathscr{L}_{n^2-4\ell}(1-z)\,\mathcal{J}_{n,\ell}(z)\,dz.\]

3. Treat \(I_{\ell}(\mathcal{T}_{\mu}g)\) similarly. Repeating the first two steps for \(I_{\ell}(\mathcal{T}_{\mu}g)\) gives \(I_{\ell}(\mathcal{T}_{\mu}g)=\Delta_\ell(\mathcal{T}_{\mu}g)+\mathcal{G}_\ell(\mathcal{T}_{\mu}g)\) with \[\mathcal{G}_\ell(\mathcal{T}_{\mu}g)=\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)} \widetilde{\mathcal{T}_{\mu}g}(1-z) \sum_{n\in\mathbb{Z}} \mathscr{L}_{n^2-4\ell}(1-z)\,\mathcal{J}_{n,\ell}(z)\,dz.\]

4. Archimedean reciprocity. For all \(n\) with \(n^2 \neq 4\ell\), the contributions to \(\mathcal{G}_\ell(g)\) and \(\mathcal{G}_\ell(\mathcal{T}_{\mu}g)\) coincide. This follows from a functional equation relating \(\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell}(z)\) to its counterpart with \(z\) replaced by \(1-z\), combined with the definition of \(\mathcal{T}_\mu\). In fact, the transform \(T_{\mu}\) is chosen so that the archimedean factor appearing after Poisson summation satisfies the same functional equation as the quadratic Dirichlet series \(\mathscr{L}_{D}(s)\).

5. Singular and diagonal corrections. When \(\ell = m^2\) is a perfect square, the remaining terms satisfy \[\sum_{n=\pm 2m}\bigl(\text{contribution from }\mathcal{G}_\ell(g)\bigr) \;+\; \Delta_\ell(g) \;=\; \sum_{n=\pm 2m}\bigl(\text{contribution from }\mathcal{G}_\ell(\mathcal{T}_{\mu}g)\bigr) \;+\; \Delta_\ell(\mathcal{T}_{\mu}g).\] This equality is verified by a direct residue calculation using the explicit forms of \(\mathscr{L}_0(s)=\zeta(2s-1)\) and \(\mathcal{J}_{\pm2m,m^2}(z)\).

The Maass case follows the same plan, with the Kuznetsov trace formula replacing the Petersson trace formula, the kernel \(\mathbb{J}_t\) replacing \(J_{2k-1}\), and the spectral-parameter-dependent factor \(\mathcal{G}_{\mu_t}\) in place of \(\mathcal{G}_\mu\). The continuous spectrum contributes an extra explicit term in Theorem 3.

Acknowledgements↩︎

T. Deng is supported by Beijing Natural Science Foundation, No. 1244042 and National Natural Science Foundation of China, No. 12401013.

2 Preliminaries↩︎

This section collects the necessary background for the proof, including special functions (Gamma, Bessel, hypergeometric, Weber–Schafheitlin integrals), trace formulas (Poisson summation, Petersson, Kuznetsov), arithmetic sums (Kloosterman sums, Ramanujan sums), and the quadratic Dirichlet series \(\mathscr{L}_D(s)\) with its meromorphic continuation and functional equation.

2.1 Special functions↩︎

2.1.1 Gamma function↩︎

The Gamma function \(\Gamma(s)\) is defined by \[\Gamma(s)=\int_0^{\infty} e^{-x}x^{s-1}dx,\quad\mathrm{Re}(s)>0.\] Integration by parts gives the recursion formula \(\Gamma(s+1)=s\Gamma(s)\). This formula yields a meromorphic extension of \(\Gamma(s)\) to \(\mathbb{C}\), with simple poles at non-positive integers \(0,-1,-2,\dots\).

We need the following properties of \(\Gamma(s)\).

• Functional equation. \[\Gamma(s)\Gamma(1-s)=\frac{\pi s}{\sin(\pi s)}.\]

• Legendre duplication formula. \[\Gamma(s)=\pi^{-\frac{1}{2}}2^{s-1}\Gamma\big(\frac{s}{2}\big)\Gamma\big(\frac{1+s}{2}\big).\]

• Stirling’s formula.In the angle \(|\mathrm{arg}(s)|<\pi-\epsilon\), we have the estimate \[\Gamma(s)=\left(\frac{2\pi}{s}\right)^{1/2}\left(\frac{s}{e}\right)^s(1+O_{\epsilon}(|s|^{-1})).\] A more useful version is that, let \(a<b\) be two real numbers, then in the vertical strip \(a\leqslant\sigma=\mathrm{Re}(s)\leqslant b\), and \(t=\mathrm{Im}(s)>0\), one has \[\Gamma(\sigma+it)=(2\pi)^{1/2}t^{\sigma-1/2}e^{-\pi t/2}\left(\frac{t}{e}\right)^{it}(1+O_{a,b}(t^{-1}))\quad\text{as}~~t\to\infty.\]

2.1.2 Bessel functions↩︎

For \(\nu\in\mathbb{C}\) and \(x>0\), the Bessel function of the first kind is defined by the series \[\label{equ:def-Bessel} J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\nu+1)}\Bigl(\frac{x}{2}\Bigr)^{2m+\nu}.\tag{1}\] When \(x\to 0^+\), we have \(J_{\nu}^{(j)}(x)\ll_{\nu} x^{\nu-j}\) if \(\nu\notin\mathbb{Z}_{\geqslant 0}\) and \(J_{\nu}^{(j)}\ll_{\nu} x^{\max(\nu-j, 0)}\) if \(\nu\in\mathbb{Z}_{\geqslant 0}\). While as \(x\to+\infty\), the Bessel function has the following asymptotic expansion \[J_{\nu}(x)=\sqrt{\frac{2}{\pi x}}\Bigl(\cos\bigl(x-\tfrac{\nu\pi}{2}-\tfrac{\pi}{4}\bigr)+O\bigl(\tfrac{1+|\nu|^2}{x}\bigr)\Bigr).\]

If we fix \(x>0\), then \(J_{\nu}(x)\) is an entire function of the parameter \(\nu\). Moreover, if \(\mathrm{Re}(\nu)\) is a half integer, then \[J_{\nu}(x)\ll_x \frac{1}{|\Gamma(\nu+1)|}\left(\frac{x}{2}\right)^{\mathrm{Re}(\nu)}.\]

For our purposes we need the following special case when \(\nu=-1/2\): \[J_{-1/2}(x)=\sqrt{\frac{2}{\pi x}}\cos(x).\] We also need the modified Bessel combination that appears in the Kuznetsov formula: \[\label{equ:mathbbJ} \mathbb{J}_t(x)=\frac{i}{\pi}\frac{t}{\cosh(\pi t)}\bigl(J_{2it}(x)-J_{-2it}(x)\bigr),\qquad t\in\mathbb{R}.\tag{2}\] This function is even in \(t\) and real for \(x>0\).

For \(-\mathrm{Re}(\nu)<\mathrm{Re}(z)<\frac{1}{2}\), we have \[\label{equ:mellin-Bessel} \int_0^{\infty}x^{z-1}J_{\nu}(x)dx=2^{z-1}\frac{\Gamma\big(\frac{\nu+z}{2}\big)}{\Gamma\big(1+\frac{\nu-z}{2}\big)}.\tag{3}\]

For a detailed treatment for Bessel functions, see [17].

2.1.3 Hypergeometric functions↩︎

For \(a,b,c\in\mathbb{C}\) with \(c\notin\{0,-1,-2,\dots\}\) and \(|x|<1\), the Gauss hypergeometric function \({}_2F_1(a,b;c;x)\) is defined by the series \[{}_2F_1(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n n!}x^n,\] where \((a)_n=\Gamma(a+n)/\Gamma(a)\) is the Pochhammer symbol. The series converges absolutely for \(|x|<1\) and defines a holomorphic function there.

A fundamental transformation formula is Euler’s transformation: \[{}_2F_1(a,b;c;x) = (1-x)^{c-a-b}{}_2F_1(c-a,c-b;c;x),\qquad |\arg(1-x)|<\pi.\] This identity plays a key role in the derivation of the functional equation of \(\mathcal{J}_{n,\ell}(x)\) (Lemma 7).

The Euler integral representation says that \[\label{equ:Euler-integral-rep} {}_2F_1(a,b;c;x)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^{b-1}(1-t)^{c-b-1}(1-xt)^{-a}dt,\tag{4}\] which holds under the additional conditions \(\operatorname{Re}(c)>\operatorname{Re}(b)>0\) and \(|\arg(1-x)|<\pi\). Under the same conditions, we have the estimate \[\label{equ:bound-hypergeo} \left|{}_2F_1(a,b;c;x)\right|\leqslant \left|\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\right| \int_0^1 t^{\mathrm{Re}(b)-1}(1-t)^{\mathrm{Re}(c-b)-1}(1-xt)^{-\mathrm{Re}(a)}dt.\tag{5}\]

Another formula that we need is \[\label{equ:hyper-geo-square-root} {}_2F_1\left(\frac{\nu}{2},\frac{\nu+1}{2};1+\nu;x\right) =\left(\frac{2}{1+\sqrt{1-x}}\right)^\nu.\tag{6}\]

For a detailed discussion of hypergeometric functions, see [18].

2.1.4 The Weber–Schafheitlin integral↩︎

The following classical result (see Watson [17]) gives the Mellin transform of a product of two Bessel functions.

2.2 Trace formulas↩︎

2.2.1 Poisson summation formula↩︎

2.2.2 Petersson trace formula of level \(1\)↩︎

Let \(S_{2k}\) be the space of holomorphic cusp forms of weight \(2k\) and level \(1\), and let \(\{f_j\}\) be a basis of Hecke eigenforms with Fourier expansions \[f_j(z)=\sum_{n\geqslant1} a_j(n)n^{\frac{2k-1}{2}} e^{2\pi i n z},\qquad a_j(1)=1.\] The Petersson trace formula states that for any positive integers \(m,n\), \[\sum_{j}\frac{\Gamma(2k-1)}{(4\pi)^{2k-1}\|f_j\|^2}a_j(m)\overline{a_j(n)} = \delta_{m,n}+2\pi i^{-2k}\sum_{c\geqslant1}\frac{S(m,n;c)}{c}J_{2k-1}\left(\frac{4\pi\sqrt{mn}}{c}\right).\] Here \(\|f_j\|\) is the Petersson inner product of \(f_j\), and \(J_{2k-1}\) is the Bessel function of the first kind.

2.2.3 Kuznetsov trace formula (same-sign)↩︎

We shall use the same-sign Kuznetsov trace formula in the following normalization. Let \(h\) be a Kuznetsov-admissible function. For such \(h\), put \[H_0(h):=\frac{1}{\pi^2}\int_{-\infty}^{\infty} t\tanh(\pi t)h(t)dt\] and \[\mathbb{J}_t(x):= \frac{i}{\pi}\frac{t}{\cosh(\pi t)} \bigl(J_{2it}(x)-J_{-2it}(x)\bigr), \qquad \mathcal{J}_h(x):=\int_{-\infty}^{\infty}h(t)\mathbb{J}_t(x)dt .\]

Then for positive integers \(m,n\) (cf. [19]), \[\label{equ:level-one-kuznetsov} \begin{align} &\sum_{u\in\mathcal{B}}\omega_u h(t_u)\lambda_u(m)\overline{\lambda_u(n)} +\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{h(t)\tau_t(m)\overline{\tau_t(n)}}{|\zeta(1+2it)|^2}dt \\ &\qquad = \delta_{m,n}H_0(h)+ \sum_{c\geqslant1}\frac{S(m,n;c)}{c} \mathcal{J}_h\left(\frac{4\pi\sqrt{mn}}{c}\right). \end{align}\tag{7}\] For level \(1\) and trivial central character the Hecke eigenvalues \(\lambda_u(n)\) and the divisor coefficients \(\tau_t(n)\) are real, so the conjugates are harmless; they are retained to make the sesquilinear normalization of Kuznetsov explicit.

2.3 Arithmetic sums↩︎

2.3.1 Kloosterman sums↩︎

For positive integers \(c\) and arbitrary integers \(m,n\), the Kloosterman sum is defined by \[S(m,n;c)=\sum_{x\bmod c,~~(x,c)=1} e\left(\frac{mx+n\bar{x}}{c}\right),\] where \(\bar{x}\) denotes the multiplicative inverse of \(x\) modulo \(c\) and \(e(z)=e^{2\pi iz}\). These sums satisfy the Weil bound \[|S(m,n;c)|\le \tau(c)\sqrt{c}\gcd(m,n,c)^{1/2},\] where \(\tau(c)\) is the divisor function. In particular, for fixed \(m,n\) one has \(S(m,n;c)=O(c^{1/2+o(1)})\) as \(c\to\infty\).

2.3.2 Ramanujan sums↩︎

The Ramanujan sum is defined for an integer \(q\geqslant 1\) by \[c_q(n)=\sum_{a\bmod q,~~(a,q)=1} e\left(\frac{an}{q}\right).\] Using the Möbius inversion one obtains the explicit formula \[c_q(n)=\sum_{d\mid (q,n)} d\mu\left(\frac{q}{d}\right).\]

A crucial arithmetic identity (used in the evaluation of \(\Sigma_c(\ell,n)\)) is \[\label{equ:Ramanujansum} \sideset{}{^*}\sum_{x\bmod q} e\left(\frac{\ell\bar{x}}{q}\right)\sum_{a\bmod q} e\left(\frac{xa^{2}+an}{q}\right) = q\sum_{d\mid q}\mu\left(\frac{q}{d}\right)\rho_d(n,\ell),\tag{8}\] where \[\label{equ:def-rhodnl} \rho_d(n,\ell)=\#\{u\bmod d: u^{2}+nu+\ell\equiv0\pmod{d}\}.\tag{9}\] Derivation. Let \(u=ax\). Then the left hand side is equal to \[\sideset{}{^\ast}\sum_{x\bmod{q}}e\left(\frac{\ell\overline{x}}{q}\right)\sum_{u\bmod{q}} e\left(\frac{u^2\overline{x}+nu\overline{x}}{q}\right)=\sum_{u\bmod{q}}~~\left[~~\sideset{}{^\ast}\sum_{x\bmod{q}}e\left(\frac{(u^2+nu+\ell)\overline{x}}{q}\right)\right]=\sum_{u\bmod{q}}c_q(u^2+nu+\ell).\] Thus the desired arithmetic identity 8 follows.

2.4 Quadratic Dirichlet series \(\mathscr{L}_D(s)\)↩︎

Let \(D\) be an integer. For \(\operatorname{Re}(s)>1\), define \[\mathscr{L}_D(s)= \frac{\zeta(2s)}{\zeta(s)}\sum_{r\geqslant 1}\frac{\rho_r(D)}{r^s},\] where \(\rho_r(D)=\#\{u\bmod r : u^2\equiv D\pmod{r}\}=\rho_{r}(n,\ell)\) with \(D=n^2-4\ell\). (see 9 ). These coefficients appear naturally after applying Poisson summation to quadratic exponential sums. For \(D=0\) one has \(\mathscr{L}_0(s)=\zeta(2s-1)\).

The series \(\mathscr{L}_D(s)\) can be expressed in terms of Dirichlet \(L\)-functions attached to quadratic characters. Write \(D=D_0f^2\) where \(D_0\) is a fundamental discriminant (with \(D_0=1\) when \(D\) is a perfect square). Then Zagier’s formula (see [20]) gives \[\label{equ:Zagier} \mathscr{L}_D(s)=L(s,\chi_{D_0})\sum_{a\mid f}\mu(a)\chi_{D_0}(a)a^{-s}\sigma_{1-2s}\left(\frac{f}{a}\right),\tag{10}\] where \(\sigma_w(n)=\sum_{d\mid n}d^w\). In particular, \(\mathscr{L}_D(s)\) is a finite linear combination of Dirichlet \(L\)-series and therefore admits a meromorphic continuation to \(\mathbb{C}\) with at most a simple pole at \(s=1\) (when \(D\) is a square, coming from the Riemann zeta factor). The functional equation is \[\label{equ:LD95fe} \mathscr{L}_D(1-s)=\Xi_D(s)\mathscr{L}_D(s), \qquad \Xi_D(s)=\left(\frac{|D|}{\pi}\right)^{s-\frac{1}{2}} \frac{\Gamma\left(\frac{s+\delta_D}{2}\right)}{\Gamma\left(\frac{1-s+\delta_D}{2}\right)},\tag{11}\] with \(\delta_D=0\) if \(D>0\) and \(\delta_D=1\) if \(D<0\). This is a standard consequence of the functional equation of Dirichlet \(L\)-functions and the properties of the divisor sum \(\sigma_{1-2s}\); see [20] for a detailed derivation.

For our purposes we only need the following uniform bound (convexity bound) on vertical strips: for any fixed \(a<b\) and any constant \(\delta>0\), there exists a constant \(A=A(a,b)>0\) (independent of \(D\)) such that \[\label{equ:convexbound} \mathscr{L}_D(s)\ll_{a,b,\delta} (1+| \mathrm{Im}(s)|)^{A}|D|^{\max(0,(1-\operatorname{Re}s)/2)+\delta},\tag{12}\] for \(a<\mathrm{Re}(s)<b\) apart from the possible simple pole at \(s=1\) when \(D\) is a square. The bound follows from the corresponding estimate for the Dirichlet \(L\)-function and the divisor sum \(\sigma_{1-2s}(f/a)\), which grows at most polynomially in \(f\) and in \(| \mathrm{Im}(s)|\).

When \(D\) is a non‑zero square, the factor \(L(s,\chi_{D_0})\) becomes \(\zeta(s)\) (up to a finite Euler product), and \(\mathscr{L}_D(s)\) has a simple pole at \(s=1\). Moreover, by 10 , we have \[\operatorname*{Res}_{s=1}\mathscr{L}_{d^2}(s)=\begin{cases} \frac{1}{2},&\quad d=0,\\ 1,&\quad d\geqslant 1. \end{cases}\]

3 The Archimedean Kernels and Their Transforms↩︎

This section defines the Schwartz space \(\mathcal{S}\) and its Mellin transform, then introduces the key archimedean transforms \(\mathcal{T}_\mu\) (holomorphic case) and \(\mathcal{T}_{\mu_t}\) (Maass case), studying their growth properties. Explicit formulas, functional equations, and singularity behaviors of the kernels \(\mathcal{J}_{n,\ell}(z)\) and \(\mathcal{J}_{n,\ell,t}(z)\) are derived in detail to prepare for the averaged Voronoi identities.

3.1 Schwartz space and Mellin transform↩︎

For an integrable function \(f\) on \((0,\infty)\) the Mellin transform is defined by \[\widetilde{f}(s)=\int_0^\infty f(x)x^{s-1}dx\] whenever the integral converges absolutely.

If \(f\in\mathcal{S}\), then \(\widetilde{f}(s)\) is entire and satisfies the following property: for any real numbers \(a<b\) and any integer \(m\ge0\), \[\sup_{a\le\operatorname{Re}(s)\le b}\bigl|s^m\widetilde{f}(s)\bigr|<\infty .\] We say \(\widetilde{f}(s)\) is essentially bounded if the above estimate holds for every \(m\in\mathbb{Z}_{\geqslant 0}\) and any real numbers \(a<b\).

A useful observation is that the converse also holds. Namely,

3.2 The holomorphic case↩︎

3.2.1 The archimedean transform \(\mathcal{T}_\mu\)↩︎

For the holomorphic case we fix an integer \(k\geqslant 2\) and set \[\mu=(1-2k,2k-1,0).\]

Recall that \[\mathcal{G}_\mu(s)=G_0(s+2k-1)G_0(s-2k+1)G_0(s),\] where \[G_0(s)=2(2\pi)^{-s}\Gamma(s)\cos\left(\frac{\pi s}{2}\right)=\frac{\Gamma_{\mathbb{R}(s)}}{\Gamma_{\mathbb{R}}(1-s)}.\] So the function \(\mathcal{G}_\mu(s)\) is holomorphic when \(\operatorname{Re}(s)>0\) and satisfies the functional equation \[\mathcal{G}_\mu(s)\mathcal{G}_\mu(1-s)=1.\] Moreover, by Stirling’s formula, \(|\mathcal{G}_\mu(s)|\) grows at most polynomially in \(|\operatorname{Im}(s)|\) on any finite vertical strip.

The transform \(\mathcal{T}_\mu g\) enjoys the following properties, which are weaker than those of \(g\) but sufficient for our purposes.

3.2.2 The holomorphic kernel \(\mathcal{J}_{n,\ell}(z)\)↩︎

For every integer \(n\), every integer \(\ell\geqslant1\) and a complex parameter \(z\) with \(-1/2<\operatorname{Re}(z)<1/2\) we define \[\mathcal{J}_{n,\ell}(z)=2\int_{0}^{\infty}y^{z-1}J_{2k-1}(4\pi\sqrt{\ell}y)\cos(2\pi ny)dy.\] More generally, for \(\nu\in\mathbb{C}\) with \(\mathrm{Re}>1\frac{1}{2}\), we define \[\mathcal{J}^{\nu}_{n,\ell}(z):=2\int_{0}^{\infty}y^{z-1}J_{\nu}(4\pi\sqrt{\ell}y)\cos(2\pi |n|y)dy,\] which converges absolutely when \(-\mathrm{Re}(\nu)<\mathrm{Re}(z)<\frac{1}{2}\).

From the explicit formulas, we can derive some useful functional equations for \(\mathcal{J}_{n,\ell}(z)=\mathcal{J}^{2k-1}_{n,\ell}(z)\).

The following lemma shows that a certain series involving quadratic \(L\)-functions and \(\mathcal{J}_{n,\ell}\) has polynomial growth in vertical strips.

3.3 The Maass form case↩︎

3.3.1 The archimedean transforms \(\mathcal{T}_{\mu_t}\)↩︎

For the Maass case, the spectral parameter is \(t\in\mathbb{R}\cup i(0,\tfrac12)\) (or \(t_u\) for a discrete Maass form). Set \(\mu_t=(2it,0,-2it)\).

The function \(\mathcal{G}_{\mu_t}(s)\) is meromorphic, and it is holomorphic for \(\operatorname{Re}(s)>0\). It satisfies \(\mathcal{G}_{\mu_t}(s)\mathcal{G}_{\mu_t}(1-s)=1\) and grows polynomially in \(|\operatorname{Im}(s)|\) on vertical strips.

These transforms are the archimedean building blocks for the Voronoi summation formulas in Theorems 1 and 3.

3.3.2 The Maass kernel \(\mathcal{J}_{n,\ell,t}(z)\)↩︎

For the Maass case, we replace the Bessel function \(J_{2k-1}(x)\) by the kernel \[\mathbb{J}_t(x)=\frac{i}{\pi}\frac{t}{\cosh(\pi t)}\bigl(J_{2it}(x)-J_{-2it}(x)\bigr),\] where \(t\in\mathbb{R}\cup i{0,\frac{1}{2}}\).

Define \[\label{equ:def-J-n-ell-t} \mathcal{J}_{n,\ell,t}(z)=2\int_{0}^{\infty}y^{z-1}\mathbb{J}_t(4\pi\sqrt{\ell}y)\cos(2\pi ny)dy.\tag{13}\] Since \(\mathbb{J}_t\) is a linear combination of \(J_{\pm2it}\), we obtain \[\mathcal{J}_{n,\ell,t}(z)=\frac{i}{\pi}\frac{t}{\cosh(\pi t)}\big(\mathcal{J}_{n,\ell}^{2it}(z)-\mathcal{J}_{n,\ell}^{-2it}(z)\big).\] By the explicit computation of \(\mathcal{J}_{n,\ell}^{\nu}(z)\), we have

3.3.3 The average kernel \(\mathcal{J}_{n,\ell,h}(z)\)↩︎

Let \(h(t)\) be a Kuznetsov-admissible test function with \(h(\pm i/2)=0\). Define \[\label{equ:J-n-ell-h-defn} \mathcal{J}_{n,\ell,h}(z)=\int_{-\infty}^{\infty}h(t)\,\mathcal{J}_{n,\ell,t}(z)\,dt.\tag{14}\] By Lemma 5, this integral is meromorphic for \(0<\operatorname{Re}(z)<\frac{3}{2}\), with a possible simple pole at \(z=\frac{1}{2}\) when \(\ell\) is a perfect square and \(n^2=4\ell\).

Exchanging the order of integration gives, for \(0<\operatorname{Re}(z)<\frac{1}{2}\), \[\mathcal{J}_{n,\ell,h}(z)=2\int_0^{\infty}y^{z-1}\mathcal{J}_h(4\pi\sqrt{\ell}y)\cos(2\pi ny)dy,\] where \(\mathcal{J}_h(y)=\int_{-\infty}^{\infty}h(t)\mathbb{J}_t(y)dt\). The absolute convergence justifies the exchange in this region.

From Lemma 13 and the integral representation, \(\mathcal{J}_{n,\ell,h}(z)\) extends meromorphically to \(-2-2\varepsilon<\operatorname{Re}(z)<\frac{3}{2}\), with a possible simple pole at \(z=\frac{1}{2}\) when \(\ell\) is a perfect square and \(n^2=4\ell\).

The averaged kernel satisfies a functional equation.

The transformed test function \(h_{1-z}\) may not be Kuznetsov-admissible due to poles of \(\mathcal{G}_{\mu_t}(1-z)\) at \(t=\pm i(1-z)/2\). Nevertheless, for \(\operatorname{Re}(z)<0\), we have the decomposition \[\label{equ:decom-Jh-1-z} \mathcal{J}_{h_{1-z}}(y)=\mathcal{Z}_{\ell,h}(z)J_{1-z}(y)+R_{\ell,h}(y),\tag{15}\] where \(J_{1-z}\) is the Bessel function of order \(1-z\), \(R_{\ell,h}^{(j)}(y)\ll y^{\alpha-j}\) as \(y\to0^+\) for any \(\alpha<2+2\varepsilon\), and \[\label{equ:Z-ell-h} \mathcal{Z}_{\ell,h}(z)=-2(1-z)h\left(\frac{1-z}{2i}\right)\frac{G_0(1-z)G_0(2-2z)}{\cosh\left(\frac{\pi(1-z)}{2i}\right)}.\tag{16}\]

4 Averaged Voronoi Identities↩︎

4.1 Holomorphic case (Theorem 1)↩︎

We prove ?? in six steps.

Step 1: Petersson trace formula.Recall that \[I_\ell(g)=\mathop{\sum\nolimits^{h}}_{f\in B_{2k}} a_f(\ell)\sum_{n\ge1} A_f(n)g(n).\] Writing \(A_f(n)=\sum_{d^2\mid n}a_f((n/d^2)^2)\) and \(n=d^2m\), we obtain \[I_\ell(g)=\sum_{d,m\ge1} g(d^2m)\mathop{\sum\nolimits^{h}}_{f\in B_{2k}} a_f(\ell)a_f(m^2).\] Applying the Petersson trace formula to \((\ell,m^2)\) gives \[\mathop{\sum\nolimits^{h}}_{f\in B_{2k}} a_f(\ell)a_f(m^2)=\delta_{\ell,m^2}+2\pi i^{-2k}\sum_{c\ge1}\frac{S(\ell,m^2;c)}{c}J_{2k-1}\left(\frac{4\pi m\sqrt\ell}{c}\right).\] Hence \[I_\ell(g)=\Delta_\ell(g)+\mathcal{G}_\ell(g),\qquad \Delta_\ell(g):=\mathbf{1}_{\ell=\square}\sum_{d\ge1}g(d^2\sqrt\ell),\] \[\mathcal{G}_\ell(g):=2\pi i^{-2k}\sum_{d,m\ge1}g(d^2m)\sum_{c\ge1}\frac{S(\ell,m^2;c)}{c}J_{2k-1}\left(\frac{4\pi m\sqrt\ell}{c}\right).\] Opening the Kloosterman sum yields \[\mathcal{G}_\ell(g)=\pi i^{-2k}\sum_{c\ge1}\frac{1}{c}\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\bar x}{c}\right)\sum_{d\ge1}\sideset{}{'}\sum_{m\in\mathbb{Z}}g(d^2|m|)J_{2k-1}\left(\frac{4\pi|m|\sqrt\ell}{c}\right)e\left(\frac{xm^2}{c}\right).\]

Step 2: Poisson summation in the quadratic variable.Since \(g\in\mathcal{S}\), Poisson summation gives \[\mathcal{G}_\ell(g)=\pi i^{-2k}\sum_{c\ge1}\frac{1}{c}\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\bar x}{c}\right)\sum_{d\ge1}\sum_{a\bmod c}e\left(\frac{xa^2+an}{c}\right)\sum_{n\in\mathbb{Z}}\widehat{W}_{c,d,\ell}\left(\frac{n}{c}\right),\] where \[\widehat{W}_{c,d,\ell}(\xi)=2\int_0^\infty g(d^2t)J_{2k-1}\left(\frac{4\pi t\sqrt\ell}{c}\right)\cos(2\pi t\xi)dt.\] Using the arithmetic identity 8 , \[\Sigma_c(\ell,n):=\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\bar x}{c}\right)\sum_{a\bmod c}e\left(\frac{xa^2+an}{c}\right)=c\sum_{r\mid c}\mu\left(\frac{c}{r}\right)\rho_r(n,\ell),\] with \(\rho_r(n,\ell)=\#\{u\bmod r:u^2+nu+\ell\equiv0\pmod r\}\). Substituting and setting \(c=rq\) leads to \[\mathcal{G}_\ell(g)=\pi i^{-2k}\sum_{r\ge1}\frac{1}{r}\sum_{n\in\mathbb{Z}}\rho_r(n,\ell)\sum_{q\ge1}\frac{\mu(q)}{q}\sum_{d\ge1}\widehat{W}_{rq,d,\ell}\left(\frac{n}{rq}\right).\]

By the Mellin inversion for \(g\), we have \[\begin{align} &\sum_{n\in\mathbb{Z}}\rho_r(n,\ell)\sum_{q\ge1}\frac{\mu(q)}{q}\sum_{d\ge1}\widehat{W}_{rq,d,\ell}\left(\frac{n}{rq}\right)\\ =&2\sum_{q\geqslant 1}\frac{\mu(q)}{q}\sum_{d\geqslant 1}\frac{1}{d^2}\int_0^\infty \left(\frac{1}{2\pi i}\int_{(\sigma)} \widetilde{g}(1-z)y^{z-1}dz\right) J_{2k-1}\left(\frac{4\pi y\sqrt{\ell}}{rqd^2}\right) \cos\left(\frac{2\pi ny}{rqd^2}\right)dy\\ =&\frac{1}{2\pi i}\int_{(\sigma)} \widetilde{g}(1-z)r^{z}\mathcal{J}_{n,\ell}(z)\sum_{q\geqslant 1}\frac{\mu(q)}{q^{1-z}}\sum_{d\geqslant 1}\frac{1}{d^{2-2z}}dz\\ =&\frac{1}{2\pi i}\int_{(\sigma)} \widetilde{g}(1-z) r^z\frac{\zeta(2-2z)}{\zeta(1-z)} \mathcal{J}_{n,\ell}(z)dz, \end{align}\] for \(\sigma\in(-1/2,0)\), where \[\label{eqn:32def-Jnl} \mathcal{J}_{n,\ell}(z) =2\int_0^\infty y^{z-1}J_{2k-1}(4\pi\sqrt{\ell}y)\cos(2\pi ny)dy.\tag{17}\]

Substituting this formula back into \(\mathcal{G}_{\ell}(g)\) and summing over \(r\) first, we obtain \[\begin{align} \mathcal{G}_\ell(g)=&\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)} \widetilde{g}(1-z)\sum_{r\geqslant1}\frac{1}{r^{1-z}} \sum_{n\in\mathbb{Z}}\rho_r(n,\ell)\frac{\zeta(2-2z)}{\zeta(1-z)} \mathcal{J}_{n,\ell}(z)dz\\ =&\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)} \widetilde{g}(1-z) \sum_{n\in\mathbb{Z}} \mathscr L_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell}(z)dz. \end{align}\] Here, we used the definition \[\mathscr{L}_{n^2-4\ell}(s)=\frac{\zeta(2s)}{\zeta(s)}\sum_{r=1}^{\infty}\frac{\rho_{r}(n,\ell)}{r^s},\quad\mathrm{Re}(z)>0.\]

Step 3: Apply the same to \(I_\ell(\mathcal{T}_\mu g)\). By Lemma 4, repeating the same steps gives \[I_\ell(\mathcal{T}_\mu g)=\Delta_\ell(\mathcal{T}_\mu g)+\mathcal{G}_\ell(\mathcal{T}_\mu g),\] with \[\mathcal{G}_\ell(\mathcal{T}_\mu g)=\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)}\widetilde{\mathcal{T}_\mu g}(1-z)\sum_{n\in\mathbb{Z}}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell}(z)dz.\]

Step 4: Archimedean reciprocity. Let \(\mathcal{G}_\ell^{\mathrm{ns}}(g)\) denote the contribution of \(n^2\neq4\ell\) in \(\mathcal{G}_\ell(g)\). By Lemma 7 and 11 , \[\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell}(z)=\mathcal{G}_\mu(z)\mathscr{L}_{n^2-4\ell}(z)\mathcal{J}_{n,\ell}(1-z)\qquad(n^2\neq4\ell).\] Combined with \(\widetilde{\mathcal{T}_\mu g}(s)=\widetilde{g}(1-s)\mathcal{G}_\mu(s)\), we obtain \[\mathcal{G}_\ell^{\mathrm{ns}}(g)=\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)}\widetilde{\mathcal{T}_\mu g}(z)\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(z)\mathcal{J}_{n,\ell}(1-z)dz.\] Shifting the contour (justified by Lemma 8 and 4) yields \[\mathcal{G}_\ell^{\mathrm{ns}}(g)=\mathcal{G}_\ell^{\mathrm{ns}}(\mathcal{T}_\mu g).\] If \(\ell\) is not a perfect square, this already gives \(I_\ell(g)=I_\ell(\mathcal{T}_\mu g)\).

Step 5: Singular and diagonal corrections. Now assume \(\ell=m^2\). Then \[I_\ell(g)-I_\ell(\mathcal{T}_\mu g)=\frac{\pi i^{-2k}}{2\pi i}\int_{(\sigma)}\left(\widetilde{g}(1-z)-\widetilde{\mathcal{T}_\mu g}(1-z)\right)\sum_{n=\pm2m}\mathscr{L}_0(1-z)\mathcal{J}_{n,m^2}(z)dz+\Delta_{m^2}(g)-\Delta_{m^2}(\mathcal{T}_\mu g).\] Since \(\mathscr{L}_0(s)=\zeta(2s-1)\) and \(\mathcal{J}_{-2m,m^2}=\mathcal{J}_{2m,m^2}\), define \[\mathcal{S}_m(g)=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(1-z)\mathcal{R}_m(z)dz,\qquad \mathcal{R}_m(z):=2\pi i^{-2k}\zeta(1-2z)\mathcal{J}_{2m,m^2}(z).\] By Lemma 6, \(\mathcal{R}_m(z)=\mathcal{G}_\mu(z)m^{-z}\zeta(2z)\). A direct computation using \(\mathcal{G}_\mu(z)\mathcal{G}_\mu(1-z)=1\) shows that the combined kernel satisfies \[\mathcal{R}_m(z)+m^{z-1}\zeta(2-2z)=\mathcal{G}_\mu(z)\bigl(\mathcal{R}_m(1-z)+m^{-z}\zeta(2z)\bigr).\] Notice that the term \(m^{z-1}\zeta(2-2z)\) comes from the Noting that \(\Delta_{m^2}(g)=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(1-z)m^{z-1}\zeta(2-2z)dz\), and using \(\widetilde{\mathcal{T}_\mu g}(z)=\widetilde{g}(1-z)\mathcal{G}_\mu(z)\), we obtain after contour shifting \[\mathcal{S}_m(\mathcal{T}_\mu g)+\Delta_{m^2}(\mathcal{T}_\mu g)=\mathcal{S}_m(g)+\Delta_{m^2}(g).\] Hence \(I_{m^2}(g)=I_{m^2}(\mathcal{T}_\mu g)\).

This completes the proof of Theorem 1.

4.2 Maass case (Theorem 3)↩︎

The proof follows the same argument as the holomorphic case with the following modifications:

1. Trace formula:Replace the Petersson trace formula with the Kuznetsov trace formula, introducing both discrete (Maass cusp forms) and continuous (Eisenstein series) spectra.

2. Archimedean transform:Replace \(T_\mu\) with \(T_{\mu_t}\), depending on the spectral parameter \(t\), where \(\mu_t = (2it,0,-2it)\) and the associated kernel \(G_{\mu_t}(s)\) replaces \(G_\mu(s)\). Since \(I^{\vee}_{\ell}(g;h)\neq I_{\ell}(\mathcal{T}_{\mu_t}g;h)\) for some fixed \(t\), we need to treat \(I^{\vee}_{\ell}(g;h)\) differently.

3. Continuous spectrum contribution:Unlike the holomorphic case, the pole at \(z=0\) coming from the square-discriminant terms \(\mathscr{L}_{n,\ell}(1-z)\) is not cancelled by the Maass kernel \(J_{n,4\ell,t}(z)\). Consequently, the contour shift crosses this pole and produces an additional residue term. After averaging over the continuous spectrum, these residues give rise to the explicit Eisenstein contribution \(P_t(g)\) appearing in the Voronoi identity.

We proceed in six steps.

Step 1: Kuznetsov trace formula. Define \(I_\ell(g;h)\) as the left-hand side of ?? . Using \[A_u(n)=\sum_{d^2\mid n}\lambda_u\left(\left(\frac{n}{d^2}\right)^2\right),\qquad A_t(n)=\sum_{d^2\mid n}\tau_t\left(\left(\frac{n}{d^2}\right)^2\right),\] and writing \(n=d^2m\), the Kuznetsov formula 7 gives \[I_\ell(g;h)=H_0(h)\Delta_\ell(g)+\mathcal{G}_{\ell,h}(g),\] where \[H_0(h):=\frac{1}{\pi^2}\int_{-\infty}^{\infty}t\tanh(\pi t)h(t)dt,\quad \Delta_\ell(g):=\mathbf{1}_{\ell=\square}\sum_{d\ge1}g(d^2\sqrt\ell),\] and \[\mathcal{G}_{\ell,h}(g)=\sum_{c\ge1}\frac{1}{c}\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\bar x}{c}\right)\sum_{d\ge1}\sideset{}{'}\sum_{m\in\mathbb{Z}}g(d^2|m|)\mathcal{J}_h\left(\frac{4\pi|m|\sqrt\ell}{c}\right)e\left(\frac{xm^2}{c}\right).\]

Step 2: Poisson summation in the quadratic variable. Repeating the Poisson summation argument from the holomorphic case (which is valid because \(g\in\mathcal{S}\)) yields \[\mathcal{G}_{\ell,h}(g)=\sum_{r\ge1}\frac{1}{r}\sum_{n\in\mathbb{Z}}\rho_r(n,\ell)\sum_{q\ge1}\frac{\mu(q)}{q}\sum_{d\ge1}\frac{1}{d^2}\int_0^{\infty}g(y)\mathcal{J}_h\left(\frac{4\pi y\sqrt\ell}{rqd^2}\right)\cos\left(\frac{2\pi ny}{rqd^2}\right)dy;\] and furthermore, \[\label{equ:maass-G-integral} \mathcal{G}_{\ell,h}(g) =\frac{1}{4\pi i}\int_{(\sigma)} \widetilde{g}(1-z) \sum_{n\in\mathbb{Z}} \mathscr L_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)dz,\tag{18}\] where \(-\frac{1}{2}<\sigma<0\).

Step 3: Apply the same to \(I_\ell^\vee(g;h)\). Unlike the holomorphic case, the transform \(\mathcal{T}_{\mu_t}\) depends on the spectral parameter \(t\), so the manipulation for \(I_\ell^\vee(g;h)\) is not identical.

Let \(\varepsilon_0>0\) such that \(h\) is holomorphic for \(|\operatorname{Im}(t)|<1+\varepsilon_0\), and take \(\sigma\in(-\varepsilon_0,0)\). By definition, \[I_\ell^\vee(g;h)=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(z)\mathcal{C}_{\ell,h}(z)dz,\] where \[\mathcal{C}_{\ell,h}(z):=\sum_{d,m\ge1}d^{2z-2}m^{z-1}\left[\sum_{u\in\mathcal{B}}\omega_u h_{1-z}(t_u)\lambda_u(\ell)\lambda_u(m^2)+\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{h_{1-z}(t)\tau_t(\ell)\tau_t(m^2)}{|\zeta(1+2it)|^2}dt\right],\] with \(h_{1-z}(t)=h(t)\mathcal{G}_{\mu_t}(1-z)\).

Applying the Kuznetsov trace formula gives \[\mathcal{C}_{\ell,h}(z)=\zeta(2-2z)\sum_{m\ge1}m^{z-1}\left[\delta_{\ell,m^2}H_0(h_{1-z})+\sum_{c\ge1}\frac{S(\ell,m^2;c)}{c}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi m\sqrt{\ell}}{c}\right)\right].\]

Define \[\mathcal{D}_{\ell,h}(z):=\sum_{m\ge1}m^{z-1}\sum_{c\ge1}\frac{S(\ell,m^2;c)}{c}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi m\sqrt{\ell}}{c}\right).\] Opening the Kloosterman sum yields \[\mathcal{D}_{\ell,h}(z)=\frac{1}{2}\sum_{c\ge1}\frac{1}{c}\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\overline{x}}{c}\right)\sideset{}{'}\sum_{m\in\mathbb{Z}}|m|^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi|m|\sqrt{\ell}}{c}\right)e\left(\frac{xm^2}{c}\right).\]

To apply Poisson summation, we use the decomposition (see 15 ) \[\mathcal{J}_{h_{1-z}}(y)=\mathcal{Z}_{\ell,h}(z)J_{1-z}(y)+R_{\ell,h}(y),\] where \(J_{1-z}\) is the Bessel function of order \(1-z\), \(R_{\ell,h}^{(j)}(y)\ll y^{\alpha-j}\) as \(y\to0^+\) for \(j=0,1,2\) and any \(\alpha<2+2\varepsilon_0\), and \[\mathcal{Z}_{\ell,h}(z)=-2(1-z)h\left(\frac{1-z}{2i}\right)\frac{G_0(1-z)G_0(2-2z)}{\cosh\left(\frac{\pi(1-z)}{2i}\right)}.\]

Applying Poisson summation (and noting the \(m=0\) contribution) gives \[\label{equ:poisson-dual} \begin{align} &\sideset{}{'}\sum_{m\in\mathbb{Z}}|m|^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi|m|\sqrt{\ell}}{c}\right)e\left(\frac{xm^2}{c}\right)+\lim_{y\to0^+}y^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi y\sqrt{\ell}}{c}\right)\\ =&\frac{2}{c}\sum_{a\bmod c}\sum_{n\in\mathbb{Z}}e\left(\frac{a^2x+an}{c}\right)\int_0^{\infty}y^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi y\sqrt{\ell}}{c}\right)\cos(2\pi ny)dy. \end{align}\tag{19}\]

Using the arithmetic identity 8 , we obtain \[\begin{align} \mathcal{D}_{\ell,h}(z)=&\sum_{r\ge1}\frac{1}{r}\sum_{n\in\mathbb{Z}}\rho_r(n,\ell)\sum_{q\ge1}\frac{\mu(q)}{q}\int_0^{\infty}y^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi y\sqrt{\ell}}{rq}\right)\cos(2\pi ny)dy\\ &+\frac{1}{2}\sum_{c\ge1}\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\overline{x}}{c}\right)\lim_{y\to0^+}y^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi y\sqrt{\ell}}{c}\right). \end{align}\]

Substituting back into \(\mathcal{C}_{\ell,h}(z)\) and simplifying, the middle term becomes \[\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n\in\mathbb{Z}}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h_{1-z}}(z)dz.\]

Finally, we decompose \(I_\ell^\vee(g;h)\) into four components: \[I_\ell^\vee(g;h)=I_\ell^{\vee,\mathrm{ns}}(g;h)+I_\ell^{\vee,\mathrm{sing}}(g;h)+I_\ell^{\vee,0}(g;h)+I_\ell^{\vee,\mathrm{diag}}(g;h),\] where \[\begin{align} I_\ell^{\vee,\mathrm{ns}}(g;h)&=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h_{1-z}}(z)dz,\\ I_\ell^{\vee,\mathrm{sing}}(g;h)&=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h_{1-z}}(z)dz,\\ I_\ell^{\vee,0}(g;h)&=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\zeta(2-2z)\sum_{c\ge1}\sideset{}{^*}\sum_{x\bmod c}e\left(\frac{\ell\overline{x}}{c}\right)\lim_{y\to0^+}y^{z-1}\mathcal{J}_{h_{1-z}}\left(\frac{4\pi y\sqrt{\ell}}{c}\right)dz,\\ I_\ell^{\vee,\mathrm{diag}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(z)\zeta(2-2z)\sum_{m\ge1}m^{z-1}\delta_{\ell,m^2}H_0(h_{1-z})dz. \end{align}\]

Step 4: Archimedean reciprocity. Denote \[\mathcal{G}_{\ell,h}^{\mathrm{ns}}(g)=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(1-z)\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)dz,\] with \(\sigma\in(-\frac{1}{2},0)\). By contour shifting and Lemma 14, \[\mathcal{G}_{\ell,h}^{\mathrm{ns}}(g)=\frac{1}{4\pi i}\int_{(1-\sigma)}\widetilde{g}(1-z)\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)dz-\frac{1}{2}\widetilde{g}(1)\mathcal{E}_{\ell,h}^{\mathrm{ns},0},\] where \(\mathcal{E}_{\ell,h}^{\mathrm{ns},0}:=\operatorname*{Res}_{z=0}\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)\). Applying Lemma 15 and \(z\mapsto1-z\) yields \[\mathcal{G}_{\ell,h}^{\mathrm{ns}}(g)=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h_{1-z}}(z)dz-\frac{1}{2}\widetilde{g}(1)\mathcal{E}_{\ell,h}^{\mathrm{ns},0},\] where \(h_{1-z}(t)=h(t)\mathcal{G}_{\mu_t}(1-z)\).

For the singular part \(n^2=4\ell\), the kernel \(\mathcal{J}_{2m,m^2,h}(z)\) has a pole at \(z=1/2\), so contour shifting gives an extra residue: \[\begin{align} \mathcal{G}_{\ell,h}^{\mathrm{sing}}(g)&:=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(1-z)\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)dz\\ &=\frac{1}{4\pi i}\int_{(1-\sigma)}\widetilde{g}(1-z)\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)dz\\ &\quad-\frac{1}{2}\widetilde{g}(1)\mathcal{E}_{\ell,h}^{\mathrm{sing},0}-\frac{1}{2}\widetilde{g}\left(\frac{1}{2}\right)\operatorname*{Res}_{z=1/2}\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)\\ &=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(z)\mathcal{J}_{n,\ell,h}(1-z)dz\\ &\quad-\frac{1}{2}\widetilde{g}(1)\mathcal{E}_{\ell,h}^{\mathrm{sing},0}-\frac{1}{2}\widetilde{g}\left(\frac{1}{2}\right)\operatorname*{Res}_{z=1/2}\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z), \end{align}\] where \(\mathcal{E}_{\ell,h}^{\mathrm{sing},0}:=\operatorname*{Res}_{z=0}\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z)\).

We decompose \(I_\ell(g;h)\) as \[I_\ell(g;h)=I_\ell^{\mathrm{ns}}(g;h)+I_\ell^{\mathrm{sing}}(g;h)+I_\ell^{0}(g;h)+I_\ell^{\mathrm{diag}}(g;h),\] with \[\begin{align} I_\ell^{\mathrm{ns}}(g;h)&=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n^2\neq4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h_{1-z}}(z)dz,\\[4pt] I_\ell^{\mathrm{sing}}(g;h)&=\frac{1}{4\pi i}\int_{(\sigma)}\widetilde{g}(z)\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(z)\mathcal{J}_{n,\ell,h}(1-z)dz\\ &\quad-\frac{1}{2}\widetilde{g}\left(\frac{1}{2}\right)\operatorname*{Res}_{z=1/2}\sum_{n^2=4\ell}\mathscr{L}_{n^2-4\ell}(1-z)\mathcal{J}_{n,\ell,h}(z),\\[4pt] I_\ell^{0}(g;h)&=-\frac{1}{2}\widetilde{g}(1)\bigl(\mathcal{E}_{\ell,h}^{\mathrm{ns},0}+\mathcal{E}_{\ell,h}^{\mathrm{sing},0}\bigr),\\[4pt] I_\ell^{\mathrm{diag}}(g;h)&=H_0(h)\Delta_\ell(g). \end{align}\]

By construction, \(I_\ell^{\mathrm{ns}}(g;h)=I_\ell^{\vee,\mathrm{ns}}(g;h)\). It remains to prove \[I_\ell^{\mathrm{sing}}(g;h)+I_\ell^{0}(g;h)+I_\ell^{\mathrm{diag}}(g;h)-I_\ell^{\vee,\mathrm{sing}}(g;h)-I_\ell^{\vee,0}(g;h)-I_\ell^{\vee,\mathrm{diag}}(g;h)=\frac{1}{\pi}\int_\mathbb{R}h(t)\tau_t(\ell) \mathcal{P}_g(t)dt,\] where \[\mathcal{P}_g(t)=\widetilde{g}(1-2it)\frac{\zeta(1-4it)}{\zeta(1+2it)} +\widetilde{g}(1) +\widetilde{g}(1+2it)\frac{\zeta(1+4it)}{\zeta(1-2it)}.\]

Step 5: Computation of \(I_\ell^{0}(g;h)-I_\ell^{\vee,0}(g;h)\). We first compute \(I_\ell^0(g;h)\).

Next we compute \(I_\ell^{\vee,0}(g;h)\).

Combining the two lemmas, we obtain \[I_\ell^0(g;h)-I_\ell^{\vee,0}(g;h)=\frac{1}{\pi}\int_{-\infty}^{\infty}h(t)\tau_t(\ell)\left\{\widetilde{g}(1-2it)\frac{\zeta(1-4it)}{\zeta(1+2it)}+\widetilde{g}(1)+\widetilde{g}(1+2it)\frac{\zeta(1+4it)}{\zeta(1-2it)}\right\}dt.\]

It remains to prove \[\label{equ:maass-singular-diag} I_\ell^{\mathrm{sing}}(g;h)+I_\ell^{\mathrm{diag}}(g;h)=I_\ell^{\vee,\mathrm{sing}}(g;h)+I_\ell^{\vee,\mathrm{diag}}(g;h).\tag{20}\]

Step 6: Proof of the singular-diagonal equality. If \(\ell\) is not a square, there is nothing to prove. Assume \(\ell=m^2\) with \(m\ge1\). For \(\sigma\in(-\frac{1}{2},0)\), \[\begin{align} I_\ell^{\mathrm{sing}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(z)\zeta(2z-1)\mathcal{J}_{2m,m^2,h}(1-z)dz-\widetilde{g}\left(\frac{1}{2}\right)\operatorname*{Res}_{z=\frac{1}{2}}\zeta(2z-1)\mathcal{J}_{2m,m^2,h}(z),\\[4pt] I_\ell^{\mathrm{diag}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}H_0(h)\widetilde{g}(1-z)\zeta(2-2z)m^{z-1}dz, \end{align}\] and \[\begin{align} I_\ell^{\vee,\mathrm{sing}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(z)\zeta(1-2z)\mathcal{J}_{2m,m^2,h_{1-z}}(z)dz,\\[4pt] I_\ell^{\vee,\mathrm{diag}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}H_0(h_{1-z})\widetilde{g}(z)\zeta(2-2z)m^{z-1}dz. \end{align}\]

Denote \(d(t):=t\tanh(\pi t)/\pi^2\). By Lemma 11, \[\label{equ:maass-critical-evaluation} G_0(2z)\mathcal{J}_{2m,m^2,t}(z)=d(t)m^{-z}\mathcal{G}_{\mu_t}(z).\tag{21}\]

From the pointwise identities, multiplying by \(h(t)\mathcal{G}_{\mu_t}(1-z)\) and integrating gives \[\zeta(1-2z)\mathcal{J}_{2m,m^2,h_{1-z}}(z)=H_0(h)m^{-z}\zeta(2z),\qquad H_0(h_{1-z})m^{z-1}\zeta(2-2z)=\zeta(2z-1)\mathcal{J}_{2m,m^2,h}(1-z).\] Since \(\mathcal{G}_{\mu_t}(1/2)=1\), we have \(H_0(h_{1/2})=H_0(h)\), and \(\zeta(0)=-1/2\) implies \[\operatorname*{Res}_{z=1/2}\mathcal{J}_{2m,m^2,h}(z)=-H_0(h)m^{-1/2}.\]

Substituting these into the expressions for \(I_\ell^{\mathrm{sing}}\) and \(I_\ell^{\mathrm{diag}}\), we obtain \[\begin{align} I_\ell^{\mathrm{sing}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(z)\zeta(2z-1)\mathcal{J}_{2m,m^2,h}(1-z)dz+\frac{1}{2}\widetilde{g}\left(\frac{1}{2}\right)H_0(h)m^{-1/2},\\ I_\ell^{\mathrm{diag}}(g;h)&=\frac{1}{2\pi i}\int_{(\sigma)}H_0(h)\widetilde{g}(z)\zeta(2z)m^{-z}dz-\frac{1}{2}\widetilde{g}\left(\frac{1}{2}\right)H_0(h)m^{-1/2}. \end{align}\] Adding them cancels the residue terms: \[I_\ell^{\mathrm{sing}}(g;h)+I_\ell^{\mathrm{diag}}(g;h)=\frac{1}{2\pi i}\int_{(\sigma)}\widetilde{g}(z)\Bigl[\zeta(2z-1)\mathcal{J}_{2m,m^2,h}(1-z)+H_0(h)\zeta(2z)m^{-z}\Bigr]dz.\]

Finally, applying ?? transforms the bracket into \(\zeta(1-2z)\mathcal{J}_{2m,m^2,h_{1-z}}(z)+H_0(h_{1-z})m^{z-1}\zeta(2-2z)\), which is exactly the integrand of \(I_\ell^{\vee,\mathrm{sing}}(g;h)+I_\ell^{\vee,\mathrm{diag}}(g;h)\). Hence \[I_\ell^{\mathrm{sing}}(g;h)+I_\ell^{\mathrm{diag}}(g;h)=I_\ell^{\vee,\mathrm{sing}}(g;h)+I_\ell^{\vee,\mathrm{diag}}(g;h).\]

This completes the proof of Theorem 3.

5 From Voronoi to Functional Equation↩︎

5.1 Holomorphic case↩︎

5.2 Maass case↩︎

For \(\operatorname{Re}(s)>1\), define \[L(s,\operatorname{Sym}^2 u)=\sum_{n=1}^{\infty}\frac{A_u(n)}{n^s}=\zeta(2s)\sum_{m=1}^{\infty}\frac{\lambda_u(m^2)}{m^s},\] and similarly \[L(s,\operatorname{Sym}^2 t)=\sum_{n=1}^{\infty}\frac{A_t(n)}{n^s}=\zeta(2s)\sum_{m=1}^{\infty}\frac{\tau_t(m^2)}{m^s},\] By checking the Euler product, one sees that \(L(s,\operatorname{Sym}^2,t)=\zeta(s+2it)\zeta(s)\zeta(s-2it)\).

Let \(t_0\) be a cuspidal spectral parameter. For \(g\in\mathcal{S}\), define \[F_{\ell,g,t_0}(x)=\sum_{u\in\mathcal{B}[t_0]} \omega_u \lambda_u(\ell)\sum_{n=1}^{\infty}A_u(n)g(nx).\] for \(x>0\). Then

Define \[A_{\ell,g,t_0}(s)=\sum_{u\in\mathcal{B}[t_0]}\omega_u \lambda_u(\ell)L(s,\operatorname{Sym}^2u),\quad\mathrm{Re}(s)>1.\]

References↩︎