Petersson Norm of Cusp Forms Associated to Real Quadratic Fields

TL;DR

This paper computes the Petersson norm of certain weight one cusp forms linked to real quadratic fields, expressing it through the Rademacher symbol and the field's regulator, advancing understanding of automorphic forms.

Contribution

It provides an explicit formula for the Petersson norm of cusp forms associated with real quadratic fields, connecting automorphic forms with algebraic invariants.

Findings

Explicit formula for Petersson norm in terms of Rademacher symbol

Connection between cusp forms and real quadratic field invariants

Enhanced understanding of automorphic forms related to quadratic fields

Abstract

In this article, we compute the Petersson norm of a family of weight one cusp forms constructed by Hecke and express it in terms of the Rademacher symbol and the regulator of real quadratic field.

Equations

$${\Vert {\vartheta }_L\Vert }_{\mathrm{Pet}}^2:={\int }_{\Gamma \backslash \mathcal{H}}v|{\vartheta }_L(\tau ){|}^{2}d\mu (\tau ),$$

$${\Vert {\vartheta }_L\Vert }_{\mathrm{Pet}}^2=-\frac{1}{12}\left(\Psi ({\gamma }_0)+\Psi ({\gamma }_1)\right)\log {\epsilon }_{\kappa }.$$

$$\Phi (t,t_0;L):={\int }_{\Gamma \backslash \mathcal{H}}v⟨{\Theta }_L(\tau ,t),{\Theta }_L(\tau ,t_0)⟩\frac{dudv}{v^2},$$

$$\{Z\in V(\mathbb{C}):Q(Z)=0,Q(\mathrm{Im}(Z))<0\}/{\mathbb{C}}^{\times }\subset \mathbb{P}(V(\mathbb{C})).$$

$${\mathcal{H}}^2$$

$$(z_1,z_2)$$

$$\begin{array}{rl}\mathbb{R}& \to {\mathcal{D}}_{F,N}^{+}\subset {\mathbb{R}}^2\\ t& \mapsto Z(t):=\sqrt{N}\left(\genfrac{}{}{0ex}{}{e^t}{-e^{-t}}\right).\end{array}$$

$$V:=\left\{\left(\begin{array}{cc}{\lambda }_1& {\lambda }_2\\ {\lambda }_2^{\prime }& {\lambda }_1^{\prime }\end{array}\right)\in M_2(F)\right\}.$$

$$w=w(t,t_0)=X(t)+iY(t_0):=\sqrt{N}\left(\begin{array}{cc}{e}^t& i{e}^{t_0}\\ i{e}^{-t_0}& -e^{-t}\end{array}\right)\in V(\mathbb{C}),$$

$$w^{\perp }=w^{\perp }(t,t_0)=X^{\perp }(t)+i{Y}^{\perp }(t_0):=\sqrt{N}\left(\begin{array}{cc}{e}^t& i{e}^{t_0}\\ -i{e}^{-t_0}& e^{-t}\end{array}\right)\in V(\mathbb{C}).$$

$$Q({\lambda }_w)=-\frac{(\lambda ,X(t){)}^2+(\lambda ,Y(t_0){)}^2}{4},Q({\lambda }_{w^{\perp }})=\frac{(\lambda ,X^{\perp }(t){)}^2+(\lambda ,Y^{\perp }(t_0){)}^2}{4},$$

$$U:=\left\{\left(\begin{array}{cc}\mu & -\mu \\ -{\mu }^{\prime }& {\mu }^{\prime }\end{array}\right)\in M_2(F)\right\},U^{\vee }:=\left\{\left(\begin{array}{cc}\mu & \mu \\ {\mu }^{\prime }& {\mu }^{\prime }\end{array}\right)\in M_2(F)\right\}.$$

$${\mathfrak{e}}_1:=\frac{\sqrt{D}}{2}\left(\begin{array}{cc}1& -1\\ 1& -1\end{array}\right),{\mathfrak{e}}_2:=\frac{1}{2}\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right),{\mathfrak{e}}_1^{\vee }:=\frac{N}{2\sqrt{D}}\left(\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right),{\mathfrak{e}}_2^{\vee }:=\frac{N}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right).$$

$$Q(\lambda )=\text{prescript}t{\lambda }_U\cdot {\lambda }_{U^{\vee }}$$

$$\begin{array}{rl}\iota :V& \stackrel{\cong }{\to }{M}_2(\mathbb{Q})\\ \lambda & \mapsto \left(\begin{array}{cc}(\lambda ,{\mathfrak{e}}_2^{\vee })& -(\lambda ,{\mathfrak{e}}_1^{\vee })\\ (\lambda ,{\mathfrak{e}}_1)& (\lambda ,{\mathfrak{e}}_2)\end{array}\right).\end{array}$$

$$z_1:=\frac{(w,{\mathfrak{e}}_1)}{(w,{\mathfrak{e}}_2)},z_2:=-\frac{(w,{\mathfrak{e}}_1^{\vee })}{(w,{\mathfrak{e}}_2)}.$$

$${\tilde{z}}_1=g\cdot z_1,{\tilde{z}}_2=\frac{z_2}{\det g}.$$

$$\begin{array}{rl}\iota (w(t,t_0))& =\frac{\mathrm{sh}(t)+i\mathrm{ch}(t_0)}{\sqrt{N}}Z(z_1(t,t_0),z_2(t,t_0)),\\ z_1(t,t_0)& =\sqrt{D}\frac{-\mathrm{sh}(2t)-\mathrm{sh}(2t_0)+2i\mathrm{ch}(t-t_0)}{2(1+{\mathrm{sh}}^{2}t+{\mathrm{sh}}^2{t}_0)},z_2(t,t_0)=\frac{N}{D}{z}_1(t,-t_0).\end{array}$$

$$z(t)=x(t)+iy(t):=\sqrt{D}\frac{-\mathrm{sh}(t)+i}{\mathrm{ch}(t)}.$$

$$\begin{array}{rl}{\iota }_g(w(t,0))& =\frac{-\overline{z(t)}}{\sqrt{N}y(t)}(cz(t)+d)Z({\tilde{z}}_1(t),{\tilde{z}}_2(t)),\\ {\iota }_g(w^{\perp }(t,0))& =\frac{\sqrt{D}}{\sqrt{N}y(t)}(c\overline{z(t)}+d)Z(\overline{{\tilde{z}}_1(t)},{\tilde{z}}_2(t))\end{array}$$

$$d{\tilde{z}}_1=\frac{i\det (g)yz}{(cz+d{)}^2\sqrt{D}}dt,d{\tilde{z}}_2=\frac{iNyz}{\det (g)D^{3/2}}dt.$$

$$\kappa :=\mathrm{Nm}(\mathfrak{a})/N\in \mathbb{N},$$

$${\epsilon }_{\kappa }-1\in \kappa \mathfrak{d}.$$

$${\Theta }_L(\tau ,t):=\sqrt{v}\sum _{\lambda \in L^{\vee }}(\lambda ,Z^{\perp }(t){)}_{F,N}\mathbf{e}\left(\frac{(\lambda ,Z^{\perp }(t){)}_{F,N}^2}{4}\tau -\frac{(\lambda ,Z(t){)}_{F,N}^2}{4}\overline{\tau }\right){\phi }_{\lambda },$$

$${\vartheta }_L(\tau ):={\int }_{{\Gamma }_{\kappa }^{+}\backslash {\mathcal{D}}_{F,N}^{+}}{\Theta }_L(\tau ,t)dt=\sum _{\begin{array}{c}\lambda \in {\Gamma }_{\kappa }^{+}\backslash L^{\vee }\\ \mathrm{Nm}(\lambda )>0\end{array}}\mathrm{sgn}(\lambda )\mathbf{e}\left(\frac{\mathrm{Nm}(\lambda )}{N}\right){\phi }_{\lambda }.$$

$$M=M_{\mathfrak{a},\kappa }:=\left\{\left(\begin{array}{cc}{\lambda }_1& {\lambda }_2\\ {\lambda }_2^{\prime }& {\lambda }_1^{\prime }\end{array}\right)\in M_2(L^{\vee }):{\lambda }_1-{\lambda }_2\in L\right\}.$$

$$\begin{array}{rl}{P}_U& :=M\cap U=\left\{\left(\begin{array}{cc}\mu & -\mu \\ -{\mu }^{\prime }& {\mu }^{\prime }\end{array}\right):\mu \in L^{\vee }\cap \frac{1}{2}L\right\},\\ P_{U^{\vee }}& :=M\cap U^{\vee }=\left\{\left(\begin{array}{cc}\mu & \mu \\ {\mu }^{\prime }& {\mu }^{\prime }\end{array}\right):\mu \in L^{\vee }\right\}.\end{array}$$

$$g_L:=\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)\in {\mathrm{GL}}_2^{+}(\mathbb{Q}).$$

$$2\det (g_L)=\mathrm{Nm}(\mathfrak{a}\cap 2(\kappa \mathfrak{d}{)}^{-1}\mathfrak{a}).$$

$$g_{\kappa }:=\{\begin{array}{ll}\left(\begin{array}{cc}2/(D\kappa )& \\ & 1\end{array}\right)& 2\mid D\kappa ,\\ \left(\begin{array}{cc}1/(D\kappa )& 1\\ & 2\end{array}\right)& 2\nmid D\kappa .\end{array}$$

$${\phi }_M(x)=\sum _{\lambda \in M/P}{\phi }_{\lambda }(x)=\sum _{\mu \in \frac{1}{2}{P}_U/P_U}{\phi }_{\mu }(x_U){\phi }_{{\mu }^{\vee }}(x_{U^{\vee }}).$$

Full text

1. Introduction

For an integral ideal $\mathfrak{a}$ in a real quadratic field $F=\mathbb{Q}(\sqrt{D})\subset\mathbb{R}$ with fundamental discriminant $D$ and a positive integer $\kappa$, we have an even integral lattice $L=L_{\mathfrak{a},\kappa}=(\mathfrak{a},\kappa{\mathrm{Nm}}/{\mathrm{Nm}}(\mathfrak{a}))$. It is anisotropic over $\mathbb{Q}$ and has signature $(1,1)$ over $\mathbb{R}$. In [6], Hecke associated to such lattice $L$ a holomorphic, vector-valued cusp form $\vartheta_{L}$ of weight one, whose Fourier expansion is given explicitly in terms of the elements in the dual lattice $L^{\vee}$ (see §2.3). In this note, we will calculate the Petersson norm of $\vartheta_{L}$, which is defined by

[TABLE]

with $\Gamma:={\mathrm{SL}}_{2}(\mathbb{Z})$ and $d\mu(\tau):=\frac{dudv}{v^{2}}$ the invariant measure. Let $\varepsilon_{\kappa}\in\mathcal{O}_{F}^{\times}$ be the totally positive unit defined in §2.3. It turns out that the Petersson norm of $\vartheta_{L}$ is a rational multiple of $\log\varepsilon_{\kappa}$, and the rational multiple can be expressed in terms of Rademacher symbols $\Psi(\gamma)$ of hyperbolic elements $\gamma\in\Gamma$ (see §2.7).

Our result is as follows.

Theorem 1.1**.**

Let $g_{L},g_{\kappa}\in{\mathrm{GL}}^{+}_{2}(\mathbb{Q})$ and $\gamma_{D,\kappa}\in\Gamma$ be defined as in (2.18), (2.20) and (2.34) respectively. Denote $\gamma_{0}:=g_{\kappa}\gamma_{D,\kappa}g_{\kappa}^{-1}$ and $\gamma_{1}:=g_{L}\gamma_{D,\kappa}g_{L}^{-1}$. Then $\gamma_{j}\in\Gamma$ and

[TABLE]

Remark 1.2*.*

In [3], we constructed a harmonic Maass form $\tilde{\vartheta}_{L}$ that maps to $\vartheta_{L}$ under the differential operator $\xi:=2iv\overline{\partial_{\overline{\tau}}}$. One can express $\|\vartheta_{L}\|^{2}_{\mathrm{Pet}}$ as a rational linear combination of the principal part Fourier coefficients of $\tilde{\vartheta}_{L}$, which are rational multiples of the fundamental unit of $F$. Therefore, $\|\vartheta_{L}\|^{2}_{\mathrm{Pet}}=c_{L}\log\varepsilon_{\kappa}$ for some $c_{L}\in\mathbb{Q}$ by the result loc. cit.

In [6], Hecke gave a well-known example of $\vartheta_{L}$ by taking $D=12,\kappa=1$ and $\mathfrak{a}=\mathfrak{d}=\sqrt{D}\mathcal{O}_{F}$. The cusp form $\vartheta_{L}(\tau)$ has four nonzero components, all equal to $\eta^{2}(\tau)$ up to sign. In this case, $\varepsilon_{\kappa}=7+2\sqrt{D}$ and $\gamma_{0}=\gamma_{1}=\left(\begin{smallmatrix}7&4\\ 12&7\end{smallmatrix}\right)$. From the definition of $\Psi$ in Eq. (2.31), we can evaluate $\Psi(\gamma_{0})=\Psi(\gamma_{1})=-2$. Therefore the theorem above implies that $\|\vartheta_{L}\|^{2}_{\mathrm{Pet}}=4\|\eta^{2}\|^{2}_{\mathrm{Pet}}=\frac{2\log(2+\sqrt{3})}{3}$. On the other hand, if one takes $D=12,\kappa=1$ and $\mathfrak{a}=\mathcal{O}_{F}$, then $\gamma_{0}$ is unchanged, but $\gamma_{1}=\left(\begin{smallmatrix}7&12\\ 4&7\end{smallmatrix}\right)$ and $\Psi(\gamma_{1})=2$, which implies that $\vartheta_{L}(\tau)$ vanishes identically. The same is true when $12$ is replaced with any positive, even fundamental discriminant.

The idea of the proof is to interchange the order of integration and evaluate the theta integral

[TABLE]

where $\Theta_{L}$ is the integral kernel used to produce $\vartheta_{L}$ (see §2.3) and $\langle,\rangle$ is the Hermitian pairing on finite dimensional complex vector spaces induced by the $L_{2}$-norm. It turns out that the integral of $\Phi$ over $t$ is a constant independent of $t_{0}$. An important observation is that this integrand is the theta function attached to a unimodular lattice $M$ containing $L\oplus-L$, which has signature $(2,2)$. We can then evaluate $\Phi(t,t_{0};L)$ using its Fourier expansion along a 1-dimensional boundary of an $\mathrm{O}(2,2)$ Shimura variety. This step is a simpler version of the calculations in [8] (see [5] for a nice example), except the infinity component of the Schwartz function being slightly different.

Since $\vartheta_{L}$ is itself a theta lift, the Rallis inner product formula is a natural tool to evaluate its Petersson norm [12]. The most general case of this formula was recently completed by Gan, Qiu and Takeda [4]. In the notation loc. cit., the case at hand is outside both the convergent range and the first term range. Gan, Qiu and Takeda extended the Rallis inner product formula to such cases by providing a missing ingredient, a second term identity of the regularized Siegel-Weil formula. Our result can be seen as a more explicit version of Theorem 1.3 loc. cit. for the dual pair $({\mathrm{SL}}_{2},\mathrm{O}(1,1))$. The proof here does not use the regularized Siegel-Weil formula, and is more straightforward using Fourier expansions of theta lifts. Furthermore, the calculations here will be useful for other theta lifts involving real-analytic modular forms that we are currently pursuing [9].

The paper is organized as follows. We recall the basic setups in §2, then proceed to the necessary calculations in §3, before proving Theorem 1.1 in §4.

Acknowledgment

We thank Ö. Imamoglu for helpful conversations about cycle integrals and the Rademacher symbol, and the referee for helpful comments. The author is partially supported by the DFG grant BR-2163/4-2 and an NSF postdoctoral fellowship.

2. Setup.

2.1. Quadratic Spaces and Embeddings.

Let $(V,Q)$ be a rational quadratic space of signature $(p,q)$. The Grassmannian ${\mathcal{D}}$ of oriented, negative definite subspaces of dimension $q$ is a symmetric space. When $q=2$, there is a complex structure on the Grassmannian by identifying it with

[TABLE]

For example, take $(V,Q)=(M_{2},\det/N)$, where $M_{2}(R)$ is the space of $2$ by $2$ matrices with entries in an additive subgroup $R\subset\mathbb{C}$, and $N\in\mathbb{Q}_{>0}$. Let ${\mathcal{D}}^{+}$ be a connected component of ${\mathcal{D}}$ isomorphic to ${\mathcal{H}}^{2}$ via the map

[TABLE]

where $Z(z_{1},z_{2}):=\left(\begin{smallmatrix}z_{1}z_{2}&z_{2}\\ z_{1}&1\end{smallmatrix}\right)\in M_{2}(\mathbb{C})$ for any $z_{1},z_{2}\in\mathbb{C}$. Under this identification, the action of $\mathrm{SO}(2,2)$ on ${\mathcal{D}}^{+}$ is the action of ${\mathrm{SL}}_{2}\times{\mathrm{SL}}_{2}$ on ${\mathcal{H}}^{2}$.

Another example comes from number field. Let $F=\mathbb{Q}(\sqrt{D})\subset\mathbb{R}$ be a real quadratic field with ′ the Galois conjugation in $\mathrm{Gal}(F/\mathbb{Q})$. The rational quadratic space $V_{F,N}:=(F,{\mathrm{Nm}}/N)$ has signature $(1,1)$. Since $F\otimes_{\mathbb{Q}}\mathbb{R}\cong\mathbb{R}^{2}$ via the two real embeddings of $F$, the symmetric space ${\mathcal{D}}_{F,N}$ associated to $V_{F,N}$ is a subset of $\mathbb{R}^{2}$, whose connected component ${\mathcal{D}}_{F,N}^{+}$ is isomorphic to $\mathbb{R}$ via

[TABLE]

The orthogonal complement of $Z(t)$ is spanned by the vector $Z^{\perp}(t):=\sqrt{N}\binom{e^{t}}{e^{-t}}$.

With the quadratic form $Q=\det/N$, the following vector space is also a rational quadratic space of signature $(2,2)$

[TABLE]

We can embed $V_{F,N}$ and $-V_{F,N}$ into $V$ via the the maps $\lambda\mapsto\left(\begin{smallmatrix}\lambda&\\ &\lambda\end{smallmatrix}\right)$ and $\lambda\mapsto\left(\begin{smallmatrix}&\lambda\\ \lambda&\end{smallmatrix}\right)$. This induces an isometry $V\cong V_{F,N}\oplus-V_{F,N}$ of rational quadratic spaces and an embedding of $({\mathcal{D}}_{F,N}^{+})^{2}\cong\mathbb{R}^{2}$ into the symmetric space ${\mathcal{D}}_{V}$ associated to $V$ via

[TABLE]

which satisfies $(w,w)=0$ and $(w,\bar{w})=-1<0$ for any $(t,t_{0})\in\mathbb{R}^{2}$. Meanwhile, we also define ${w^{\perp}}(t,t_{0})\in V(\mathbb{C})$ by

[TABLE]

Then $\{X(t),Y(t_{0}),X^{\perp}(t),Y^{\perp}(t_{0})\}$ form an orthogonal basis of $V(\mathbb{R})$. For ${\lambda}\in V(\mathbb{R})$, we denote ${\lambda}_{w}$ and ${\lambda}_{w^{\perp}}$ the projections of ${\lambda}$ onto the plane spanned by $\{X(t),Y(t_{0})\}$ and $\{X^{\perp}(t),Y^{\perp}(t_{0})\}$ respectively. It is then easy to see that

[TABLE]

where $(,)$ is the bilinear form associated to $Q$. We define $Q_{w}({\lambda}):=Q({\lambda}_{w^{\perp}})-Q({\lambda}_{w})$ to be the majorant associate to $w$.

2.2. Witt Decomposition.

The quadratic space $V$ has a natural Witt decomposition $V=U+U^{\vee}$, where $U,U^{\vee}$ are isotropic $\mathbb{Q}$-subspaces of $V$ defined by

[TABLE]

We fix a basis $\{\mathfrak{e}_{1},\mathfrak{e}_{2}\}$ of $U$ with corresponding dual basis $\{\mathfrak{e}^{\vee}_{1},\mathfrak{e}^{\vee}_{2}\}$ of $U^{\vee}$ by

[TABLE]

For ${\lambda}\in V$, write ${\lambda}_{U}$ and ${\lambda}_{U^{\vee}}$ the $U$ and $U^{\vee}$ component of ${\lambda}$ respectively. With respect to the basis (2.5), we can view ${\lambda}_{U}$ and ${\lambda}_{U^{\vee}}$ as column vectors in $\mathbb{Q}^{2}$ and write

[TABLE]

Therefore, $V$ is isomorphic to $M_{2}(\mathbb{Q})$ as rational quadratic spaces via

[TABLE]

This isometry allows us to identify ${\mathcal{D}}^{+}_{V}$ with ${\mathcal{D}}^{+}\cong{\mathcal{H}}^{2}$ by sending $w\in{\mathcal{D}}^{+}_{V}$ to $(w,\mathfrak{e}_{2})^{-1}\iota(w)=Z(z_{1},z_{2})\in M_{2}(\mathbb{C})$, where

[TABLE]

Suppose $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2}\}=\{a\mathfrak{e}_{1}+b\mathfrak{e}_{2},c\mathfrak{e}_{1}+d\mathfrak{e}_{2}\}$ is a different rational basis of $U$ with $g:=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in{\mathrm{GL}}^{+}_{2}(\mathbb{Q})$ and dual basis $\{{\tilde{\mathfrak{e}}}^{\vee}_{1},{\tilde{\mathfrak{e}}}^{\vee}_{2}\}=\{\tfrac{d\mathfrak{e}^{\vee}_{1}-c\mathfrak{e}^{\vee}_{2}}{\det{g}},\tfrac{-b\mathfrak{e}^{\vee}_{1}+a\mathfrak{e}^{\vee}_{2}}{\det{g}}\}$. Then the corresponding point $({\tilde{z}}_{1},{\tilde{z}}_{2})\in{\mathcal{H}}^{2}$ defined as in (2.7) satisfies

[TABLE]

Let $\iota_{g}:V\to M_{2}(\mathbb{Q})$ be the isomorphism with respect to this basis.

With respect to the basis (2.5), $\mathbb{C}\cdot\iota(w)$ is a point on ${\mathcal{D}}$ where

[TABLE]

Furthermore, $\iota({w^{\perp}})=\frac{(\mathrm{ch}(t)+i\mathrm{sh}(t_{0}))}{\sqrt{N}}\left(\begin{smallmatrix}\overline{z_{1}}z_{2}&z_{2}\\ \overline{z_{1}}&1\end{smallmatrix}\right)$. For $t\in\mathbb{R}$, we define

[TABLE]

Note that $|z(t)|=\sqrt{D}$ for all $t\in\mathbb{R}$. When $t_{0}=0$, we have $z_{1}(t,0)=Dz_{2}(t,0)/N=z(t)$. For any $g=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in{\mathrm{GL}}_{2}^{+}(\mathbb{Q})$, the vectors $w(t,0)$ and ${w^{\perp}}(t,0)$ become

[TABLE]

with ${\tilde{z}}_{1}(t)=gz(t),{\tilde{z}}_{2}(t)=\frac{Nz(t)}{D\det{g}}$. It is straightforward to check that

[TABLE]

2.3. Hecke’s Theta Function.

Let $N\in\mathbb{Q}_{>0}$ be the same as in the previous two sections. For an $\mathcal{O}_{F}$-ideal $\mathfrak{a}$ such that

[TABLE]

the lattice $L=L_{\mathfrak{a},\kappa}:=(\mathfrak{a},{\mathrm{Nm}}/N)\subset V_{F,N}$ is even integral, and its dual $L^{\vee}$ is given by the fractional ideal $(\kappa\mathfrak{d})^{-1}\mathfrak{a}$. Let $\Gamma_{\kappa}^{+}\cong\mathbb{Z}$ be subgroup of the discriminant kernel of this lattice that also fixes ${\mathcal{D}}^{+}_{F,N}$. It is generated by the smallest totally positive unit $\varepsilon_{\kappa}\in\mathcal{O}_{F}^{\times}$ satisfying $\varepsilon_{\kappa}>1$ and $(\varepsilon_{\kappa}-1)\mathfrak{a}^{\vee}\subset\mathfrak{a}$. The last condition is equivalent to

[TABLE]

Let $\mathcal{S}_{L}$ be the space of $\mathbb{C}$-valued Schwartz-Bruhat functions on $L\otimes\mathbb{A}_{\mathbf{f}}$ with support on $L^{\vee}\otimes{\widehat{\mathbb{Z}}}$ and that are translation-invariant under $L\otimes{\widehat{\mathbb{Z}}}$. It is a $\mathbb{C}$-vector space of dimension $|L^{\vee}/L|=D\kappa$. There is the usual Weil representation $\omega$ of ${\mathrm{SL}}_{2}(\mathbb{A}_{\mathbf{f}})$ on $\mathcal{S}(V(\mathbb{A}_{\mathbf{f}}))$ with respect to the standard additive character $\psi_{\mathrm{f}}$ of $\mathbb{A}_{\mathbf{f}}$. It becomes the Weil representation $\rho_{L}$ in [2] when restricted to the subspace $\mathcal{S}_{L}$ [7]. To define Hecke’s theta function, we begin with a theta kernel with value in $\mathcal{S}_{L}$

[TABLE]

where $(\cdot,\cdot)_{F,N}$ is the bilinear form associated to ${\mathrm{Nm}}/N$ and $\varphi_{\lambda}$ is the characteristic function of $\lambda+L\otimes{\widehat{\mathbb{Z}}}\subset V_{F,N}(\mathbb{A}_{\mathbf{f}})$. It is $\Gamma^{+}_{\kappa}$ invariant as a function of $t\in{\mathcal{D}}_{F,N}^{+}$, and transforms in $\tau\in{\mathcal{H}}$ with respect to the Weil representation $\omega$ of weight 1 on $\Gamma$. Integrating $\Theta_{L}(\tau,t){dt}{}$ then defines a weight one modular form $\vartheta_{L}(\tau)$ with the following Fourier expansion (see [6, 3])

[TABLE]

From this, it is clear that $\vartheta_{L}\in S_{1}(\Gamma,\rho_{L})\otimes\mathcal{S}_{L}$, and that it only depends on the class of $\mathfrak{a}$ in the narrow class group ${\mathrm{Cl}}^{+}(F)$ and the integer $\kappa$. Finally, we can identify $\mathcal{S}_{L}$ with $\mathbb{C}^{D\kappa}$ by sending $\varphi\in\mathcal{S}_{L}$ to $(\varphi(\mu))_{\mu\in L^{\vee}/L}$, and let $\langle\cdot,\cdot\rangle$ be the standard Hermitian pairing induced by the $L^{2}$-norm $|\cdot|$. Then $v\left|\vartheta_{L}(\tau)\right|^{2}$ is an integrable, $\Gamma$-invariant function on ${\mathcal{H}}$ and the Petersson norm of $\vartheta_{L}(\tau)$ is defined as in Eq. (1.1).

2.4. Lattice in $V$.

From the lattice $L=L_{\mathfrak{a},\kappa}$ in the previous section, we can construct the following unimodular lattice in $V$

[TABLE]

Let $\Gamma_{M}$ be its discriminant kernel and $\Gamma_{M}^{+}$ the subgroup fixing ${\mathcal{D}}^{+}$. Then $\Gamma_{M}^{+}$ contains $(\Gamma_{\kappa}^{+})^{2}$. As in section 1.3 of [8], we can define a sublattice $P=P_{U}+P_{U^{\vee}}\subset M$ by

[TABLE]

Via the map $\left(\begin{smallmatrix}\lambda_{1}&\lambda_{2}\\ \lambda^{\prime}_{1}&\lambda^{\prime}_{2}\end{smallmatrix}\right)\mapsto\lambda_{1}-\lambda_{2}$, the group $M/P$ is isomorphic to $L/(2L^{\vee}\cap L)$, which is then isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{2}$ if $2\nmid D\kappa$ and trivial otherwise. In the former case, $P^{\vee}=\frac{1}{2}P$.

Suppose $L\cap 2L^{\vee}$ has the $\mathbb{Z}$-basis $\{a\sqrt{D}+b,d\}$ with $a,b,d\in\mathbb{Q}$ and $a,d>0$. Then $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2}\}=\{a\mathfrak{e}_{1}+b\mathfrak{e}_{2},d\mathfrak{e}_{2}\}$ is a $\mathbb{Z}$-basis of $P_{U}$. We will denote

[TABLE]

Note that

[TABLE]

In addition, we also define $g_{\kappa}\in{\mathrm{GL}}^{+}_{2}(\mathbb{Q})$ by

[TABLE]

Then $g_{\kappa}\cdot\binom{\sqrt{D}/2}{1/2}$ is a $\mathbb{Z}$-basis of the inverse different $\mathfrak{d}_{\kappa}^{-1}$ of the order $\kappa\mathbb{Z}+\frac{\kappa^{2}D+\kappa\sqrt{D}}{2}\mathbb{Z}$.

For an element $\lambda\in V$, suppose $\lambda_{U}=r,\lambda_{U^{\vee}}=\eta_{1}$ with respect to the basis $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2},{\tilde{\mathfrak{e}}}^{\vee}_{1},{\tilde{\mathfrak{e}}}^{\vee}_{2}\}$, then $\lambda_{U^{\vee}}\cdot P_{U}\subset\mathbb{Z}$ if and only if $\eta_{1}\in\mathbb{Z}^{2}$. We use $\varphi_{\lambda}\in\mathcal{S}(V(\mathbb{A}_{\mathbf{f}}))$ to denote the characteristic function of the coset $\lambda+P\otimes{\widehat{\mathbb{Z}}}$.

Example 2.1**.**

Suppose $2\nmid D\kappa$ and $\mathfrak{a}=\mathbb{Z}\frac{\sqrt{D}+B}{2}+\mathbb{Z}A$ with $A$ odd, i.e. $a=1,b=B,d=2A$. Then $\det(g)=2A=2{\mathrm{Nm}}(\mathfrak{a})$ and $\kappa=A/N$. Any $\lambda\in P^{\vee}/P$ can be written as $\lambda_{U}+\lambda_{U^{\vee}}$ with $\lambda_{U}\in{\tfrac{1}{2}}P_{U}/P_{U},\lambda_{U^{\vee}}\in{\tfrac{1}{2}}P_{U^{\vee}}/P_{U^{\vee}}$. Suppose $\lambda\in M/P$, then $\lambda_{U}\in P_{U}$ if and only if $\lambda_{U^{\vee}}\in P_{U^{\vee}}$. Therefore the map $\lambda\mapsto\lambda_{U}$ (resp. $\lambda\mapsto P_{U^{\vee}}$) induces an isomorphism from $M/P$ to ${\tfrac{1}{2}}P_{U}/P_{U}$ (resp. ${\tfrac{1}{2}}P_{U^{\vee}}/P_{U^{\vee}}$). For $\mu\in{\tfrac{1}{2}}P_{U}/P_{U}$, let $\mu^{\vee}\in{\tfrac{1}{2}}P_{U^{\vee}}/P_{U^{\vee}}$ be its image under the isomorphism above. Let $\varphi_{M}\in\mathcal{S}(V(\mathbb{A}_{\mathbf{f}}))$ be the characteristic function of $M$. Then it can be rewritten as

[TABLE]

The basis $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2},{\tilde{\mathfrak{e}}}_{1}^{\vee},{\tilde{\mathfrak{e}}}_{2}^{\vee}\}$ is given by

[TABLE]

We can identify ${\tfrac{1}{2}}P_{U}/P_{U}$, resp. ${\tfrac{1}{2}}P_{U^{\vee}}/P_{U^{\vee}}$, with $({\tfrac{1}{2}}\mathbb{Z}/\mathbb{Z})^{2}$, resp. $(\mathbb{Z}/2\mathbb{Z})^{2}$, via the bases $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2}\}$, resp. $\{{\tilde{\mathfrak{e}}}_{1}^{\vee},{\tilde{\mathfrak{e}}}_{2}^{\vee}\}$. Under this identification, we have $\mu^{\vee}=\binom{0}{0},\binom{1}{0},\binom{0}{1},\binom{1}{1}$ when $\mu=\binom{0}{0},\binom{0}{1/2},\binom{1/2}{0},\binom{1/2}{1/2}$ respectively.

2.5. Weil Representation and Theta Distribution.

We quickly recall the Weil representation following §4.1 in [8]. Recall that $\Gamma={\mathrm{SL}}_{2}(\mathbb{Z})$ and $\omega$ is the Weil representation of ${\mathrm{SL}}_{2}(\mathbb{A})$ acting on the space of Schwartz functions $\mathcal{S}(V(\mathbb{A}))$, which comes from the usual polarization $V\otimes W=V\otimes X+V\otimes Y$ for $W=X+Y$ a 2-dimensional symplectic vector space. On the other hand, the Witt decomposition $V=U+U^{\vee}$ gives another polarization $V\otimes W=U\otimes W+U^{\vee}\otimes W$, which provides another model of the Weil representation on $\mathcal{S}(U^{\vee}\otimes W(\mathbb{A}))$. For a choice of basis, we can identify $U^{\vee}(\mathbb{A})$ and $U(\mathbb{A})$ with $\mathbb{A}^{2}$ and view $\varphi\in\mathcal{S}(V(\mathbb{A}))$ as a function in $(\eta_{1},r)$ with $\eta_{1},r\in\mathbb{A}^{2}$. Let $\psi$ be the standard additive character on $\mathbb{A}/\mathbb{Q}$ that is trivial on ${\widehat{\mathbb{Z}}}$ and restricts to $x\mapsto{\mathbf{e}}(x)$ on $\mathbb{R}$. For $\varphi\in\mathcal{S}(V(\mathbb{A}))$, we can define its partial Fourier transform

[TABLE]

where $\eta=[\eta_{1}\;\eta_{2}]\in M_{2}(\mathbb{A})=\mathrm{Hom}(U^{\vee},W)(\mathbb{A})=U^{\vee}\otimes W(\mathbb{A})$. This defines an intertwining map from $\mathcal{S}(V(\mathbb{A}))$ to $\mathcal{S}(U^{\vee}\otimes W(\mathbb{A}))$ with respect to $\omega$, i.e.

[TABLE]

for $g\in{\mathrm{SL}}_{2}(\mathbb{A})$ and $\eta\in M_{2}(\mathbb{A})$. There is an equality of theta distribution

[TABLE]

with $\Gamma_{\eta}\subset\Gamma$ the stabilizer of $\eta\in M_{2}(\mathbb{Q})/\Gamma$.

Let $g_{\tau}=n(u)m(\sqrt{v})\in{\mathrm{SL}}_{2}(\mathbb{A})$ be the element corresponding to $\tau=u+iv\in{\mathcal{H}}$. It acts on the theta distribution and defines a distribution on $S(V(\mathbb{A}_{\mathbf{f}}))$

[TABLE]

where $\varphi_{\infty}\in S(V(\mathbb{R}))\otimes A^{0}({\mathcal{D}})$. For a function $f$ on $\Gamma\backslash{\mathcal{H}}$ valued in $\mathcal{S}(V(\mathbb{A}_{\mathbf{f}}))$ having mild growth near the cusp, the pairing $\langle f,\theta(\tau,w,\varphi_{\infty})\rangle$ is still $\Gamma$-invariant and can be integrated on $\Gamma\backslash{\mathcal{H}}$ to define

[TABLE]

When $f$ has exponential growth at the cusp, the theta integral above can be regularized and is a special case considered in [2] (also see §1 in [7]).

2.6. Cycle Integral of Theta Lift.

Suppose $M=M_{\mathfrak{a},\kappa}$ and $w=w(t,t_{0})$ as above. Then $\Phi(w;\varphi_{\infty},f)$ is also invariant when $\varepsilon\in\Gamma_{\kappa}$ acts on $t\in\mathbb{R}$ via translation by $\log\varepsilon$. Integrating it with respect to the invariant measure ${dt}$ over a fundamental domain $[0,\log\varepsilon_{\kappa})$ of $\Gamma_{\kappa}\backslash{\mathcal{D}}_{F}^{+}$ defines

[TABLE]

Let $\varphi_{M}\in\mathcal{S}(V(\mathbb{A}_{\mathbf{f}}))$ be the characteristic function of $M\otimes\hat{\mathbb{Z}}$ and

[TABLE]

In this case, we denote

[TABLE]

Notice that $Q({\lambda}_{w^{\perp}})\tau+Q({\lambda}_{w})\overline{\tau}=Q({\lambda})u+Q_{w}({\lambda})iv$ with $\tau=u+iv$. Then

[TABLE]

The first result concerning this function of $t_{0}$ is as follows.

Proposition 2.2**.**

The function $I(t_{0};\phi,\varphi_{M_{\mathfrak{a},\kappa}})$ is a constant that only depends on the class of $\mathfrak{a}$ in the narrow class group ${\mathrm{Cl}}^{+}(F)$ of $F$ and the positive integer $\kappa$.

Proof.

Since $I(t_{0};\phi,\varphi_{M})$ is differentiable in $t_{0}$, it suffices to show that $\partial_{t_{0}}I(t_{0})=0$. To see this, consider the following elements in $\mathcal{S}(V(\mathbb{R}))$

[TABLE]

Notice that

[TABLE]

From this, it is easy to check that $\partial_{t_{0}}Q_{w}({\lambda})=({\lambda},Y(t_{0}))({\lambda},Y^{\perp}(t_{0}))$. Therefore, we have

[TABLE]

which implies $v^{2}\partial_{t_{0}}\Theta_{M}(\tau,t,t_{0};\phi)d\mu(\tau)dt=d\omega$ with

[TABLE]

It is clear that $\omega$ is a $(\Gamma\times\Gamma_{\kappa})$-invariant 2-form on ${\mathcal{H}}\times{\mathcal{D}}_{F}^{+}$. Applying Stokes’ Theorem then proves that $\partial_{t_{0}}I(t_{0})=0$. It is now easy to check from the definition that $\Theta_{M_{\mathfrak{a},\kappa}}=\Theta_{M_{\mathfrak{a}(\mu),\kappa}}$ for a totally positive element $\mu\in\mathcal{O}_{F}$. Therefore, $I(t_{0};\phi,\varphi_{M_{\mathfrak{a},\kappa}})$ only depends on the class of $\mathfrak{a}$ in ${\mathrm{Cl}}^{+}(\mathcal{O}_{F})$ and $\kappa$. ∎

Remark 2.3*.*

For ${\mathcal{A}}\in{\mathrm{Cl}}^{+}(\mathcal{O}_{F})$ and $\kappa\in\mathbb{N}$, the proposition above implies that the quantity

[TABLE]

is well-defined with $\mathfrak{a}$ any representative of ${\mathcal{A}}$. Here we denote $\Phi(t;\mathfrak{a},\kappa):=\Phi(w(t,0);\phi,\varphi_{M_{\mathfrak{a},\kappa}})$.

From (2.9), we know that when restricting to $t_{0}=0$, the image of $D_{F}^{+}\times\{0\}$ lies on the twisted diagonal $z_{1}=D\kappa z_{2}/{\mathrm{Nm}}(\mathfrak{a})$ in $\Gamma_{M}^{+}\backslash{\mathcal{H}}^{2}$. On this embedded ${\mathcal{H}}$, the image of $\Gamma_{\kappa}\backslash{\mathcal{D}}^{+}_{F}$ is the geodesic connecting $\sqrt{D}i$ and $\sqrt{D}\frac{-\beta\sqrt{D}+i}{\alpha}$ if $\varepsilon_{\kappa}=\alpha+\beta\sqrt{D}$. Therefore, $I(0;\phi,\varphi_{M})$ can be considered as the cycle integral of the (twisted) diagonal restriction of $\Phi(w;\phi,\varphi_{M})$.

The second result of this section relates the Petersson norm of $\vartheta_{L}$ to the quantity ${\tilde{\Psi}}({\mathcal{A}},\kappa)$.

Proposition 2.4**.**

For an integral ideal $\mathfrak{a}\subset\mathcal{O}_{F}$ and positive integer $\kappa$, let $L=L_{\mathfrak{a},\kappa},\varepsilon_{\kappa}\in\mathcal{O}_{F}^{\times}$ and $\vartheta_{L}\in S_{1}(\Gamma,\rho_{L})\otimes\mathcal{S}_{L}$ be as in §2.3. Then $\|\vartheta_{L}(\tau)\|^{2}_{\mathrm{Pet}}={\tilde{\Psi}}([\mathfrak{a}\mathfrak{d}],\kappa)\log\varepsilon_{\kappa}$.

Proof.

If we unfold the definition of $\vartheta_{L}$ and its Petersson norm in Eqs. (2.16) and (1.1), then it is clear that

[TABLE]

where $\phi_{2}$ is defined in the proof of Prop. 2.2. The proposition now follows from the fact $\Theta_{M_{\mathfrak{a},\kappa}}(\tau,t,t_{0};\phi_{2})=\Theta_{M_{\mathfrak{a}\mathfrak{d},\kappa}}(\tau,t,t_{0};\phi)$ and Prop. 2.2. ∎

2.7. Rademacher symbol.

For each $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma={\mathrm{SL}}_{2}(\mathbb{Z})$, Rademacher defined in [11] the following function

[TABLE]

where $s(h,k):=\sum_{\mu\bmod k}(\!(\frac{\mu}{k})\!)(\!(\frac{\mu h}{k})\!)$ is the Dedekind sum with $(\!(x)\!):=x-\lfloor x\rfloor-{\tfrac{1}{2}}$ if $x\in\mathbb{R}\backslash\mathbb{Z}$ and zero otherwise. He showed that $\Psi$ is integer-valued, invariant under conjugation by elements in $\Gamma$ and

[TABLE]

for any $g\in{\mathrm{GL}}_{2}(\mathbb{Z})$. We call $\Psi$ the Rademacher symbol. It is closely related to the transformation formula of the Dedekind eta function and appears in many places in mathematics (see e.g. [1, 13]).

An element $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma$ is called hyperbolic if $|\mathrm{tr}(\gamma)|>2$. It fixes the semi-circle $C_{\gamma}:=\{z\in{\mathcal{H}}:|2cz-(a-d)|^{2}=(a+d)^{2}-4\}$. Define the weight 2 real-analytic Eisenstein series $E^{*}_{2}(z)$ by

[TABLE]

A Theorem of Meyer [10] connects the cycle integral of $E^{*}_{2}(z)dz$ with the Rademacher symbol.

Theorem 2.5** (see §III of [13]).**

Let $\gamma\in\Gamma$ be a hyperbolic element and $z_{0}\in C_{\gamma}$ an arbitrary point. Then the integral $\int^{\gamma z_{0}}_{z_{0}}E^{*}_{2}(z)dz$ is independent of $z_{0}$ and equals to $\Psi(\gamma)$.

Let $F=\mathbb{Q}(\sqrt{D})$ and $\varepsilon_{\kappa}=\alpha+\beta\sqrt{D}\in\mathcal{O}_{F}^{\times}$ as in §2.3. Then

[TABLE]

is a hyperbolic element, and $C_{\gamma}=\{z\in{\mathcal{H}}:|z|=\sqrt{D}\}$. Recall $z(t):=\sqrt{D}\frac{-\mathrm{sh}(t)+i}{\mathrm{ch}(t)}$ from §2.2, which is an isomorphism from $\mathbb{R}$ to $C_{\gamma}$. It is straightforward to check that

[TABLE]

Therefore, Meyer’s theorem implies that

[TABLE]

3. Calculations

3.1. Partial Fourier Transform.

Fix a class ${\mathcal{A}}\in{\mathrm{Cl}}^{+}(\mathcal{O}_{F})$, a representative $\mathfrak{a}\subset\mathcal{O}_{F}$ of ${\mathcal{A}}$, a positive integer $\kappa\in\mathbb{N}$ and denote $M=M_{\mathfrak{a},\kappa}$. In order to evaluate ${\tilde{\Psi}}({\mathcal{A}},\kappa)$, we will calculate the Fourier expansion of $\Phi(w;\phi,\varphi_{M})$ along the 1-dimensional boundary coming from the stabilizer of $U$. This was done in [8] with $\varphi_{\infty}$ being the Gaussian. We follow the same calculations with $\varphi_{\infty}$ a polynomial times a Gaussian. The key step in the evaluation is a 2-dimensional partial Fourier transform. For this, we need the analogue of Lemma 4.3 in [8] for $\varphi_{\tau,w}\in\mathcal{S}(V(\mathbb{R}))$ defined in Eq. (2.28). The spaces $U(\mathbb{R})$ and $U^{\vee}(\mathbb{R})$ are both isomorphic to $\mathbb{R}^{2}$ with respect to a choice of basis. Therefore, we view $\varphi_{\tau,w}$ as a function in $(\eta_{1},r)$ with $\eta_{1},r$ column vectors in $\mathbb{R}^{2}$. The goal is to calculate the partial Fourier transform of $\varphi_{\tau,w}$ defined by

[TABLE]

where $\eta=[\eta_{1}\;\eta_{2}]\in M_{2}(\mathbb{R})$. This is the infinity component of the partial Fourier transform in (2.22).

For the bases of $U$ and $U^{\vee}$, we will use $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2}\}$ and $\{{\tilde{\mathfrak{e}}}^{\vee}_{1},{\tilde{\mathfrak{e}}}^{\vee}_{2}\}$ in §2.2 with $g=g_{L}$ given in Eq. (2.18). Denote $J:=\left(\begin{smallmatrix}&1\\ -1&\end{smallmatrix}\right)$, which satisfies $J=-\prescript{t}{}{J}$ and

[TABLE]

Recall $z(t),z_{1}(t,t_{0}),z_{2}(t,t_{0}),{\tilde{z}}_{1}(t),{\tilde{z}}_{2}(t)$ as in (2.8), (2.9) and (2.10), which satisfy

[TABLE]

After setting $t_{0}=0$ and omitting the argument $t$ in all the notations above, we can express $({\lambda},w),({\lambda},{w^{\perp}})$ as

[TABLE]

Then $\varphi_{\tau,w}({\lambda})$ becomes

[TABLE]

where $r^{\prime}:=r-Cz_{2}$ with $R=R(z_{1},g):=g\binom{z_{1}}{1}$. In the notations above, we have the following lemma.

Lemma 3.1**.**

For $t_{0}=0$ and $\eta\in M_{2}(\mathbb{Q})$ with columns $\eta_{1}$ and $\eta_{2}$, the function $\widehat{\varphi_{\tau,w}}(\eta)$ is given by

[TABLE]

Proposition 3.2**.**

The theta function $\Theta(\tau,t,t_{0};\phi)$ can be written as

[TABLE]

This result follows from changing the model of the Weil representation (see §4.1 in [8] for detailed discussions). Recall the lattice $P\subset M$ defined in (2.17) with $M/P\cong L/(L\cap 2L^{\vee})$. By (4.10) in [8] and Example (2.1) above, we have for $\eta=[\eta_{1}\;\eta_{2}]\in M_{2}(\mathbb{Q})\cong U^{\vee}\times W$

[TABLE]

3.2. Orbital Integrals.

In this section, we will decompose the integral defining $\Phi(w(t,0);\phi,\varphi_{M})$ into orbits according to the rank of $\eta\in M_{2}(\mathbb{Q})$ and calculate the orbital integral

[TABLE]

for various $\eta\in M_{2}(\mathbb{Q})/\Gamma$. Here, we used the notation $z=x+iy=z(t)$ as in Eq. (2.10). The procedure now follows that of [8].

Lemma 3.3**.**

Suppose $g=\left(\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right)\in{\mathrm{GL}}_{2}^{+}(\mathbb{Q})$. Let $\eta=\left(\begin{smallmatrix}m&k\\ 0&\alpha\end{smallmatrix}\right)\in M_{2}(\mathbb{Q})$ with $m>0$ and $\alpha\neq 0$. Then

[TABLE]

Proof.

Since $\eta$ has rank 2, $\Gamma_{\eta}$ is trivial and $\Gamma_{\eta}\backslash{\mathcal{H}}={\mathcal{H}}$. Substituting in $\eta=\left(\begin{smallmatrix}m&k\\ 0&\alpha\end{smallmatrix}\right)$ gives us $\eta_{\tau}=\binom{m\tau+k}{\alpha}$ and

[TABLE]

Make the change of variable $u^{\prime}=d(mu+k)-\alpha(ax+b)$ and $v^{\prime}=dmv$ gives us

[TABLE]

Therefore, the integral defining $\Phi_{\eta}(z,0)$ becomes

[TABLE]

We can apply the identities $\int_{\mathbb{R}}\left(\tfrac{1}{2\pi A}-Bu^{2}\right)\exp(-\pi ABu^{2})du=0,\int_{\mathbb{R}}\exp(-\pi Au^{2})du=1/\sqrt{A}$ and $\int^{\infty}_{0}e^{-Aw-B/w}dw/\sqrt{w}=\sqrt{\pi/A}e^{-2\sqrt{AB}}$ to simplify the expression above to

[TABLE]

This finishes the proof. ∎

Lemma 3.4**.**

Suppose $g=\left(\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right)$ and ${\tilde{z}}=\tilde{x}+i\tilde{y}:=gz=(az+b)/d$. When $\eta=\left(\begin{smallmatrix}0&m\\ 0&n\end{smallmatrix}\right)\in M_{2}(\mathbb{Q})$, the orbital integral $\Phi_{\eta}(z,s)$ is given by

[TABLE]

for $s\in\mathbb{C}$.

Proof.

In this case, we have $\tilde{\eta}_{\tau}=\frac{1}{ad}\binom{dm-bn}{an}$ and

[TABLE]

Integrating this against $v^{2-s}\frac{dudv}{v^{2}}$ over $\Gamma_{\eta}\backslash{\mathcal{H}}=\Gamma_{\infty}\backslash{\mathcal{H}}$ gives us the desired result. ∎

Proposition 3.5**.**

When $\eta=\left(\begin{smallmatrix}0&0\\ 0&0\end{smallmatrix}\right)$, we have $\Gamma_{\eta}\backslash{\mathcal{H}}=\Gamma\backslash{\mathcal{H}}$ and

[TABLE]

Finally, we will state a lemma that follows readily from differentiating the Kronecker limit formula,

[TABLE]

where $\sideset{}{{}^{\prime}}{\sum}$ means summing over $(m,n)\in\mathbb{Z}^{2}\backslash\{(0,0)\}$, $\gamma$ is the Euler constant and $\eta$ is the Dedekind eta function.

Lemma 3.6**.**

For $w\in{\mathcal{H}}$ and $s$ near $0\in\mathbb{C}$, we have

[TABLE]

where $E^{*}_{2}(w)$ is the real-analytic Eisenstein series defined in Eq. (2.33).

4. Proof.

After calculating the orbital integrals, we can now add all the contributions together to calculate ${\tilde{\Psi}}({\mathcal{A}},\kappa)$. For this, we need the following key proposition.

Proposition 4.1**.**

Let $M=M_{\mathfrak{a},\kappa}$, $g_{0}=g_{\kappa},g_{1}=g_{L}$ and $z(t)=\sqrt{D}\frac{-\mathrm{sh}(t)+i}{\mathrm{ch}(t)}$ as in sections 2.2 and 2.4. Then

[TABLE]

where ${\tilde{z}}_{j}(t)=g_{j}\cdot z(t)$ for $j=0,1$.

Proof.

First, we can write $\Phi(t;\mathfrak{a},\kappa)=\lim_{s\to 0}\int_{\Gamma\backslash{\mathcal{H}}}v^{2-s}\Theta_{M}(\tau,t,0;\phi)\frac{dudv}{v^{2}}$ using the definition of $\Phi(t;\mathfrak{a},\kappa)$ (see Remark 2.3). By the absolute convergence of this integral for $\mathrm{Re}(s)>0$ and Eq. (3.5), we can write

[TABLE]

where $\Gamma_{\mathcal{H}}=\{\pm\left(\begin{smallmatrix}1&0\\ 0&1\end{smallmatrix}\right)\}\subset\Gamma$. Suppose $2\mid D\kappa$ and we choose $g=g_{L}$ such that $\{{\tilde{\mathfrak{e}}}_{1},{\tilde{\mathfrak{e}}}_{2}\}$ is a $\mathbb{Z}$-basis of $P_{U}$ defined in Eq. (2.17). Then Eqs. (2.19) and (3.6) tell us that $N/\det(g)=2/\kappa$ and $\widehat{\varphi_{M}}(\eta)=\varphi_{M_{2}({\widehat{\mathbb{Z}}})}(\eta)$. In this case, we have $M_{2}(\mathbb{Z})/\Gamma=S_{2}\sqcup S_{1}\sqcup\{\left(\begin{smallmatrix}0&0\\ 0&0\end{smallmatrix}\right)\}$, where

[TABLE]

It is easy to see that $\frac{|\Gamma_{\mathcal{H}}|}{|\Gamma_{\mathcal{H}}\cap\Gamma_{\eta}|}=2$ if $\eta\in S_{2}$, and is 1 otherwise. By Eq. (2.12) and Lemma 3.3, we have

[TABLE]

Similarly for the sum over $S_{1}$, we can apply Lemma 3.3 and 3.6 to obtain

[TABLE]

Finally, we can rewrite $\Phi_{\left(\begin{smallmatrix}0&0\\ 0&0\end{smallmatrix}\right)}(z,0)=\frac{d{{\tilde{z}}_{0}}+d\overline{{\tilde{z}}_{0}}}{12}$. Adding everything together with the index $\frac{|\Gamma_{\mathcal{H}}|}{|\Gamma_{\mathcal{H}}\cap\Gamma_{\eta}|}$ then finishes the proof for $2\mid D\kappa$. If $2\nmid D\kappa$, then $\widehat{\varphi_{M}}(\eta)=\varphi_{M_{2}(\mathbb{Z})}(\eta){\mathbf{e}}(-\det(\eta)/2)$ for $\eta\in M_{2}(\mathbb{Q})$ by Eq. (3.6). Calculating exactly as before, we obtain Eq. (4.1) in this case. ∎

To prove Theorem 1.1, we will also need the following proposition.

Proposition 4.2**.**

Recall $\gamma=\gamma_{D,\kappa}\in\Gamma$ from Eq. (2.34) and let $g_{0},g_{1}\in{\mathrm{GL}}^{+}_{2}(\mathbb{Q})$ be the same as in Prop. 4.1. Denote $\gamma_{j}:=g_{j}\gamma g_{j}^{-1}$ for $j=0,1$. Then $\gamma_{j}\in\Gamma$ for $j=0,1$ and

[TABLE]

where ${\tilde{\Psi}}$ is the invariant defined in Eq. (2.30) and $\Psi$ is the Rademacher symbol defined in §2.7.

Proof.

From Prop. 4.1 and the definition of ${\tilde{\Psi}}({\mathcal{A}},\kappa)$ in Eq. (2.30), it is clear that

[TABLE]

where ${\tilde{z}}_{0}(t),{\tilde{z}}_{1}(t)$ were defined in Prop. 4.1. By Eq. (2.35), we know that $\gamma_{j}\cdot{\tilde{z}}_{j}(\log\varepsilon_{\kappa})={\tilde{z}}_{j}(-\log\varepsilon_{\kappa})$ and $\gamma_{j}\cdot\left(-\overline{{\tilde{z}}_{j}(\log\varepsilon_{\kappa})}\right)=-\overline{{\tilde{z}}_{j}(-\log\varepsilon_{\kappa})}$ for $j=0,1$. For any $g=\left(\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right)\in{\mathrm{GL}}_{2}^{+}(\mathbb{Q})$, it is easy to check that

[TABLE]

One can check that $\gamma_{j}\in\Gamma$ after substituting in $g=g_{j}$ for $j=0,1$. Therefore, we will denote ${\tilde{z}}_{j}={\tilde{z}}_{j}(\log\varepsilon_{\kappa})$ and can write for $j=0,1$

[TABLE]

where $w^{\prime}=-\overline{w}$. Since $-\gamma_{j}\overline{{\tilde{z}}_{j}}=\gamma_{j}^{\prime}(-\overline{{\tilde{z}}_{j}})$ with $\gamma^{\prime}_{j}=\left(\begin{smallmatrix}1&\\ &-1\end{smallmatrix}\right)\gamma_{j}\left(\begin{smallmatrix}1&\\ &-1\end{smallmatrix}\right)$, Meyer’s Theorem implies that the last integral is $\Psi(\gamma^{\prime}_{j})=-\Psi(\gamma_{j})$ and this finishes the proof. ∎

Proof of Theorem 1.1.

In view of Props. 2.4, 4.1 and 4.2, it suffices to calculate ${\tilde{\Psi}}([\mathfrak{a}\mathfrak{d}],\kappa)$. From the proposition above, we see that $\gamma_{0}$ only depends on $\kappa$. For $\gamma_{1}=\gamma_{1}(\mathfrak{a}\mathfrak{d})$, if $\{a\sqrt{D}+b,d\}$ is a basis of $\mathfrak{a}\cap 2(\kappa\mathfrak{d})^{-1}\mathfrak{a}$ as in §2.4, then $\{a{D}+b\sqrt{D},d\sqrt{D}\}$ is a basis of $\mathfrak{d}\mathfrak{a}\cap 2\kappa^{-1}\mathfrak{a}$. Let $r=\gcd(b,d)$, $b^{\prime}=b/r,d^{\prime}=d/r$ and $g_{2}=\left(\begin{smallmatrix}a^{\prime}&c^{\prime}\\ d^{\prime}&-b^{\prime}\end{smallmatrix}\right)\in{\mathrm{GL}}_{2}(\mathbb{Z})$ with $\det(g_{2})=-1$. Then $\{r\sqrt{D}+aa^{\prime}D,ad^{\prime}D\}$ is a basis of $\mathfrak{a}\mathfrak{d}\cap 2\kappa^{-1}\mathfrak{a}$ and we choose $g_{1}=\left(\begin{smallmatrix}r&aa^{\prime}D\\ &ad^{\prime}D\end{smallmatrix}\right)$. It is straightforward to check that

[TABLE]

Since $\left(\begin{smallmatrix}&D\\ 1&\end{smallmatrix}\right)\gamma\left(\begin{smallmatrix}&1\\ 1/D&\end{smallmatrix}\right)=\gamma$, we have $\Psi(\gamma_{1}(\mathfrak{a}\mathfrak{d}))=\Psi(g_{2}\gamma_{1}(\mathfrak{a})g_{2}^{-1})=-\Psi(\gamma_{1}(\mathfrak{a}))$. This finishes the proof. ∎