2.1. Setup

Let B be a quaternion algebra over $\mathbb {Q}$ . We denote by $d_B$ its reduced discriminant, or equivalently, the product of the primes at which B ramifies. We write G for the linear algebraic group over $\mathbb {Q}$ given by for any $\mathbb {Q}$ -algebra L. Then G is an inner form of $\operatorname {PGL}_2$ , and all rational forms of $\operatorname {PGL}_2$ arise in this way. Denote by $[G]$ the adelic quotient $G(\mathbb {Q}) \backslash G(\mathbb {A})$ . We fix the probability Haar measure on $[G]$ . Let $K_\infty $ be a compact maximal torus of $G(\mathbb {R})$ . We assume that $K_\infty $ comes equipped with a choice of isomorphism $\kappa : \mathbb {R} / \pi \mathbb {Z} \xrightarrow {\sim } K_\infty $ . In the split case $B=\operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Q})$ , we identify $G=\operatorname {PGL}_2$ and set $\kappa (\theta )=\left (\begin {smallmatrix} \cos (\theta ) & \sin (\theta ) \\ -\sin (\theta ) & \cos (\theta ) \end {smallmatrix}\right )$ .

Let R be an Eichler order in B, that is, an intersection of two maximal orders. We denote by N the level of R. It is a natural number, coprime to $d_B$ , characterized as follows: For each prime $p \nmid d_B$ , there is an isomorphism under which maps to the order $\left ( \begin {smallmatrix} \mathbb {Z}_p &\mathbb {Z}_p \\ N \mathbb {Z}_p& \mathbb {Z}_p \end {smallmatrix} \right )$ . We may then identify $G(\mathbb {Q}_p)$ with $\operatorname {PGL}_2(\mathbb {Q}_p)$ and the image of $R_p^\times $ with a finite index subgroup of $\operatorname {PGL}_2(\mathbb {Z}_p)$ . We assume that N is squarefree so that $d_B N$ is likewise squarefree. We denote by $K_R$ the compact open subgroup of $G(\mathbb {A}_f)=\prod _p ' G(\mathbb {Q}_p)$ given by the image of $\prod _p R_p^\times $ .

Fix $k \in 2\mathbb {Z}$ . Let $\mathcal {A}$ denote the set of cusp forms $\varphi : [G] \rightarrow \mathbb {C}$ having the following properties:

• ∘ $\varphi (g \kappa (\theta )) = e^{i k \theta } \varphi (g)$ for all $\theta $ .

• ∘ $\varphi $ is an eigenfunction for some fixed Casimir operator for $G(\mathbb {R})$ , with eigenvalue $\lambda _{\varphi }$ . For the sake of concreteness, we scale the Casimir operator such that it agrees with the standard Laplace operator on the locally symmetric space $G(\mathbb {R}) / K_{\infty }$ , which identifies with either $\mathbb {H}$ or $S^2$ .

• ∘ $\varphi $ is $K_R$ -invariant: $\varphi (g k) = \varphi (g)$ for $k \in K_R$ .

• ∘ $\varphi $ belongs to the newspace for R, that is, $K_R$ is the largest subgroup of $G(\mathbb {A}_f)$ keeping $\varphi $ invariant. Equivalently, $\varphi $ is orthogonal the space of $K_{R'}$ -invariant cusp forms for every Eichler order $R'$ strictly containing R.

• ∘ $\varphi $ is an eigenform for almost all Hecke operators.

If $k \ge 2$ , then we write $\mathcal {A}^{\operatorname {hol}} \subseteq \mathcal {A}$ for the subspace of automorphic lifts of holomorphic forms or, equivalently, the kernel of the raising (resp. lowering) operator attached to $K_\infty $ if B is definite (resp. indefinite).

Denote by $\mathcal {F}$ a maximal orthonormal subset of $\mathcal {A}$ . Analogously, we define $\mathcal {F}^{\operatorname {hol}} \subseteq \mathcal {A}^{\operatorname {hol}}$ if $k \ge 2$ . Because of the multiplicity-one theorem for $\operatorname {GL}_2$ and its inner forms, the bases $\mathcal {F}, \mathcal {F}^{\operatorname {hol}}$ are unique up to rescaling each element by a scalar of unit magnitude. We note that the sets $\mathcal {A}$ , $\mathcal {A}^{\operatorname {hol}}$ , $\mathcal {F}$ and $\mathcal {F}^{\operatorname {hol}}$ depend on k; while we suppress this dependence from the notation, k is one of the main parameters of interest.

We will consider several subfamilies of $\mathcal {F}$ and $\mathcal {F}^{\operatorname {hol}}$ . Here, a minus sign in the exponent signifies the indefinite case, a plus sign the definite case.

• ∘ If B is indefinite and $k=0$ , then we take and let $\mathcal {F}^{-}_{\lambda }$ (resp. $\mathcal {F}^{-}_{\le L}$ ) denote the subsets defined by taking the Casimir eigenvalue equal to $- \lambda $ (respectively at most L in magnitude).

• ∘ If B is indefinite and $k \ge 2$ , then we take .

• ∘ If B is definite and $k=0$ , then we let $\mathcal {F}^{+}_m \subset \mathcal {F}$ be the subset of forms, whose associated automorphic representation at infinity is isomorphic to the unique irreducible unitary representation of $\operatorname {\mathrm {SU}}_2(\mathbb {C})$ of degree $m+1$ . In other words, their eigenvalue with respect to the Casimir operator equals to $-m(m+1)$ .

• ∘ If B is definite and $k \ge 2$ , then we let $\mathcal {F}^{+, \operatorname {hol}}=\mathcal {F}^{\operatorname {hol}}$ .

2.2. The split case

Assume for the moment that B is split. We may suppose then that

(2.1) $$ \begin{align} B = \operatorname{\mathrm{Mat}}_{2 \times 2}(\mathbb{Q}), \quad G = \operatorname{PGL}_2, \quad R = \begin{pmatrix} \mathbb{Z} & \mathbb{Z} \\ N \mathbb{Z} & \mathbb{Z} \end{pmatrix}, \end{align} $$

(2.2) $$ \begin{align} K_\infty = \operatorname{PSO}_2(\mathbb{R}), \quad \kappa(\theta) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \end{align} $$

and may identify

$$\begin{align*}[G] / K_\infty K_R \cong \Gamma_0(N) \backslash \mathbb{H}. \end{align*}$$

We define

$$\begin{align*}H : [G] / K_\infty K_R \rightarrow \mathbb{R}_{>0}, \end{align*}$$

as follows. Let $A_0(N)<\operatorname {GL}_2(\mathbb {Q})^+$ denote the group generated by $\Gamma _0(N)$ and all Atkin–Lehner operators. If $g \in [G] / K_\infty K_R$ identifies with $z \in \Gamma _0(N) \backslash \mathbb {H}$ , then we set

Since the Atkin–Lehner operators constitute scaling matrices for the various cusps of $\Gamma _0(N)$ (cf. §4.3.1), the function H may be understood as a normalized height or as quantifying closeness to the cusps. Let $\mathfrak {a} \in P^{1}(\mathbb {Z})$ be a cusp of $\Gamma _0(N)$ , and let $\sigma _{\mathfrak {a}}\in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ such that $\sigma _{\mathfrak {a}} \infty = \mathfrak {a}$ . Then,

(2.3) $$ \begin{align} H(z) = \max_{\mathfrak{a}} \frac{\Im(z_{\mathfrak{a}})}{w_{\mathfrak{a}}}, \end{align} $$

where $\mathfrak {a}$ runs over all cusps of $\Gamma _0(N)$ , $z_{\mathfrak {a}}= \sigma _{\mathfrak {a}}^{-1} z$ and $w_{\mathfrak {a}}$ is the cusp width of $\mathfrak {a}$ .

2.3. Results on forms

We adopt the following asymptotic notation $\preccurlyeq $ :

$$ \begin{align*}A_1 \preccurlyeq A_2 \quad \iff \quad A_1 \ll_{\varepsilon} \left(d_BN(1+k)(1+\mu)\right)^{\varepsilon} A_2, \end{align*} $$

where $\mu $ is a quantity relating to the eigenvalues with respect to the Casimir operator of the automorphic forms of relevance to the inequality. Concretely, when talking about the families $\mathcal {F}^{-}_{\lambda },\mathcal {F}^{-}_{\le L}, \mathcal {F}^{+}_m, \mathcal {F}^{\pm ,\operatorname {hol}}$ we mean $\mu =|\lambda |,L,m, k$ , respectively.

Theorem 2.1. Let $g_1,g_2 \in [G]$ . If B is indefinite, then

(2.4) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{-}_{\le L} } (|\varphi(g_1)|^2-|\varphi(g_2)|^2)^2 \preccurlyeq_{L} d_B N \left( 1+ d_BN \left[H(g_1)^2+H(g_2)^2 \right] \right), \end{align} $$

for $L> 0$ , and

(2.5) $$ \begin{align} \nonumber & \sum _{\varphi \in \mathcal{F}^{-,\operatorname{hol}} } (|\varphi(g_1)|^2-|\varphi(g_2)|^2)^2 \\ &\qquad\qquad\quad\qquad \preccurlyeq d_B N k \left( 1+(d_BN)^{\frac{1}{2}} \left[H(g_1)+H(g_2)\right]+ d_B N k^{-\frac{1}{2}} \left[H(g_1)^2+H(g_2)^2 \right] \right), \end{align} $$

for $k \ge 2$ even. In both cases, the term involving $H(g_{1,2})$ is only present if B is split.

If B is definite, then

(2.6) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{+}_{m} } (|\varphi(g_1)|^2-|\varphi(g_2)|^2)^2 \preccurlyeq d_B N(m+1)^2, \end{align} $$

for $m \in \mathbb {N}_0$ , and

(2.7) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{+,\operatorname{hol}} } |\varphi(g_1)|^4 \preccurlyeq d_B N k, \end{align} $$

for $k \in 2 \mathbb {N}$ .

Corollary 2.3. For $k \ge 2$ and $\varphi \in \mathcal {F}^{\operatorname {hol}}$ , we have

$$ \begin{align*} \| \varphi\|_{\infty} \preccurlyeq (d_BN k)^{\frac{1}{4}}. \end{align*} $$

For $k=0$ and $\varphi \in \mathcal {F}$ , we have

$$ \begin{align*} \| \varphi \|_{\infty} \preccurlyeq_{\lambda_{\varphi}} (d_BN)^{\frac{1}{4}}. \end{align*} $$

If B is definite, then we have more precisely

$$ \begin{align*} \| \varphi \|_{\infty} \preccurlyeq (d_BN)^{\frac{1}{4}} (1+|\lambda_{\varphi}|)^{\frac{1}{4}}. \end{align*} $$

By a well-known procedure, these statements may be translated into the classical language, thus giving rise to the theorems in the introduction. For further details; see, for example, [Reference BumpBum97, §3.2 & §3.6] for the indefinite case and [Reference Blomer and MichelBM13] for the definite case.

2.4. Counting problems: setup

2.4.3. Coordinates tailored to $K_\infty $

Define . If B is indefinite, then $B_\infty \cong \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {R})$ is split; otherwise, $B_\infty $ is isomorphic to the real Hamilton quaternions. The exponential series identifies $B_\infty ^0$ with the Lie algebra of $G(\mathbb {R})$ . We write ${\mathbf{i}} \in B_\infty ^0$ for the derivative at the identity of $\kappa $ so that $\kappa (\theta ) = \exp ( \theta {\mathbf{i}} )$ . Then, ${\mathbf{i}} ^2 = -1$ . We may find ${\mathbf{j}} \in B_\infty ^0$ with ${\mathbf{j}} ^2 = \pm 1$ ( $+1$ if B is indefinite, $-1$ if B is definite) so that $B_\infty = \mathbb {R}({\mathbf{i}} ) \oplus \mathbb {R}({\mathbf{i}} ) {\mathbf{j}} $ . We note that ${\mathbf{j}} $ is not uniquely determined, but any two choices differ by multiplication by a norm one element of $\mathbb {R}({\mathbf{i}} )$ . We set ${\mathbf{k}} = {\mathbf{i}} {\mathbf{j}} $ . Then, ${\mathbf{i}} ,{\mathbf{j}} ,{\mathbf{k}} $ give an $\mathbb {R}$ -basis of $B_\infty ^0$ . For real numbers $a,b,c$ , we set . A general element of $B_\infty $ may then be written $[a,b,c] + d$ , where we identify the real number d with a scalar element of $B_\infty $ . In these coordinates,

(2.8) $$ \begin{align} \operatorname{\mathrm{tr}}([a,b,c]+d) = 2 d, \quad \det([a,b,c]+d) = a^2 \mp (b^2 + c^2) + d^2. \end{align} $$

Example 2.4. Suppose that $B_\infty = \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {R})$ and that $\kappa $ is as in Equation (2.2). Then, with suitable choices,

$$\begin{align*}[a,b,c] + d = \begin{pmatrix} d + c & b + a \\ b- a & d-c \end{pmatrix}. \end{align*}$$

2.4.4. Archimedean regions

For $T> 0$ and $\delta \in (0,1]$ , we denote by $\Omega (\delta ,T)$ the set of all elements $[a,b,c] + d$ of $B_\infty $ for which

$$\begin{align*}a^2 + b^2 + c^2 + d^2 \le T^2, \quad b^2 + c^2 \le \delta T^2. \end{align*}$$

With $\Omega ^{\star }(\delta ,T)$ , we denote the subset of nonzero elements of $\Omega (\delta ,T)$ . Likewise, for $T>0$ and $\delta \in (0,1]$ , we let $\Psi (\delta ,T)$ denote the set of all elements $[a,b,c]+d$ of $B_{\infty }$ for which

$$\begin{align*}a^2 + b^2 + c^2 + d^2 \le T^2, \quad a^2 + d^2 \le \delta T^2, \end{align*}$$

and $\Psi ^{\star }(\delta ,T)$ its subset consisting of nonzero elements.

2.5. Counting problems: results

We adopt the following asymptotic notation for counting estimates (compare with the notation $\preccurlyeq $ introduced in §2.3):

$$\begin{align*}A_1 \prec A_2 \quad \iff \quad A_1 \ll_\varepsilon (d_B N (1+T))^\varepsilon A_2. \end{align*}$$

Recall from §2.2 the height function H defined in the split case. In the nonsplit case, we adopt the convention in the following results that any terms involving H (in minima or sums) should be omitted.

Theorem 2.5 (Type I estimates).

Let $g \in G(\mathbb {R})$ . Then, the first successive minima (see Definition 6.1) of $g^{-1} R(\ell )^0 g$ with respect to $\Omega (\delta , 1)\cap B_{\infty }^0$ is $\gg \min \left \{ \ell ^{-\frac {1}{2}} , \ell ^{-1} \delta ^{-\frac {1}{2}} H(g)^{-1} \right \}$ . Furthermore, we have

$$\begin{align*}|g^{-1} R(\ell)^0 g \cap \Omega(\delta, T)| \prec 1 + \left(\ell^{\frac{1}{2}} +\ell \delta^{\frac{1}{2}} H(g) \right) T + \left( \frac{\ell^{\frac{3}{2}} \delta^{\frac{1}{2}} }{(d_B N)^{\frac{1}{2}}} + \ell \delta H(g) \right) T^2 + \frac{\ell^2 \delta}{d_B N} T^{3}. \end{align*}$$

If B is nonsplit, we further have that the first successive minima of $g^{-1} R(\ell )^0 g$ with respect to $\Psi (\delta , 1)\cap B_{\infty }^0$ is at least $\gg \ell ^{-\frac {1}{2}}$ and

$$\begin{align*}|g^{-1} R(\ell)^0 g \cap \Psi(\delta, T)| \prec 1 + \ell^{\frac{1}{2}} T + \frac{\ell^{\frac{3}{2}} }{(d_B N)^{\frac{1}{2}}} T^2 + \frac{\ell^2 \delta^{\frac{1}{2}}}{d_B N} T^{3}. \end{align*}$$

Theorem 2.6 (Type II estimates).

Let $g \in G(\mathbb {R})$ and $n \in \frac {1}{\ell } \mathbb {Z}$ . We have

$$\begin{align*}|g^{-1} R(\ell)^0 g \cap \Omega(\delta, T) \cap \det{}^{-1}(\{n\})| \prec 1 + \ell \delta^{\frac{1}{2}} H(g) T +\frac{\ell^2}{d_B N} \delta T^2. \end{align*}$$

The proof of these results occupies §7 onwards. In §3, we explain how these results imply our main fourth moment bound, Theorem 2.1.

3.1. Traversing the genus

Recall that $K_R$ is defined as the image of the subgroup $\prod _p R_p^{\times }$ in $G(\mathbb {A}_f)$ ; it is a compact open subgroup of $G(\mathbb {A}_f)$ . In due course, we will consider the conjugated sets $h_f K_R h_f^{-1}$ , for $h_f \in G(\mathbb {A}_f)$ . These are precisely the compact open subgroups $K_{R'}$ associated to the Eichler orders $R'$ in the genus of R. We note that $R'$ has the same level as R and may be given explicitly by the following intersection:

$$ \begin{align*}R' = h_f (R \otimes \widehat{\mathbb{Z}}) h_f^{-1} \cap B(\mathbb{Q}), \end{align*} $$

where $\widehat {\mathbb {Z}}$ denotes the closure of $\mathbb {Z}$ inside $\mathbb {A}_f$ . We further note that the action of $G(\mathbb {A}_f)$ on the genus of R commutes with partial dualization in the sense that

$$ \begin{align*}R'(\ell) = h_f (R(\ell) \otimes \widehat{\mathbb{Z}}) h_f^{-1} \cap B(\mathbb{Q}). \end{align*} $$

This observation permits us to formulate the required $L^2$ -estimates for our differences of theta kernels in terms of integration over Archimedean, rather than adelic, arguments. To that end, we introduce the notation

for $h=(h_\infty ,h_f) \in G(\mathbb {A})$ . We note that for $h \in G(\mathbb {R})$ (i.e., $h_f = 1$ ), the set $R(\ell; h)$ is just $h^{-1}R(\ell )h$ . Since taking the trace commutes with conjugation, we may extend the notation to kernels of the reduced trace without concern for confusion regarding the order of operation, that is,

$$ \begin{align*}R(\ell; h)^0 = (h_\infty^{-1} R'(\ell) h_\infty)^0 = h_\infty^{-1} R'(\ell)^0 h_\infty = h_\infty^{-1} ( h_f ( R(\ell)^0 \otimes \widehat{\mathbb{Z}} ) h_f^{-1} \cap B(\mathbb{Q})) h_\infty. \end{align*} $$

If B is split, then the class number of R is one and we have fixed the representative as in Equation (2.1). In this case, we find for $h \in G(\mathbb {A})$ that $h^{-1}R h=h^{\prime -1}Rh'$ , where $h' \in G(\mathbb {R})$ has the same image under the isomorphism $[G]/ K_{\infty }K_R \cong \Gamma _0(N) \backslash \mathbb {H}$ as h does. In particular, we have the equality of height functions (see §2.2) $H(h)=H(h')$ .

3.2. Estimating fourth moments via lattice sums

In §5, we introduce certain theta kernels. A spectral expansion of their $L^2$ -norms will yield the fourth moments of interest, while a ‘geometric’ expansion, using Siegel domains and Fourier expansions, bounds those $L^2$ -norms in terms of certain lattice sums. We now state the latter bounds.

Proposition 3.2. Suppose B is indefinite. Then, for $g_1, g_2 \in [G]$ , there exists $\ell |d_BN$ , $g \in \{g_1, g_2\}$ , and $0< T \preccurlyeq \frac {(d_B N(k+1))^{\frac {1}{2}}}{\ell }$ (here, the notation $\preccurlyeq $ is as in §2.3) so that, for $k=0$ ,

(3.1) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{-}_{\le L} } (|\varphi(g_1)|^2 - |\varphi(g_2)|^2)^2 \preccurlyeq_{L} 1+ \frac{d _B N}{\ell T^2} \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(1, T): \\ \det(\gamma_1) = \det(\gamma_2) } } 1, \end{align} $$

while for $k> 0$ ,

(3.2) $$ \begin{align} \sum _{\varphi \in \mathcal{F^{-,\operatorname{hol}}} } (|\varphi(g_1)|^2 - |\varphi(g_2)|^2)^2 \preccurlyeq 1+ \frac{d _B N k}{\ell T^2} \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(1, T): \\ \det(\gamma_1) = \det(\gamma_2) } } 1. \end{align} $$

Proposition 3.3. Suppose B is indefinite. Let $g \in [G]$ , and assume that $k \gg (d_BN)^{\eta }$ for some arbitrarily small $\eta>0$ . Then, there exists $\ell |d_BN$ and $ 0<T \preccurlyeq \frac {(d_B N k)^{\frac {1}{2}}}{\ell }$ so that

(3.3) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{-,\operatorname{hol}} } |\varphi(g)|^4 \preccurlyeq_{\eta} 1+ \frac{d _B N k}{\ell T^2} \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(k^{-1+\varepsilon}, T): \\ \det(\gamma_1) = \det(\gamma_2)>0 } } 1. \end{align} $$

Proposition 3.4. Suppose B is definite and the weight is $k=0$ . Then, for $g_1,g_2 \in [G]$ and $m \in \mathbb {N}_0$ , there exists $\ell |d_BN$ , $ 0< T \preccurlyeq \frac {(d_B N (m+1))^{\frac {1}{2}}}{\ell }$ and $\frac {1}{m^2+1} \preccurlyeq \delta \le 1$ so that

(3.4) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{+}_{m}} \left(|\varphi(g_1)|^2-|\varphi(g_2)|^2\right)^2 \preccurlyeq 1+ \frac{d_B N}{\ell \delta^{\frac{1}{2}} T^2} \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \\ \gamma_1, \gamma_2 \in \Omega^{\star}(\delta, T)\cup \Psi^{\star}(\delta, T): \\ \det(\gamma_1) = \det(\gamma_2) } } 1. \end{align} $$

Proposition 3.5. Suppose B is definite. Then, for $g \in [G]$ , there exists $\ell |d_BN$ and $ 0< T \preccurlyeq \frac {(d_B N k)^{\frac {1}{2}}}{\ell }$ so that

(3.5) $$ \begin{align} \sum _{\varphi \in \mathcal{F}^{+,\operatorname{hol}}} |\varphi(g)|^4 \preccurlyeq 1+\frac{d _B N k}{\ell T^2}\left( \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(k^{-1+\varepsilon}, T): \\ \det(\gamma_1) = \det(\gamma_2) } }1 + \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(1, T): \\ \det(\gamma_1) = \det(\gamma_2) } } k^{-2027}\right). \end{align} $$

3.3. Reduction to ternary lattices

In this section, we reduce the vital counting problem involving quaternary quadratic form to problems involving only ternary quadratic forms. The key observation is that we may orthogonally decompose the quaternion algebra $B_{\infty }$ into its trace part and its traceless part $B_{\infty }^{0}$ . Thus, for any $\alpha = \frac {1}{2}\operatorname {\mathrm {tr}}(\alpha ) + \alpha ^0 \in \mathbb {R} \oplus B_{\infty }^{0}$ , we have

(3.6) $$ \begin{align} \det(\alpha)= \tfrac{1}{4}\operatorname{\mathrm{tr}}(\alpha)^2+\det(\alpha^0). \end{align} $$

We further note that the trace is invariant under conjugation. Hence, we have $\operatorname {\mathrm {tr}}(R(\ell ;g)) \subseteq \mathbb {Z}$ . We conclude that

(3.7) $$ \begin{align} R(\ell;g) \subseteq \tfrac{1}{2} \mathbb{Z} \oplus \tfrac{1}{2} R(\ell;g)^{0} \end{align} $$

is a sublattice of the direct sum of the lattices $\frac {1}{2}\mathbb {Z}$ in $\mathbb {R}$ and $\tfrac {1}{2} R(\ell ;g)^{0}$ in $B_{\infty }^0$ . Using this decomposition, we deduce

(3.8) $$ \begin{align} \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g) \cap\Omega^{\star}(\delta,T) \\ \det(\gamma_1) = \det(\gamma_2)}} 1 \ll_{\varepsilon} T^{\varepsilon} \left|R(\ell;g)^0 \cap \Omega(\delta,2T) \right|{}^2 + T \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g)^0 \cap\Omega(\delta,2T) \\ \det(\gamma_1) = \det(\gamma_2)}}1 \end{align} $$

by distinguishing the two cases of equal and nonequal trace and applying the divisor bound to the equality

$$ \begin{align*}\det(\gamma_1)=\det(\gamma_2) \Leftrightarrow \tfrac{1}{4}\operatorname{\mathrm{tr}}(\gamma_1)^2 - \tfrac{1}{4}\operatorname{\mathrm{tr}}(\gamma_2)^2 = \det(\gamma_2^0)-\det(\gamma_1^0). \end{align*} $$

We remark that we have forfeited the congruence condition $\det (\gamma _1^0) \equiv \det (\gamma _2^0) \ \mathrm {mod}\,(1)$ , and this forfeiture will be reflected in the suboptimality of our final counting estimates on larger scales when $\ell>1$ . We circumnavigate these larger scales by an appropriate choice of a covering domain (cf. Lemma 4.1).

Note that we may further bound the diagonal contribution by considering its largest fiber:

(3.9) $$ \begin{align} \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g)^0 \cap\Omega(\delta,2T) \\ \det(\gamma_1) = \det(\gamma_2)}}1 \le |R(\ell;g)^0 \cap\Omega(\delta,2T)| \times \max_{\substack{n \in \frac{1}{\ell} \mathbb{Z} \\ |n|\le 4T^2}} |R(\ell;g)^0 \cap\Omega(\delta,2T) \cap \det{}^{-1}(\{n\})|. \end{align} $$

Arguing along the same lines, we also arrive at

(3.10) $$ \begin{align} \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g) \cap\Psi^\star(\delta,T) \\ \det(\gamma_1) = \det(\gamma_2)}} 1 \ll_{\varepsilon} T^{\varepsilon} \left|R(\ell;g)^0 \cap \Psi(\delta,2T) \right|{}^2 + \delta^{\frac{1}{2}}T \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g)^0 \cap\Psi(\delta,2T) \\ \det(\gamma_1) = \det(\gamma_2)}}1 \end{align} $$

and

(3.11) $$ \begin{align} \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g)^0 \cap\Psi(\delta,2T) \\ \det(\gamma_1) = \det(\gamma_2)}}1 \le |R(\ell;g)^0 \cap\Psi(\delta,2T)| \times \max_{\substack{n \in \frac{1}{\ell} \mathbb{Z} \\ |n|\le 4T^2}} |R(\ell;g)^0 \cap\Omega(1,2T) \cap \det{}^{-1}(\{n\})|. \end{align} $$

Note that in this last inequality, we have passed from $\Psi (\delta ,2T)$ to the larger set $\Omega (1,2T)=\Psi (1,2T)$ ; the resulting bound remains adequate for us thanks to the additional saving of $\delta ^{\frac {1}{2}}$ in Equation (3.10).

3.4. Proof of Theorem 2.1

Theorem 2.1 is an immediate consequence of the following pair of lemmas together with Propositions 3.2 through 3.5.

Lemma 3.6. We have

(3.12) $$ \begin{align} \nonumber \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g) \cap\Omega^{\star}(\delta,T) \\ \det(\gamma_1) = \det(\gamma_2)}} 1 \prec \ell T^2 \left(1 + \ell^{\frac{1}{2}} \delta^{\frac{1}{2}} H(g) + \ell^{\frac{1}{2}} \delta H(g) T + \frac{\ell^2 }{d_B N} \delta T^2 \right) \\ \times \left(1 + \ell^{\frac{1}{2}} \delta^{\frac{1}{2}} H(g) + \ell^{\frac{1}{2}} \delta^{\frac{1}{2}} H(g) T + \frac{\ell^2 }{d_B N} \delta T^2 \right). \end{align} $$

Proof. Recall, from the discussion of Section §3.1, that we may express $R(\ell ;g)^0$ , for $g \in G(\mathbb {A})$ , as $(g')^{-1}R'(\ell )^0g'$ , where $R'$ is an Eichler order of the same level and $g' \in G(\mathbb {R})$ , with $H(g)=H(g')$ in the case that B is split. We may thus apply the results of Section §2.5.

Since $\operatorname {\mathrm {tr}}(R(\ell ;g)) \subseteq \mathbb {Z}$ , we find that the first successive minimum of $R(\ell ;g)$ with respect to $\Omega (\delta ,1)$ is at least the minimum of $1$ and the first successive minimum of $R(\ell ;g)^0$ with respect to $\Omega (\delta ,1)\cap B_{\infty }^{0}$ . The latter is $\gg \min \{ \ell ^{-\frac {1}{2}}, \ell ^{-1} \delta ^{-\frac {1}{2}} H(g)^{-1} \} =: \Lambda $ by Theorem 2.5, where the term involving $H(g)$ is to be omitted if B is nonsplit. Thus, we find that $R(\ell ;g) \cap \Omega ^{\star }(\delta ,T)$ is empty for $T \ll \Lambda $ , in which case there is nothing to show. Next, assume instead that $T \gg \Lambda $ . Then, by Theorem 2.5, we have

(3.13) $$ \begin{align} \begin{aligned} |g^{-1} R(\ell)^0 g \cap \Omega(\delta, 2T)| &\prec 1 + \left(\ell^{\frac{1}{2}} +\ell \delta^{\frac{1}{2}} H(g) \right) T + \left( \frac{\ell^{\frac{3}{2}} \delta^{\frac{1}{2}} }{(d_B N)^{\frac{1}{2}}} + \ell \delta H(g) \right) T^2 + \frac{\ell^2 }{d_B N} \delta T^{3} \\ &\prec \ell^{\frac{1}{2}} T \left( 1 +\ell^{\frac{1}{2}} \delta^{\frac{1}{2}} H(g) + \frac{\ell }{(d_B N)^{\frac{1}{2}}} \delta^{\frac{1}{2}} T + \ell^{\frac{1}{2}} \delta H(g) T + \frac{\ell^{2} }{d_B N} \delta T^{2} \right), \end{aligned} \end{align} $$

where we have used $1 \ll \ell ^{\frac {1}{2}}T+\ell \delta ^{\frac {1}{2}}H(g)T$ and $\ell ^{\frac {3}{2}} \le \ell ^{2}$ . Further note that the middle term in the bracket is dominated by the sum of the first and last term in the bracket. We also find by Theorem 2.6 that

(3.14) $$ \begin{align} \begin{aligned} \max_{n \in \frac{1}{\ell} \mathbb{Z}} |g^{-1} R(\ell)^0 g \cap \Omega(\delta, 2T) \cap \det{}^{-1}(\{n\})| &\prec 1 + \ell \delta^{\frac{1}{2}} H(g) T +\frac{\ell^2}{d_B N} \delta T^2 \\ &\prec \ell^{\frac{1}{2}} \left( 1 + \ell^{\frac{1}{2}} \delta^{\frac{1}{2}}H(g)T+\frac{\ell^{2}}{d_BN}\delta T^2 \right). \end{aligned} \end{align} $$

We conclude the Lemma by further appealing to the inequalities (3.8) and (3.9).

Lemma 3.7. Assume that B is nonsplit. Then

(3.15) $$ \begin{align} \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g) \cap\Psi^{\star}(\delta,T) \\ \det(\gamma_1) = \det(\gamma_2)}} 1 \prec \ell T^2 \left(1 + \frac{\ell}{(d_BN)^{\frac{1}{2}}} T + \frac{\ell^2 }{d_B N} \delta^{\frac{1}{2}} T^2 \right)^2. \end{align} $$

Proof. As in the proof of Lemma 3.6, we find that the first successive minimum of $R(\ell ;g)$ with respect to $\Psi (\delta ,1)$ is at least the minimum of $\delta ^{-\frac {1}{2}}$ and the first successive minimum of $R(\ell ;g)^0$ with respect to $\Psi (\delta ,1)\cap B_{\infty }^{0}$ . The latter is $\gg \ell ^{-\frac {1}{2}}$ by Theorem 2.5. Therefore, $R(\ell ;g) \cap \Psi ^{\star }(\delta ,T)$ is empty for $T \ll \ell ^{-\frac {1}{2}} \le 1 \le \delta ^{-\frac {1}{2}}$ , in which case there is nothing to show. If $T \gg \ell ^{-\frac {1}{2}}$ , then, by Theorem 2.5, we have

(3.16) $$ \begin{align} \begin{aligned} |g^{-1} R(\ell)^0 g \cap \Psi(\delta, 2T)| &\prec 1 + \ell^{\frac{1}{2}} T + \frac{\ell^{\frac{3}{2}} }{(d_B N)^{\frac{1}{2}}} T^2 + \frac{\ell^2 }{d_B N} \delta^{\frac{1}{2}} T^{3} \\ &\prec \ell^{\frac{1}{2}} T \left( 1 + \frac{\ell }{(d_B N)^{\frac{1}{2}}} T + \frac{\ell^2 }{d_B N} \delta^{\frac{1}{2}} T^{2} \right), \end{aligned} \end{align} $$

where we have used $1 \ll \ell ^{\frac {1}{2}}T$ and $\ell ^{\frac {3}{2}} \le \ell ^{2}$ . Furthermore, by Theorem 2.6, we have

(3.17) $$ \begin{align} \max_{n \in \frac{1}{\ell} \mathbb{Z}} |g^{-1} R(\ell)^0 g \cap \Omega(1, 2T) \cap \det{}^{-1}(\{n\})| \prec 1 +\frac{\ell^2}{d_B N} T^2 \prec \ell^{\frac{1}{2}} \delta^{-\frac{1}{2}} \left( 1 + \frac{\ell^2 }{d_B N} \delta^{\frac{1}{2}} T^{2} \right), \end{align} $$

where we have used $1 \le \ell ^{\frac {1}{2}}$ and $1 \le \delta ^{-\frac {1}{2}}$ . We conclude the lemma by further appealing to the inequalities (3.10) and (3.11).

3.5. Proof of Corollary 2.3

Let $\varphi \in \mathcal {F}$ , respectively $\mathcal {F}^{\operatorname {hol}}$ , be $L^2$ -normalized. Assume first that B is nonsplit. Then, since $[G] / K_{\infty } K_{R}$ is compact and equipped with a probability measure, we may find $g_2$ in $[G]$ such that $|\varphi (g_2)| \le 1$ . Hence, Corollary 2.3 follows immediately from Theorem 2.1 by positivity and the particular choice of  $g_2$ .

We now turn our attention to the case that B is split, in other words when $d_B=1$ . Here, we need to supplement Theorem 2.1 with the additional information that for $H(g) \ge N^{-\frac {1}{2}}$ , we have

(3.18) $$ \begin{align} \!|\varphi(g)| &\preccurlyeq_{\lambda_{\varphi}} N^{\frac{1}{4}} &&\text{if}\ \varphi \in \mathcal{F}^{-},\quad \end{align} $$

(3.19) $$ \begin{align} |\varphi(g)| &\preccurlyeq (kN)^{\frac{1}{4}} &&\text{if}\ \varphi \in \mathcal{F}^{-,\operatorname{hol}}. \end{align} $$

The former is recorded in [Reference TemplierTem15, Prop. 3.1 & 3.2], for example, and the latter may be deduced from the Fourier expansion along the lines of Xia [Reference XiaXia07]. We include a brief proof here for the sake of completeness.

Lemma 3.8. Assume B is split, and let $\varphi \in \mathcal {F}^{-,\mathrm {hol}}$ be an $L^2$ -normalized holomorphic cuspidal newform of squarefree level N and even weight $k \ge 2$ . Then, we have for all $g \in [G]$ ,

(3.20) $$ \begin{align} |\varphi(g)| \preccurlyeq H(g)^{o(1)} \left( k^{\frac{1}{4}}H(g)^{-\frac{1}{2}}+ k^{-\frac{1}{4}}H(g)^{\frac{1}{2}} \right). \end{align} $$

If $H(g) \ge \frac {k}{2 \pi }$ , then we have the stronger bound

(3.21) $$ \begin{align} |\varphi(g)| \preccurlyeq 1. \end{align} $$

Proof. Suppose that g corresponds to $z=x+iy \in \Gamma _0(N) \backslash \mathbb {H}$ . As $|\varphi (g)|$ is further invariant under the Atkin–Lehner operators we may further assume that z has maximal imaginary part under the action of the group $A_0(N)$ generated by the Atkin–Lehner operators and $\Gamma _0(N)$ , thus $H(g)=y$ . We shall subsequently make use of the Fourier expansion of $\varphi $ at $\infty $ :

$$ \begin{align*} |\varphi(g)| = \left| y^{\frac{k}{2}} \sum_{n=1}^{\infty} a_n e(n(x+iy)) \right |. \end{align*} $$

We may bound the Fourier coefficients by appealing to Deligne’s bound for the Hecke eigenvalues [Reference DeligneDel71, Reference DeligneDel74]. This implies $|a_n| \ll _{\varepsilon } n^{\frac {k-1}{2}+\varepsilon } |a_1|$ .Footnote ¹ We find

$$ \begin{align*} |\varphi(g)| \ll_{\varepsilon} |a_1| (2\pi)^{-\frac{k}{2}} y^{\frac{1}{2}-\varepsilon} \sum_{n=1}^{\infty} (2 \pi n y)^{\frac{k-1}{2}+\varepsilon} e^{-2 \pi n y}. \end{align*} $$

The above sum, we may bound by comparison to the corresponding integral. For this manner, we note that the function $x^{\alpha }e^{-x}$ increases up to $x= \alpha $ and then decreases. We may also bound the first Fourier coefficient $a_1$ by a result of Hoffstein–Lockhart [Reference Hoffstein and LockhartHL94] (cf. [Reference Harcos and MichelHM06, Eq. (31)]Footnote ²). The bound reads $|a_1| \ll _{\varepsilon } (Nk)^{\varepsilon } (4\pi )^{\frac {k}{2}} \Gamma (k)^{-\frac {1}{2}}$ . We thus arrive at

$$ \begin{align*}\begin{aligned} |\varphi(g)| &\ll_{\varepsilon} (Nk)^{\varepsilon}y^{-\varepsilon} \frac{2^{\frac{k}{2}}y^{\frac{1}{2}}}{\Gamma(k)^{\frac{1}{2}}} \left( \frac{1}{y} \Gamma\left( \frac{k+1}{2} \right) + \left(\frac{k-1}{2}\right)^{\frac{k-1}{2}+\varepsilon}e^{-\frac{k-1}{2}} \right) \\ &\ll_{\varepsilon} (Nk)^{\varepsilon} y^{-\varepsilon} \left( k^{\frac{1}{4}}y^{-\frac{1}{2}}+ k^{-\frac{1}{4}}y^{\frac{1}{2}} \right), \end{aligned}\end{align*} $$

where we have made use of Stirling’s approximation. If $y \ge \frac {k}{2 \pi }$ , then the maximum summand occurs when $n=1$ and we may derive the improved bound

$$ \begin{align*}|\varphi(g)| &\ll_{\varepsilon} (Nk)^{\varepsilon} y^{-\varepsilon} \frac{2^{\frac{k}{2}}y^{\frac{1}{2}}}{\Gamma(k)^{\frac{1}{2}}} \left( \frac{1}{y} \Gamma\left( \frac{k+1}{2} \right) + \left(2 \pi y\right)^{\frac{k-1}{2}+\varepsilon}e^{-2 \pi y } \right) \\ & \ll_{\varepsilon} (Nk)^{\varepsilon} y^{-\varepsilon}.\\[-38pt] \end{align*} $$

To deduce Equation (3.19) from the lemma, we consider separately the cases $N^{-\frac {1}{2}} \le H(g) \le \frac {k}{ 2 \pi }$ and $H(g) \ge \frac {k}{2 \pi }$ , applying Equation (3.20) in the former case and Equation (3.21) in the latter.

We may now deduce the split case of Corollary 2.3, as follows. Our task is to bound $\varphi (g_1)$ suitably for $g_1 \in [G]$ . We may assume that $H(g_1) \le N^{-\frac {1}{2}}$ , as otherwise the estimates (3.18) and (3.19) are adequate. In that case, we choose another point $g_2 \in [G]$ arbitrarily with $H(g_2) = N^{-\frac {1}{2}}$ such that $|\varphi (g_2)| \preccurlyeq _{\lambda _{\varphi }} N^{\frac {1}{4}}$ , respectively $|\varphi (g_2)| \preccurlyeq (kN)^{\frac {1}{4}}$ , by Equation (3.18), respectively Equation (3.19). We apply Theorem 2.1 with these choices of $g_1$ and $g_2$ . Upon recalling that $d_B=1$ in the split case, we find by positivity that Equation (2.4), respectively Equation (2.5), yield

$$ \begin{align*} \left| |\varphi(g_1)|^2- |\varphi(g_2)|^2 \right| &\preccurlyeq_{\lambda_{\varphi}} N^{\frac{1}{2}}, \quad \text{respectively} \\ \left| |\varphi(g_1)|^2- |\varphi(g_2)|^2 \right| &\preccurlyeq (kN)^{\frac{1}{2}}. \end{align*} $$

We conclude by the triangle inequality and taking square roots:

$$ \begin{align*} \left|\varphi(g_1)\right|{}^2 \le \left| |\varphi(g_1)|^2- |\varphi(g_2)|^2 \right| + \left| \varphi(g_2) \right |^2. \end{align*} $$

4.1. Measure normalizations

For indefinite B, we fix an isomorphism $G(\mathbb {R})\cong \operatorname {PGL}_2(\mathbb {R})$ sending $K_\infty $ to $\operatorname {PSO}_2(\mathbb {R})$ . We fix the Haar measure $\,\mathrm {d} g = \frac {\,\mathrm {d} y \,\mathrm {d} x}{y^2}\frac {\,\mathrm {d} \theta }{2 \pi }$ for $g=\left (\begin {smallmatrix} y^{1/2} & xy^{-1/2} \\ 0 & y^{-1/2} \end {smallmatrix}\right ) \kappa (\theta )$ on $\operatorname {SL}_2(\mathbb {R})$ . The push-forward of this measure to the hyperbolic plane is then the measure $\frac {\,\mathrm {d} x \,\mathrm {d} y}{y^2}$ . The Haar measure on $\operatorname {PGL}_2(\mathbb {R})$ is fixed so that its restriction to $\operatorname {PSL}_2(\mathbb {R})$ coincides with the push-forward of the Haar measure from $\operatorname {SL}_2(\mathbb {R})$ .

If B is definite, we fix an isomorphism $G(\mathbb {R})\cong \operatorname {SO}_3(\mathbb {R})$ sending $K_\infty $ to $\operatorname {SO}_2(\mathbb {R})$ . We fix a Haar measure on $\operatorname {SO}_3(\mathbb {R})$ so that the measure of the $2$ -sphere is $4\pi $ .

4.2. Volumes

Recall, that we fixed the measure on $[G]$ to be the probability Haar measure. Hence, the volume of the quotient $[G]/K_R$ is $1$ . In due course, we shall also require the volume of said quotient when viewed as a real manifold with respect to our fixed Haar measure on $G(\mathbb {R})$ . More specifically, we will need the volume with respect to the measure on $G'(\mathbb {R})$ , where $G'$ is the linear algebraic group defined over $\mathbb {Q}$ whose rational points are the proper unit quaternions $B^1$ . There is an obvious isogeny map $G'\to G$ , where $G'$ is the simply connected form and G is the adjoint one. Define $R_p^1=R_p\cap G'(\mathbb {Q}_p)$ to be the proper unit quaternions in the local order $R_p$ , and set $K_R^1=\prod _{p} R_p^1$ . Then, the map $[G']/K_R^{1}\to [G]/K_R$ is a homeomorphism that pushes forward the probability Haar measure on $[G']/K_R^{1}$ to the probability Haar measure on $[G]/K_R$ ; see Lemma A.2. In general, this map is not bijective if $K_R$ is replaced by a general compact open subgroup of $G(\mathbb {A}_f)$ and the fact that the map is indeed a homeomorphism is due to $K_R$ being the projectivized group of units of an Eichler order.

By Borel’s finiteness of class numbers [Reference BorelBor63], $[G']/K_R^1$ is a finite collection of $G'(\mathbb {R})$ -orbits with representatives $\delta _1,\ldots ,\delta _h\in G'(\mathbb {A})$ . Define $\Gamma _i=G'(\mathbb {Q})\cap \delta _i K_R^1 \delta _i^{-1}$ ; the intersection is taken in $G'(\mathbb {A}_f)$ but regarded as a subset of $G'(\mathbb {Q})$ and hence also of $G'(\mathbb {R})$ . In particular, $\Gamma _i$ is a lattice in $G'(\mathbb {R})$ . It follows that

$$ \begin{align*} [G]/K_R\cong [G']/K_R^{1}=\bigsqcup_{i=1}^h {\Gamma_i}\backslash{G'(\mathbb{R})}. \end{align*} $$

This is a finite disjoint union of finite-volume homogeneous spaces for the real Lie group $G'(\mathbb {R})$ . We define $\operatorname {covol}(\Gamma _i)$ to be the measure of a fundamental domain for the action of $\Gamma _i$ on $G'(\mathbb {R})$ with respect to the fixed Haar measure on $G'(\mathbb {R})$ , either $\frac {\,\mathrm {d} x \,\mathrm {d} y}{y^2} \frac {\,\mathrm {d} \theta }{2\pi }$ in the indefinite case or the measure giving volume $4\pi $ to in the definite case. We finally set

$$ \begin{align*} V = V_{d_B,N}=\sum_{i=1}^h \operatorname{covol}(\Gamma_i). \end{align*} $$

If B is indefinite, then $G'(\mathbb {R})\cong \operatorname {SL}_2(\mathbb {R})$ is noncompact and strong approximation implies that $h=1$ and we can write $[G]/K_R\cong \Gamma \backslash \operatorname {SL}_2(\mathbb {R})$ . In this case, V is the volume of the hyperbolic surface with respect to the volume form $\frac {\,\mathrm {d} x \,\mathrm {d} y}{y^2}$ . If B is definite, then in general h can be large and $[G]/K_\infty K_R$ is a finite collection of quotients of $2$ -spheres by discrete rotation groups.

Recall that we have denoted by $d_B$ the reduced discriminant of B and by N the squarefree level of the Eichler order R. The volume is given in both cases by

(4.1) $$ \begin{align} V = V_{d_B,N} = \frac{\pi}{3} d_BN \prod_{p | d_B} \left(1-\frac{1}{p}\right) \prod_{p | N} \left(1+\frac{1}{p} \right) = (d_BN)^{1+o(1)}. \end{align} $$

This follows from a corresponding mass formula; see [Reference VoightVoi18, Thm 39.1.8] in the indefinite case and [Reference VoightVoi18, Thm 25.1.1 & Thm 25.3.18] in the definite one. The space is furthermore compact if and only if B nonsplit, that is, $d_B> 1$ .

4.3. Siegel domains

The main purpose of this section is to provide a specific Siegel-domain covering in order to bound the $L^2$ -norm of the (difference of) theta kernels in §5.2. Let M be a squarefree natural number and set $U = \left (\begin {smallmatrix} \widehat {\mathbb {Z}} & \widehat {\mathbb {Z}} \\ M \widehat {\mathbb {Z}} & \widehat {\mathbb {Z}} \end {smallmatrix}\right ) \cap \operatorname {\mathrm {SL}}_{2}(\mathbb {A}_f)$ . The theta functions of interest will turn out to be right invariant in the symplectic variable under U with $M = d_B N$ , but the present discussion applies to any squarefree M.

4.3.1. Cusps and Atkin–Lehner operators

Since M is squarefree, a representative set of cusps for $\Gamma _0(M)$ is given by the ratios $\frac {\ell }{M}$ , where $\ell $ runs through the positive divisors of M. The width of the cusp $\frac {\ell }{M}$ (understood here as with respect to the group $\Gamma _0(M)$ ) is given by $\ell $ [Reference IwaniecIwa97, §2.4]. For each $\ell |M$ , we choose an element $\tau _{\ell } \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ satisfying

(4.2) $$ \begin{align} \tau_{\ell} \equiv \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \mathrm{mod}\,{\ell}, \quad \tau_{\ell} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mathrm{mod}\,{ M / \ell}. \end{align} $$

Then, the cusp $\tau _{\ell } \infty $ is $\Gamma _0(M)$ -equivalent to the cusp $\frac {\ell }{M}$ . Hence, writing $n(x)=\left (\begin {smallmatrix} 1 & x\\ & 1 \end {smallmatrix}\right )$ , we see that the elements $\tau _{\ell }n(j)$ , where $j=0,\dots ,\ell -1$ and $\ell |M$ , give a complete system of representatives for $\Gamma _0(M) \backslash \operatorname {\mathrm {SL}}_{2}(\mathbb {Z})$ . The normalized matrices , where $a(y)= \operatorname {\mathrm {diag}}(y^{\frac {1}{2}},y^{-\frac {1}{2}})$ , are scaling matrices for the respective cusps. Furthermore, the matrices $\tilde {\tau _{\ell }}$ are Atkin–Lehner operators for $\Gamma _0(M)$ and give a set of representatives for $A_0(M) / \Gamma _0(M)$ [Reference Atkin and LehnerAL70, Lemma 9].

4.3.2. Coverings

The basic idea of the following lemma is to apply the Fricke involution to the tiling of $\Gamma _0(M) \backslash \mathbb {H}$ by translates of the standard fundamental domain for $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ .

Lemma 4.1. Let $F : [\operatorname {\mathrm {SL}}_2] \rightarrow \mathbb {R}_{\ge 0}$ be a measurable function that is right U invariant and of weight $0$ . Then,

$$ \begin{align*} \int_{[\operatorname{\mathrm{SL}}_2]} F(g) dg \le \frac{1}{V_{1,M}} \sum_{\ell|M} \int_{\frac{\sqrt{3}}{2} \frac{l^2}{M}}^{\infty} \int_0^{\ell} F \left( (\tau_{\ell})_{\infty} \left(\begin{smallmatrix} y^{1/2} & x y^{-1/2} \\ & y^{-1/2} \end{smallmatrix}\right) K_{\infty} U) \right) dx \frac{dy}{y^2}. \end{align*} $$

Here, $(\tau )_{\infty }$ denotes the image of $\tau $ in the Archimedean coordinate of $\operatorname {\mathrm {SL}}_{2}(\mathbb {A})$ .

Proof. Let $f: \mathbb {H} \to \mathbb {R}_{\ge 0}$ be given by

$$ \begin{align*}f(z)=F \left( \left(\begin{smallmatrix} y^{1/2} & x y^{-1/2} \\ & y^{-1/2} \end{smallmatrix}\right) K_{\infty} U \right).\end{align*} $$

Then, f is $\Gamma _0(M)$ invariant on the left and we have

$$ \begin{align*} \int_{[\operatorname{\mathrm{SL}}_2]} F(g) dg = \frac{1}{V_{1,M}} \int_{\Gamma_0(M)\backslash \mathbb{H}} f(z) \frac{dxdy}{y^2} = \frac{1}{V_{1,M}} \int_{\Gamma_0(M)\backslash \mathbb{H}} f( \tilde \tau_{M}z) \frac{dxdy}{y^2}. \end{align*} $$

The standard Siegel set $\{z \in \mathbb {H} | \ 0\le \Re (x)\le 1 \text { and } \Im (z) \ge \frac {\sqrt {3}}{2}\}$ contains a fundamental domain for $\operatorname {\mathrm {SL}}_2({\mathbb {Z}})$ . Using that the $\tau _{\ell }n(j)$ , for $j=0,\dots ,\ell -1$ and $\ell |M$ , form a representative set for $\Gamma _0(M) \backslash \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and that f is nonnegative, we may bound

$$ \begin{align*}\int_{\Gamma_0(M)\backslash \mathbb{H}} f( \tilde \tau_{M}z) \frac{dxdy}{y^2} \le \sum_{\ell|M} \int_{\frac{\sqrt{3}}{2}}^{\infty} \int_0^{\ell} f( \tilde{\tau}_{M}\tau_{\ell} z) dx \frac{dy}{y^2}. \end{align*} $$

Since $\tilde {\tau }_{M} \tilde {\tau }_{\ell } = \gamma \tilde {\tau }_{\frac {M}{\ell }}$ for some $\gamma \in \Gamma _0(M)$ , we have the identity

$$ \begin{align*} f(\tilde{\tau}_{M}\tau_{\ell} z) = f( \tilde{\tau}_{\tfrac{M}{\ell}}a(\ell)^{-1} z) = f(\tau_{\tfrac{M}{\ell}} a(\tfrac{M}{\ell})a(\ell)^{-1} z). \end{align*} $$

Substituting this identity above and applying the change of variables $\frac {M}{l^2}z \mapsto z$ gives the desired result.

5.1. Theta kernels and lifts

In this section, we summarize the required results on theta kernels and their lifts. The necessary theory is developed in Appendix A.

5.1.1. Theta functions

The theta kernels constructed in Appendix A are modular functions on $\operatorname {O}_{\det }(\mathbb {A})\times \operatorname {\mathrm {SL}}_2(\mathbb {A})$ . The group G acts by conjugation on the quadratic space $(B,\det )$ , preserving the quadratic form. This gives an embedding $G\hookrightarrow \operatorname {O}_{\det }$ . We are mainly concerned here with the pullback of the theta kernels to $G(\mathbb {A})\times \operatorname {\mathrm {SL}}_2(\mathbb {A})$ . We denote that pullback by $\Theta (g,s)$ . The function $\Theta (g,s)$ is right $K_R \times U_R^1$ invariant, where and of moderate growth. We caution that it is not a theta kernel for the Howe dual pair of the orthogonal group of the traceless quaternions and $\operatorname {\mathrm {SL}}_2$ .

We shall require several explicit expressions of the theta kernels $\Theta $ . Define functions $P, u, X$ on $B_\infty $ by setting, for $\gamma = [a,b,c] + d \in B_\infty $ ,

(5.1)

In other words, by identifying ${\mathbf{i}} \in B_{\infty }$ with $i \in \mathbb {C}$ , we have that

• ∘ X is the projection from $B_\infty = \mathbb {C} \oplus \mathbb {C} {\mathbf{j}} $ to the summand $\mathbb {C}$ ,

• ∘ u is the squared magnitude of the projection onto the other summand $\mathbb {C} {\mathbf{j}} $ and

• ∘ P is the sum of the squared magnitudes of the two projections.

Upon recalling the notation $R(\ell ;g)$ from Section §3.1, we define for $g \in G(\mathbb {A})$ and $z=x+iy \in \mathbb {H}$ the theta functions

(5.2)

if $k \ge 6$

(5.3)

(5.4)

(5.5)

where $P_{m}$ is the m-th Legendre polynomial. When $\ell = 1$ , we abbreviate by dropping the subscript, for example, . We are now ready to express $\Theta $ by means of strong approximation. Set

$$ \begin{align*}s_{\infty}= \left(\begin{smallmatrix} y^{1/2} & xy^{-1/2} \\ & y^{-1/2} \end{smallmatrix}\right) \left(\begin{smallmatrix} \cos(\theta) & \sin(\theta) \\ - \sin(\theta) & \cos(\theta) \end{smallmatrix}\right) \in \operatorname{\mathrm{SL}}_2(\mathbb{R}).\end{align*} $$

Then, $\Theta (g, s_{\infty }U_R^1) = \theta _g(z) e^{i \kappa \theta }$ for some $\kappa \in 2\mathbb {Z}$ and a choice of $\theta _{g}$ from Equations (5.2)–(5.5). The precise choice and value of $\kappa $ depends on the family $\mathcal {G}$ under consideration and may be read off Table 1. (For our study of $\mathcal {F}^{-,\operatorname {hol}}$ , the precise choice of $\Theta $ depends upon the size of k.)

Table 1 Families and the choice of $\Theta $ .

Antipating the application of Lemma 4.1, we further require Fourier–Whittaker expansions of $\Theta $ at all of the cusps. They are expressable in terms of the $\theta _{g,\ell }$ from Equations (5.2)–(5.5) and a weight $\kappa $ , the choice of which are given by the Table 1 as before. We have

(5.6) $$ \begin{align} \Theta(g,(\tau_{\ell})_{\infty}s_{\infty}U_R^1) = \frac{\mu(\gcd(\ell,d_B))}{\ell} \theta_{g,\ell}(z)e^{i \kappa \theta}, \end{align} $$

for $\ell | d_B N$ with $\mu $ the Möbius function, $\tau _{\ell }$ as in Equation (4.2), and where $(\tau _{\ell })_{\infty }$ denotes the image of $\tau _{\ell }$ in the Archimedean coordinate of $\operatorname {\mathrm {SL}}_{2}(\mathbb {A})$ .

5.1.2. Jacquet–Langlands lifts

Set $U_R$ to be the image of

$$ \begin{align*} \left\{g\in \left(\begin{smallmatrix} \widehat{\mathbb{Z}} & \widehat{\mathbb{Z}} \\ d_BN \widehat{\mathbb{Z}} & \widehat{\mathbb{Z}} \end{smallmatrix}\right) \colon \det g \in \widehat{\mathbb{Z}}^\times\right\} \end{align*} $$

in $\operatorname {PGL}_2(\mathbb {A}_f)$ . This is a compact open subgroup of $\operatorname {PGL}_2(\mathbb {A}_f)$ . For each $\varphi $ in the families $\mathcal {F}^{-}, \mathcal {F}^{-,\operatorname {hol}}, \mathcal {F}^{+}_m, \mathcal {F}^{+,\operatorname {hol}}$ , we consider the Jacquet–Langlands transfer $\pi ^{\operatorname {\mathrm {JL}}}$ to $[\operatorname {PGL}_2]$ of the representation $\pi $ generated by $\varphi $ . In the case that G is split, we let $\pi ^{\operatorname {\mathrm {JL}}}=\pi $ . The space of vectors in $\pi ^{\operatorname {\mathrm {JL}}}$ that are $U_R$ -invariant and $K_{\infty }$ -isotypical of minimal nonnegative weight is one-dimensional [Reference Jacquet and LanglandsJL70, Reference CasselmanCas73]. We define the arithmetically normalized Jacquet–Langlands lift $\varphi ^{\operatorname {\mathrm {JL}}}\in L^2([\operatorname {\mathrm {SL}}_2])$ of $\varphi $ to be the $U_R^1$ -invariant restrictionFootnote ³ to $[\operatorname {\mathrm {SL}}_2]$ of a vector in this one-dimensional space, that has a Whittaker function at $\left (\begin {smallmatrix} y^{1/2} & xy^{-1/2} \\ & y^{-1/2} \end {smallmatrix}\right ) \in \operatorname {\mathrm {SL}}_2(\mathbb {R}) \hookrightarrow \operatorname {\mathrm {SL}}_2(\mathbb {A})$ given by

• ∘ $2 \sqrt {y} K_{it}(2 \pi y)e(x) \text { if } \varphi \in \mathcal {F}^{-}_{\frac {1}{4}+t^2}$ , and

• ∘ $y^{\frac {\kappa }{2}} e(x+iy)$ if $\varphi $ is in either of the families $\mathcal {F}^{-,\operatorname {hol}}$ , $ \mathcal {F}^{+}_m$ , $\mathcal {F}^{+,\operatorname {hol}}$ , where $\kappa =k$ , $2m+2$ , $k+2$ depends on the family as before.

The bounds by Hoffstein–Lockhart [Reference Hoffstein and LockhartHL94] and Iwaniec [Reference IwaniecIwa90] then imply the following bounds for the $L^2$ -norm of the arithmetically normalized Jacquet–Langlands lift (cf. [Reference Harcos and MichelHM06, (30), (31)]Footnote ⁴). One may also compare with the geometric normalization in [Reference Petrow and YoungPY19, Thm. 6.1]. If B is indefinite and $\varphi \in \mathcal {F}^{-}_{\frac {1}{4}+t^2}$ , we have

(5.7) $$ \begin{align} \|\varphi^{\operatorname{\mathrm{JL}}}\|_2^2 = (d_BN(1+|t|))^{o(1)} \cosh(\pi t)^{-1}. \end{align} $$

In the other cases, that is, when $\varphi $ lies in either of the families $\mathcal {F}^{-,\operatorname {hol}}$ , $ \mathcal {F}^{+}_m$ or $\mathcal {F}^{+,\operatorname {hol}}$ , we have

(5.8) $$ \begin{align} \|\varphi^{\operatorname{\mathrm{JL}}}\|_2^2 = (d_BN\kappa)^{o(1)} \frac{\Gamma(\kappa)}{(4 \pi)^{\kappa}}, \end{align} $$

where $\kappa $ depends on the family in accord with Table 1.

5.1.3. Explicit theta lifting

The key identity is summarized in the following proposition.

Proposition 5.1. Let $g \in [G]$ . Let $\mathcal {G}$ , $\Theta $ and $\kappa $ according to Table 1. Then, for $\varphi \in \mathcal {G}$ , we have

(5.9) $$ \begin{align} \frac { \langle \Theta(g,\cdot), \varphi^{ \operatorname{\mathrm{JL}}} \rangle } { \langle \varphi^{\operatorname{\mathrm{JL}}}, \varphi^{\operatorname{\mathrm{JL}}} \rangle } = \frac{|\varphi(g)|^2}{V_{d_B,N}}. \end{align} $$

Proof. The proof is carried out in the appendix. In short, Proposition A.16 implies that for $\varphi \in \mathcal {G}$ , the theta lift $\varphi _{\Phi }$ of $\varphi $ – defined in Equation (A.5), depending upon the precise family $\mathcal {G}$ – satisfies $\varphi _{\Phi }=(V_{d_B,N})^{-1}\varphi ^{\mathrm {JL}}$ . The claim then follows from Propositions A.15 and A.12.

5.2. $L^2$ -norms of theta kernels

5.2.1. Proofs of Propositions 3.2 through 3.5

The proofs are similar, so we discuss the first in detail and then explain the nonoverlapping parts of the rest. Recall the notation $\preccurlyeq $ from §2.3. We denote by $\Theta ^-, \Theta ^{-,\operatorname {hol}}, \Theta ^{+,m}, \Theta ^{+,\operatorname {hol}}$ the various functions ‘ $\Theta $ ’ defined as in §5.1.1.

Proof of Proposition 3.2.

Let $\mathcal {G}$ denote either $\mathcal {F}^{-}_{\le L}$ or $\mathcal {F}^{-,\operatorname {hol}}$ according to whether $k=0$ or $k \ge 2$ . By Proposition 5.1, we may write

$$ \begin{align*} \frac{\|\varphi^{\operatorname{\mathrm{JL}}}\|^2}{V_{d_B,N}} \left( |\varphi(g_1)|^2 - |\varphi(g_2)|^2 \right) = \langle \Theta^-(g_1,\cdot) - \Theta^-(g_2,\cdot), \varphi^{\operatorname{\mathrm{JL}}} \rangle. \end{align*} $$

By Bessel’s inequality, it follows that

(5.10) $$ \begin{align} \sum_{\varphi \in \mathcal{G}} \frac{\|\varphi^{\operatorname{\mathrm{JL}}}\|^2}{(V_{d_B,N})^2} \left(|\varphi(g_1)|^2-|\varphi(g_2)|^2\right)^2 \le \|\Theta^{-}(g_1,\cdot)-\Theta^{-}(g_2,\cdot)\|_2^2. \end{align} $$

We now bound the right-hand side of Equation (5.10) (and in particular, verify that it is finite). Since $\Theta ^{-}(g,\cdot )$ is $K_{\infty }$ -isotypical, Lemma 4.1 and Equation (5.6) give the bound

(5.11) $$ \begin{align} \ll \frac{1}{V_{1,d_BN}} \sum_{\ell|d_BN} \int_{\frac{\sqrt{3}}{2} \frac{\ell^2}{d_BN}}^{\infty} \int_0^{\ell} \frac{1}{\ell^2} |\theta_{g_1,\ell}^{-}(z)-\theta_{g_2,\ell}^{-}(z)|^2 dx \frac{dy}{y^2}. \end{align} $$

We insert the definition (5.2) into the inner integral and evaluate, giving

$$ \begin{align*} \frac{1}{\ell^2} \int_0^{\ell} |\theta_{g_1,\ell}^{-}(z)-\theta_{g_2,\ell}^{-}(z)|^2 dx = \frac{1}{\ell} y^{2+k} \sum_{n \in \frac{1}{\ell} \mathbb{Z}} \left| \sum_{i=1}^{2} (-1)^{i} \sum_{\substack{\gamma \in R(\ell;g_i) \\ \det(\gamma)=n}} X(\gamma)^k e^{-2\pi y P(\gamma)} \right|^2. \end{align*} $$

Note that the sum over i kills the contribution from $\gamma = 0$ , so we may omit that contribution in what follows. We separate the two sums by Cauchy–Schwarz and bound $X(\alpha )$ by $P(\alpha )^{\frac {1}{2}}$ , giving

(5.12) $$ \begin{align} \frac{1}{\ell^2} \int_0^{\ell} |\theta_{g_1,\ell}^{-}(z)-\theta_{g_2,\ell}^{-}(z)|^2 dx \ll \sum_{i=1}^{2} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell} \mathbb{Z}} \left| \sum_{\substack{ 0 \neq\gamma \in R(\ell;g_i) \\ \det(\gamma)=n}} P(\gamma)^{\frac{k}{2}} e^{-2\pi y P(\gamma)} \right|^2 y^{2+k}. \end{align} $$

We now treat the contributions from $i=1,2$ individually. We commence with the integral in the variable y. Let

$$ \begin{align*} Q(s,x)= \frac{1}{\Gamma(s)} \int_x^{\infty} t^{s} e^{-t} \frac{dt}{t} \le 1 \end{align*} $$

denote the normalized incomplete gamma function. Setting , we find

(5.13) $$ \begin{align} \nonumber \frac{(4 \pi)^{k+1}}{\Gamma(k+1)} & \int_{Y}^{\infty} \left| \sum_{\substack{0 \neq \gamma \in R(\ell;g) \\ \det(\gamma)=n}} P(\gamma)^{\frac{k}{2}} e^{-2\pi y P(\gamma)} \right|^2 y^{2+k} \frac{dy}{y^2} \\ &\nonumber = \sum_{\substack{0 \neq\gamma_1,\gamma_2 \in R(\ell;g) \\ \det(\gamma_1)=\det(\gamma_2)=n}} \frac{2}{P(\gamma_1)+P(\gamma_2)} \left( \frac{2\sqrt{P(\gamma_1)P(\gamma_2)}}{P(\gamma_1)+P(\gamma_2)} \right)^{k} Q\left(k+1, 2\pi Y (P(\gamma_1)+P(\gamma_2) ) \right) \\ &\qquad\ \qquad\qquad \le \sum_{\substack{0 \neq \gamma_1,\gamma_2 \in R(\ell;g) \\ \det(\gamma_1)=\det(\gamma_2)=n}} \frac{2}{P(\gamma_1)+P(\gamma_2)} Q\left(k+1, 2\pi Y (P(\gamma_1)+P(\gamma_2)) \right). \end{align} $$

Since $Q(s,x)$ is superpolynomially small in both s and x as soon as $x \gg s$ , we see by dyadically partitioning $\max _i\{P(\gamma _i)^{\frac {1}{2}}\}$ that Equation (5.13) is further bounded by

$$ \begin{align*} \ll_{A} \sum_{j} \frac{1}{T_j^2} \left(1+\frac{T_j^2Y}{k+1} \right)^{-A} \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(1, T_j) \\ \det(\gamma_1) = \det(\gamma_2)=n}} 1 \end{align*} $$

for any $A \ge 0$ , where $T_j=2^j, j \in \mathbb {Z}$ . By putting all of these estimates together, we arrive at

$$ \begin{align*} \frac{1}{(V_{d_B,N})^2}\sum_{\varphi \in \mathcal{G}} & \|\varphi^{\operatorname{\mathrm{JL}}}\|^2 \left(|\varphi(g_1)|^2-|\varphi(g_2)|^2\right)^2 \\ & \preccurlyeq_{A} \frac{\Gamma(k+1)(4\pi)^{-k}}{V_{1,d_BN}} \sum_{i=1}^{2} \sum_{\ell|d_B N} \frac{1}{\ell} \sum_j \frac{1}{T_j^2} \left(1+ \frac{\ell^2}{d_BN} \frac{T_j^2}{k+1} \right)^{-A} \sum _{\substack{ \gamma_1, \gamma_2 \in R(\ell;g_i) \cap \Omega^{\star}(1, T_j) \\ \det(\gamma_1) = \det(\gamma_2)}} 1 \end{align*} $$

for any $A \ge 0$ . Let us recall from Equation (4.1) that $V_{d_B,N}, V_{1,d_BN} = (d_BN)^{1+o(1)}$ and that for $\varphi \in \mathcal {F}^{-}_{\le L}$ we have $\|\varphi ^{\operatorname {\mathrm {JL}}}\| \succcurlyeq _{L} 1$ (see Equation (5.7)). In order to conclude the first part of the proposition, we note that the range of T may be limited from above to $\preccurlyeq (d_BN)^{\frac {1}{2}} \ell ^{-1}$ by the superpolynomial decay and any polynomial bound on the second moment matrix count, which was noted in §3.4 and is the subject of the remaining sections §6 through §10. The second part of the proposition follows along the same lines, but we need to use the bound $\|\varphi ^{\operatorname {\mathrm {JL}}}\|^2 \succcurlyeq \Gamma (k) (4 \pi )^{-k}$ for $\varphi \in \mathcal {F}^{-,\operatorname {hol}}$ instead (see Equation (5.8)).

Proof of Proposition 3.3.

We follow the recipe of the previous proof, only this time for the family $\mathcal {F}^{-,\operatorname {hol}}$ to which the theta function $\Theta ^{-,\operatorname {hol}}$ corresponds. As we shall see, the latter already possesses a finite $L^2$ -norm. Hence, we need not consider a difference of theta functions. After the initial steps, we arrive at

(5.14) $$ \begin{align} \nonumber \sum_{\varphi \in \mathcal{F}^{-,\operatorname{hol}}} \frac{\|\varphi^{\operatorname{\mathrm{JL}}}\|^2}{(V_{d_B,N})^2} |\varphi(g)|^4 & \ll \frac{1}{V_{1,d_BN}} \frac{k^2 \Gamma(k-1)}{(4 \pi )^k} \\ & \times \sum_{\ell |d_BN} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell} \mathbb{N}} \frac{1}{n} \left| \sum_{\substack{\gamma \in R(\ell;g)\\ \det(\gamma)=n}} \left( \frac{\det(\gamma)^{\frac{1}{2}}}{\overline{X(\gamma)}} \right)^k \right|^2 Q(k-1, 2 \sqrt{3} \pi n \tfrac{\ell^2}{d_BN}). \end{align} $$

We further simplify using the lower bound $\|\varphi ^{\operatorname {\mathrm {JL}}}\|^2 \succcurlyeq \Gamma (k) (4 \pi )^{-k}$ (see Equation (5.8)), the approximations $V_{d_B,N}, V_{1,d_BN} = (d_B N)^{1+o(1)}$ and the superpolynomial decay of normalized incomplete gamma function, as well as the identities

$$ \begin{align*} |X(\gamma)|^2 = \det(\gamma)+u(\gamma) \quad \text{ and } \quad 2u(\gamma)+\det(\gamma) = P(\gamma). \end{align*} $$

We obtain

(5.15) $$ \begin{align} \frac{1}{d_B N k}\sum_{\varphi \in \mathcal{F}^{-,\operatorname{hol}}} |\varphi(g)|^4 \preccurlyeq_A \sum_{\ell|d_BN} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell}\mathbb{N}} \frac{1}{n} \left(1+ \frac{\ell^2}{d_BN} \frac{n}{k} \right)^{-A} \left|\sum_{\substack{\gamma \in R(\ell;g) \\ \det(\gamma)=n}} \left(1+ \frac{u(\gamma)}{n} \right)^{-\frac{k}{2}} \right|^2, \end{align} $$

for any $A \ge 0$ . By the triangle inequality, we reduce to estimating similar expressions but with the sum over $\gamma $ restricted by one of the following conditions:

1. (i) $u(\gamma ) \le k^{-1+\varepsilon } \det (\gamma )$ ,

2. (ii) $k^{-1+\varepsilon }\det (\gamma ) \le u(\gamma ) \le \det (\gamma )$ or

3. (iii) $\det (\gamma ) \le u(\gamma )$ .

In Case (i), we bound $(1+u(\gamma )/n)^{-\frac {k}{2}} \le 1$ . Furthermore, we have $\det (\gamma ) \asymp P(\gamma )$ and $u(\gamma ) \ll k^{-1+\epsilon } P(\gamma )$ . Hence, after dyadically partitioning the range of $P(\gamma )^{\frac {1}{2}}$ , we arrive at

$$ \begin{align*} \preccurlyeq_A \sum_{\ell|d_BN} \frac{1}{\ell} \sum_j \frac{1}{T_j^2} \left(1+ \frac{\ell^2}{d_BN} \frac{T_j^2}{k} \right)^{-A} \sum _{\substack{\gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(k^{-1+\varepsilon}, T_j) \\ \det(\gamma_1) = \det(\gamma_2)>0}} 1. \end{align*} $$

In Case (ii), we use that $(1+u(\gamma )/n)^{-\frac {k}{2}}\le (1+k^{\varepsilon -1})^{-\frac {k}{2}}$ has superpolynomial decay in k (and hence also in $(d_BN)$ as $k \gg _{\eta } (d_BN)^{\eta }$ by assumption). As in Case (i), we have $ \det (\gamma ) \asymp P(\gamma )$ , but this time only $u(\gamma )\le P(\gamma )$ . We arrive at a contribution of

$$ \begin{align*} \preccurlyeq_{A,\eta} (kd_BN)^{-A} \sum_{\ell|d_BN} \frac{1}{\ell} \sum_j \frac{1}{T_j^2} \left(1+ \frac{\ell^2}{d_BN} \frac{T_j^2}{k} \right)^{-A} \sum _{\substack{\gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(1, T_j) \\ \det(\gamma_1) = \det(\gamma_2)>0}} 1. \end{align*} $$

In Case (iii), we bound

$$ \begin{align*} \left(1 + \frac{u(\gamma)}{n} \right)^{-\frac{k}{2}} \le 2^{-\frac{k}{4}} \left(1 + \frac{u(\gamma)}{n} \right)^{-\frac{k}{4}}. \end{align*} $$

The factor $2^{-\frac {k}{4}}$ we use for superpolynomial decay in $(kd_BN)$ as before. The other factor we use as follows

$$ \begin{align*} \frac{1}{n} \left(1+\frac{u(\gamma)}{n}\right)^{-1} \left(1+\frac{\ell^2}{d_BN}\frac{n}{k}\right)^{-\frac{k}{4}+1} \left(1+\frac{u(\gamma)}{n} \right)^{-\frac{k}{4}+1} \le \frac{1}{u(\gamma)} \left(1+\frac{\ell^2}{d_BN}\frac{u(\gamma)}{k}\right)^{-\frac{k}{4}+1}. \end{align*} $$

Hence, Equation (5.15) is bounded by

$$ \begin{align*}\preccurlyeq_{A,\eta} (kd_BN)^{-A} \sum_{\ell|d_BN} \frac{1}{\ell} \sum_{\substack{\gamma_1, \gamma_2 \in R(\ell;g)\\ \det(\gamma_1)=\det(\gamma_2)}} \frac{1}{u(\gamma_1)+u(\gamma_2)} \left(1+ \frac{\ell^2}{d_BN} \frac{u(\gamma_1)+u(\gamma_2)}{k} \right)^{-\frac{k}{4}+1} \\ \preccurlyeq_{A,\eta} (kd_BN)^{-A} \sum_{\ell|d_BN} \frac{1}{\ell} \sum_j \frac{1}{T_j^2} \left(1+\frac{\ell^2}{d_BN} \frac{T_j^2}{k} \right)^{-\frac{k}{4}+1} \sum _{\substack{\gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(1, T_j) \\ \det(\gamma_1) = \det(\gamma_2)>0}} 1, \end{align*} $$

where we have dyadically partitioned $\max _i\{P(\gamma _i)^{\frac {1}{2}}\} \asymp \sqrt {u(\gamma _1)+u(\gamma _2)}$ . The proof of the proposition is now concluded as in the previous case.

Proof of Proposition 3.4.

We treat the definite spherical case in the same spirit as the indefinite spherical case. We readily arrive at the estimate

(5.16) $$ \begin{align} &\nonumber \sum_{\varphi \in \mathcal{F}^{+}_{m}} \frac{\|\varphi^{\operatorname{\mathrm{JL}}}\|^2}{(V_{d_B,N})^2} \left( |\varphi(g_1)|^2 - |\varphi(g_2)|^2 \right)^2 \ll \frac{1}{V_{1,d_BN}} \frac{(2m+1)^2 \Gamma(2m+1)}{(4 \pi )^{2m+2}} \\ &\qquad\qquad\times \sum_{i=1}^2 \sum_{\ell |d_BN} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell} \mathbb{N}} \frac{1}{n} \left| \sum_{\substack{\gamma \in R(\ell;g_i)\\ \det(\gamma)=n}} P_m\left( \frac{|X(\gamma)|^2-u(\gamma)}{n} \right) \right|^2 Q(2m+1, 2 \sqrt{3} \pi n \tfrac{\ell^2}{d_BN}). \end{align} $$

We simplify the inequality by using the lower bound $\|\varphi ^{\operatorname {\mathrm {JL}}}\|^2 \succcurlyeq \Gamma (2m+2) (4 \pi )^{-2m-2}$ (see Equation (5.8)), the approximations $V_{d_B,N}, V_{1,d_BN}=(d_BN)^{1+o(1)}$ and the superpolynomial decay of the normalized incomplete Gamma function Q. We obtain

(5.17) $$ \begin{align} \nonumber \frac{1}{d_BN (m+1)} & \sum_{\varphi \in \mathcal{F}^{+}_{m}} \left( |\varphi(g_1)|^2 - |\varphi(g_2)|^2 \right)^2 \\ & \preccurlyeq_A \sum_{i=1}^2 \sum_{\ell |d_BN} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell} \mathbb{N}} \frac{1}{n} \left(1+\frac{\ell^2}{d_BN} \frac{n}{m+1} \right)^{-A} \left| \sum_{\substack{\gamma \in R(\ell;g_i)\\ \det(\gamma)=n}} P_m\left( \frac{|X(\gamma)|^2-u(\gamma)}{n} \right) \right|^2, \end{align} $$

for any $A \ge 0$ . We proceed further by appealing to the Bernstein inequality [Reference BernsteinBer31] for the Legendre polynomials:

(5.18) $$ \begin{align} P_m(t) \le \min\left\{1 , \sqrt{\frac{2}{\pi m}} \frac{1}{(1-t^2)^{\frac{1}{4}}} \right\}, \quad \text{for } |t| \le 1. \end{align} $$

We recall that $\det (\gamma )= |X(\gamma )|^2 + u(\gamma )$ so that, with $\gamma $ and n as in the above sum,

$$ \begin{align*} t := \frac{|X(\gamma)|^2 - u(\gamma)}{n} = \frac{|X(\gamma)|^2 - u(\gamma)}{|X(\gamma)|^2 + u(\gamma) }, \quad 1 - t^2 = \frac{4 |X(\gamma)|^2 \cdot u (\gamma) }{n^2} \ge 0. \end{align*} $$

Dyadically partitioning $P(\gamma )^{\frac {1}{2}}=\det (\gamma )^{\frac {1}{2}}$ , we conclude that Equation (5.17) is bounded by

(5.19) $$ \begin{align} \nonumber \preccurlyeq_{A} \sum_{i=1}^2 \sum_{\ell|d_BN} \frac{1}{\ell} & \sum_{j} \frac{1}{T_j^2} \left(1+\frac{\ell^2}{d_BN} \frac{T_j^2}{m+1} \right)^{-A} \\ & \times \sum_{T_j^2 \le n < 4T_j^2} \left( \sum_{\substack{\gamma \in R(\ell;g_i) \\ \gamma \in \Omega^{\star}(1,2T_j)-\Omega^{\star}(1,T_j) \\ \det(\gamma)=n}} \min\left\{1, \frac{1}{(m+1)^{\frac{1}{2}}} \frac{T_j}{(|X(\gamma)|^2\cdot u(\gamma))^{\frac{1}{4}}} \right\} \right)^2, \end{align} $$

where $T_j = 2^j$ as before. The minimum in Equation (5.19) lies between $\asymp (m+1)^{-\frac {1}{2}}$ and $1$ . Let us consider the $\gamma \in \Omega ^{\star }(1,2T_j)-\Omega ^{\star }(1,T_j)$ for which

(5.20) $$ \begin{align} \min\left\{1, \frac{1}{(m+1)^{\frac{1}{2}}} \frac{T_j}{(|X(\gamma)|^2\cdot u(\gamma))^{\frac{1}{4}}} \right\} \asymp \frac{1}{(m+1)^{\frac{1}{2}}} \frac{1}{\delta^{\frac{1}{4}}}, \end{align} $$

for some given $\delta $ with $1/(m+1)^2 \ll \delta \le 1$ . In particular, $|X(\gamma )|^2 \cdot u(\gamma ) \ll \delta T_j^4$ . Since $|X(\gamma )|^2+u(\gamma )=P(\gamma ) \asymp T_j^2$ , both cannot be simultaneously small. Hence,

$$ \begin{align*} \min \{|X(\gamma)|^2, u(\gamma)\} = \frac{|X(\gamma)|^2 \cdot u(\gamma)}{ \max\{|X(\gamma)|^2, u(\gamma)\}} \ll \delta T_j^2. \end{align*} $$

Thus, after replacing $\delta $ with its multiple by a scalar of the form $\asymp 1$ if needed (which has no affect on Equation (5.20)), we may assume that $\min \{|X(\gamma )|^2, u(\gamma )\} \le \delta (2T_j)^2$ , that is, that $\gamma $ lies in either $\Omega ^{\star }(\delta ,2T_j)$ or $\Psi ^{\star }(\delta ,2T_j)$ . We now consider dyadic scales $\delta _a$ of $\delta $ ’s between $\asymp 1/(m^2+1)$ and $1$ . The just mentioned arguments then allow us to bound second line in Equation (5.19) by

$$ \begin{align*} \sum_{T_j^2 \le n < 4T_j^2} \Biggl( \sum_{a} \sum_{\substack{\gamma \in R(\ell;g_i) \\ \gamma \in \Omega^{\star}(\delta_a,2T_j)\cup \Psi^{\star}(\delta_a,2T_j) \\ \det(\gamma)=n}} (m+1)^{-\frac{1}{2}}\delta_a^{-\frac{1}{4}} \Biggr)^2. \end{align*} $$

There are at most $ \preccurlyeq 1$ dyadic scales in the range $1/(m^2+1) \ll \delta \le 1$ . Thus, after applying Cauchy–Schwarz in order to pull out the sum over $\delta _a$ , we bound Equation (5.19) by

(5.21) $$ \begin{align} \preccurlyeq_{A} \sum_{i=1}^2 \sum_{\ell|d_BN} \frac{1}{\ell} \sum_{j} \frac{1}{T_j^2} \left(1+\frac{\ell^2}{d_BN} \frac{T_j^2}{m+1} \right)^{-A} \sum_a \sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g_i) \\ \gamma_1,\gamma_2 \in \Omega^{\star}(\delta_a,2T_j) \cup \Psi^{\star}(\delta_a,2T_j))\\ \det(\gamma_1)=\det(\gamma_2)}} (m+1)^{-1} \delta_a^{-\frac{1}{2}}. \end{align} $$

The proof is once more concluded as it was for the first proposition.

Proof of Proposition 3.5.

One final time we iterate the initial steps for the holomorphic family $\mathcal {F}^{\operatorname {hol}}$ in the definite case. We are again in the situation where the theta kernel $\Theta ^{+,\operatorname {hol}}$ has finite $L^2$ -norm, so we obtain, without having to take differences,

(5.22) $$ \begin{align} \nonumber \sum_{\varphi \in \mathcal{F}^{+,\operatorname{hol}}} \frac{\|\varphi^{\operatorname{\mathrm{JL}}}\|^2}{(V_{d_B,N})^2} |\varphi(g)|^4 & \ll \frac{1}{V_{1,d_BN}} \frac{(k+1)^2 \Gamma(k+1)}{(4 \pi )^{k+2}} \\ & \times \sum_{\ell |d_BN} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell} \mathbb{N}} \frac{1}{n} \left| \sum_{\substack{\gamma \in R(\ell;g)\\ \det(\gamma)=n}} \left( \frac{X(\gamma)^2}{n} \right)^{\frac{k}{2}} \right|^2 Q(k+1, 2 \sqrt{3} \pi n \tfrac{\ell^2}{d_BN}). \end{align} $$

We simplify this estimate using the lower bound $\|\varphi ^{\operatorname {\mathrm {JL}}}\|^2 \succcurlyeq \Gamma (k+2) (4 \pi )^{-k-2}$ (see Equation (5.8)), the approximations $V_{d_B,N}, V_{1,d_BN}=(d_BN)^{1+o(1)}$ , the superpolynomial decay of the normalized incomplete gamma function Q and the identity $|X(\gamma )|^2=\det (\gamma )-u(\gamma )$ . We thereby obtain

(5.23) $$ \begin{align} \frac{1}{d_BN k} \sum_{\varphi \in \mathcal{F}^{+,\operatorname{hol}}} |\varphi(g)|^4 \preccurlyeq_A \sum_{\ell |d_BN} \frac{1}{\ell} \sum_{n \in \frac{1}{\ell} \mathbb{N}} \frac{1}{n} \left(1+\frac{\ell^2}{d_BN} \frac{n}{k} \right)^{-A} \left| \sum_{\substack{\gamma \in R(\ell;g)\\ \det(\gamma)=n}} \left( 1- \frac{u(\gamma)}{n} \right)^{\frac{k}{2}} \right|^2, \end{align} $$

for any $A \ge 0$ . We dyadically partition $P(\gamma )^{\frac {1}{2}}=\det (\gamma )^{\frac {1}{2}}$ and distinguish the two cases:

1. (i) $u(\gamma ) \le k^{-1+\varepsilon } \det (\gamma )$ , and

2. (ii) $k^{-1+\varepsilon }\det (\gamma ) \le u(\gamma )$ .

We separate them by the triangle inequality. Using the inequality $(1-u(\gamma )/n)\le 1$ , we see that the contribution of the first case is bounded by

$$ \begin{align*} \preccurlyeq_A \sum_{\ell|d_BN} \frac{1}{\ell} \sum_j \frac{1}{T_j^2} \left(1+ \frac{\ell^2}{d_BN} \frac{T_j^2}{k} \right)^{-A} \sum _{\substack{\gamma_1, \gamma_2 \in R(\ell;g) \cap \Omega^{\star}(k^{-1+\varepsilon}, T_j) \\ \det(\gamma_1) = \det(\gamma_2)}} 1, \end{align*} $$

where $T_j=2^j, j \in \mathbb {Z}$ . In the second case, we see that $(1-k^{\varepsilon -1})^{\frac {k}{2}}$ enjoys superpolynomial decay in k. The contribution of the second case is thus bounded by

$$ \begin{align*} \preccurlyeq_{A} k^{-A} \sum_{\ell|d_BN} \frac{1}{\ell} \sum_j \frac{1}{T_j^2} \left(1+ \frac{\ell^2}{d_BN} \frac{T_j^2}{k} \right)^{-A} \sum _{\substack{\gamma_1, \gamma_2 \in R(\ell; g )\cap \Omega^{\star}(1, T_j) \\ \det(\gamma_1) = \det(\gamma_2)}} 1. \end{align*} $$

The proof is now concluded as it was for the previous propositions.

6.1. Bounds on successive minima

Lemma 6.2. Let $z\in \mathbb {H}$ with maximal imaginary part under the orbit of the Atkin–Lehner operators $A_0(N)$ of $\Gamma _0(N)$ with N squarefree. Then, we have

$$ \begin{align*} \Im(z) \ge \frac{\sqrt{3}}{2N} \quad \text{ and } \quad |cz+d|^2 \ge \frac{(c,N)}{N} \end{align*} $$

for any $(c,d) \in \mathbb {Z}^2$ distinct from $(0,0)$ .

6.2. Lattice counting

Lemma 6.4. Let $\mathcal {K}\subseteq \mathbb {R}^n$ be a closed convex $0$ -symmetric set of positive volume. Let $\Lambda \subset \mathbb {R}^n$ be a lattice, and let $ \lambda _1 \le \lambda _2 \le \dots \le \lambda _n$ denote the successive minima of $\mathcal {K}$ on $\Lambda $ . Then

$$ \begin{align*} |\mathcal{K} \cap \Lambda| \asymp_n \prod_{i=1}^n \left( 1+\frac{1}{\lambda_i} \right). \end{align*} $$

Proof. The lower bound follows from van der Corput’s generalization of Minkowski’s first theorem [Reference van der CorputvdC36]. It states that for $\mathcal {K}' \subset \mathbb {R}^d$ a closed convex $0$ -symmetric set and $\Lambda \subset \mathbb {R}^d$ a lattice, one has

(6.1) $$ \begin{align} |\mathcal{K}'\cap \Lambda'|+1 \ge |\mathrm{int}\,\mathcal{K}'\cap \Lambda'|+1 \ge 2^{1-d} \frac{\operatorname{\mathrm{vol}}(\mathcal{K}')}{\operatorname{\mathrm{vol}}(\mathbb{R}^d / \Lambda')}. \end{align} $$

Let d be the largest integer such that $\lambda _d \le 1$ . Let $v_i\in \Lambda $ , for $i=1,\dots ,d$ , be a set of linearly independent vectors such that $\lambda _i^{-1}v_i \in \mathcal {K}$ . Let $\mathcal {K}'$ be the convex hull of the vectors $\pm \lambda _i^{-1}v_i$ and $\Lambda '$ the span of the vectors $v_i$ . In particular, $\mathcal {K} '$ is nonempty, hence $0 \in \mathcal {K}' \cap \Lambda '$ , and so

$$ \begin{align*} 2 |\mathcal{K}' \cap \Lambda'| \ge |\mathcal{K}' \cap \Lambda'| + 1. \end{align*} $$

Using Equation (6.1), it follows now that

$$ \begin{align*} |\mathcal{K}\cap \Lambda| \ge |\mathcal{K}'\cap \Lambda'| \ge 2^{-d} \frac{\operatorname{\mathrm{vol}}(\mathcal{K}')}{\operatorname{\mathrm{vol}}(\mathbb{R}^d / \Lambda')} = \frac{1}{d!} \prod_{i=1}^d \frac{1}{\lambda_i}. \end{align*} $$

For the upper bound, we refer to [Reference Betke, Henk and WillsBHW93, Prop. 2.1].

Lemma 6.5. Let $\Lambda \subset \mathbb {R}^2$ be a lattice of rank $2$ and $B\subseteq \mathbb {R}^2$ a ball of radius R (not necessarily centred at $0$ ). If $\lambda _1 \le \lambda _2$ are the successive minima of $\Lambda $ , then

$$ \begin{align*} |B \cap \Lambda| \ll 1+\frac{R}{\lambda_1}+\frac{R^2}{\lambda_1\lambda_2}. \end{align*} $$

7.1. Quadratic preliminaries

Let F be a non-Archimedean local field of characteristic $\neq 2$ . Let $E/F$ be a separable quadratic extension, thus E is either the split quadratic extension $F \oplus F$ or a quadratic field extension. We write $\mathfrak {o}$ (resp. $\mathfrak {o}_E$ ) for the ring of integers in F (resp. E), $x \mapsto \bar {x}$ for the canonical involution on E and

$$ \begin{align*} \operatorname{\mathrm{nr}}(x) = x \bar{x}, \quad \operatorname{\mathrm{tr}}(x) = x + \bar{x} \end{align*} $$

for the norm and trace. Recall that the different ideal $\mathfrak {d}$ for this extension is the smallest $\mathfrak {o}_E$ -ideal for which $\operatorname {\mathrm {tr}}(\mathfrak {d}^{-1}) \subseteq \mathfrak {o}$ , and in fact $\mathfrak {d}^{-1} = \{ x \in E: \operatorname {\mathrm {tr}}(x \mathfrak {o}_E) \subseteq \mathfrak {o} \}$ . If $E/F$ is split or unramified, then $\mathfrak {d} = \mathfrak {o}_E$ .

We may regard E as a two-dimensional vector space over F.

Let $q : E \rightarrow F$ be a nondegenerate binary quadratic form with the property that for all $e,x \in E$ , we have $q(e x) =\operatorname {\mathrm {nr}}(e) q(x)$ . In other words, q is an F-multiple of $\operatorname {\mathrm {nr}}$ , specifically $q = q(1)\operatorname {\mathrm {nr}}$ .

For $x,y \in E$ , we set so that $q(x) = \langle x, x \rangle /2$ .

Let $\mathfrak {a} \subset E$ be a fractional $\mathfrak {o}_E$ -ideal. Write $\mathfrak {a}^\vee $ for the dual of $\mathfrak {a}$ with respect to the quadratic form q, that is, .

Let $\mathfrak {n}$ denote the fractional $\mathfrak {o}$ -ideal generated by $q(\mathfrak {a})$ .

7.2. Quaternionic preliminaries: general case

Let F be a non-Archimedean local field, let B be a quaternion F-algebra and let E be a separable quadratic F-subalgebra of B. We equip B with the quadratic form $q : B \rightarrow F$ given by the reduced norm, whose bilinearization $\langle \, , \rangle $ as above is described by the reduced trace and the main involution on B via the formula $\langle x, y \rangle = \operatorname {\mathrm {tr}}(x \bar {y})$ . We have a canonical decomposition $B = E \oplus E^\perp $ , where $E^\perp = \{ x \in B : \langle x, y \rangle = 0 \text { for all } y \in E \}$ .

Let $\mathfrak {o}$ and $\mathfrak {o}_E$ denote the respective maximal orders of F and E. We write $\mathfrak {d}$ for the different ideal, as before.

Let us say that an order R in B is E-adapted if it is of the form $R = \mathfrak {o}_E \oplus \mathfrak {a}$ for some $\mathfrak {o}_E$ -submodule $\mathfrak {a}$ of $E^\perp $ (for the action by either left or right multiplication – it doesn’t matter which because they are conjugates of each other).

Consider such an order R. Its traceless submodule is given by

$$\begin{align*}R^0 = \mathfrak{o}_E^0 \oplus \mathfrak{a}. \end{align*}$$

We aim to compute the dual lattice $(R^0)^\vee $ with respect to q. To that end, it suffices to dualize each summand in the above decomposition because $(R^0)^\vee = (\mathfrak {o}_E^0)^\vee \oplus \mathfrak {a}^\vee $ . We generally have ${(\mathfrak {d}^{-1})^0 \subseteq (\mathfrak {o}_E^0)^\vee \subseteq \frac {1}{2} (\mathfrak {d}^{-1})^0}$ . If E is unramified or split, then $(\mathfrak {o}_E^0)^\vee = \frac {1}{2} \mathfrak {o}_E^0 = \frac {1}{2} (\mathfrak {d}^{-1})^0$ . On the other hand, we can compute $\mathfrak {a}^\vee $ using the results of the previous section. Indeed, the choice of any invertible element $j \in E^\perp $ defines an isomorphism $E \rightarrow E^\perp $ , $x \mapsto x j$ . Transporting q and $\mathfrak {a}$ via the inverse of this isomorphism gives us a fractional ideal in E and a quadratic form on E that satisfy the hypotheses of that section. We obtain

$$ \begin{align*} \mathfrak{a}^\vee = \mathfrak{d}^{-1} \mathfrak{n}^{-1} \mathfrak{a}, \end{align*} $$

where $\mathfrak {n}$ is the integral $\mathfrak {o}$ -ideal characterized by either of the following properties:

• ∘ $\mathfrak {n}$ is generated by $q(\mathfrak {a})$ .

• ∘ $\mathfrak {D} \mathfrak {n}^2$ is the discriminant ideal of $(q,\mathfrak {a})$ .

Let $\mathcal {D}$ denote the discriminant ideal of R. The discriminant ideal of the summand $(q,\mathfrak {o}_E)$ is $\mathfrak {D}$ . Since the discriminant ideal is multiplicative with respect to direct sums, we obtain

(7.1) $$ \begin{align} \mathcal{D} = \mathfrak{D}^2 \mathfrak{n}^2. \end{align} $$

We may regard this last identity as a formula for $\mathfrak {n}$ .

7.3. Quaternionic preliminaries: unramified case

Let us restrict henceforth to the case that $E/F$ is unramified. (The ramified case would be relevant for studying, for example, the ‘minimal vectors’ considered in [Reference Hu, Nelson and SahaHNS19, Reference Hu and NelsonHN18, Reference SahaSah20].)

The above formula then simplifies to

$$\begin{align*}\mathfrak{n}^2 = \mathcal{D},\end{align*}$$

and we obtain

$$\begin{align*}\mathfrak{a}^\vee / \mathfrak{a} \cong (\mathfrak{o}/\mathfrak{n})^2.\end{align*}$$

There are three possibilities for $F \subset E \subset B$ , up to isomorphism:

1. (i) B is split and $E \cong F \oplus F$ . We may then find an isomorphism $B \cong \operatorname {\mathrm {Mat}}_{2 \times 2}(F)$ under which E identifies with the diagonal subalgebra.

2. (ii) B is split and $E/F$ is the unique unramified quadratic field extension.

3. (iii) B is nonsplit and $E/F$ is the unique unramified quadratic field extension.

In Case (i), the E-adapted orders R are just the Eichler orders. An Eichler order of level $\mathfrak {q}$ has discriminant $\mathfrak {q}^2$ (cf. [Reference VoightVoi18, §23.4]), hence the ideal $\mathfrak {n}$ defined above is $\mathfrak {q}$ .

Case (ii) corresponds to another type of ‘minimal vectors’, which we do not consider in this paper.

In Case (iii), the maximal order R is E-adapted and has discriminant ideal $\mathfrak {p}^2$ (cf. [Reference VoightVoi18, §15.2.11, §23.4]), hence $\mathfrak {n} = \mathfrak {p}$ .

7.4. Bounds for commutators of elements of $R^0$

Proof. By the previous discussion, we may write $\gamma _i=\alpha _i+\beta _i$ with $\alpha _i \in \mathfrak {o}_E$ and $\beta _i \in \mathfrak {a}$ if $\gamma _i \in R$ , respectively $\mathfrak {n}^{-1}\mathfrak {a}$ if $\gamma _i \in R^{\vee }$ . We have $[\alpha _1,\alpha _2]=0$ and $[\alpha _1,\beta _2] = 2\alpha _1\beta _2 \in \mathfrak {a}$ , respectively $\in \mathfrak {n}^{-1}\mathfrak {a}$ . Lastly, we have $\mathfrak {a}^2=\mathfrak {n} \mathfrak {a}^{\vee } \mathfrak {a} = \mathfrak {n}$ and $(\mathfrak {a}^{\vee })^2=\mathfrak {n}^{-1} \mathfrak {a} \mathfrak {a}^{\vee } = \mathfrak {n}^{-1}$ . Hence,

$$ \begin{align*} [\beta_1,\beta_2] = \beta_1\beta_2 - \beta_2\beta_1 \in \mathfrak{n}, \text{ respectively } \mathfrak{n}^{-1}. \end{align*} $$

By the bilinearity of the commutator, we obtain Claim (ii) and subsequently Claim (iii). Similarly, we have $q([\beta _1,\beta _2]) \in \mathfrak {n}^2$ for $\beta _i \in \mathfrak {a}$ . Thus, by orthogonality, we have

$$ \begin{align*} q([\gamma_1,\gamma_2]) = q(2\alpha_1\beta_2-2\alpha_2\beta_1+[\beta_1,\beta_2]) = q(2\alpha_1\beta_2-2\alpha_2\beta_1)+q([\beta_1,\beta_2]) \in \mathfrak{n}, \end{align*} $$

for $\gamma _i \in R$ , which gives Claim (i).

8.1. Non-Archimedean invariants

Let $L \subseteq V$ be a lattice, that is, a $\mathbb {Z}$ -submodule whose rank is the dimension of V. Define

• ∘ the content C of L to be the greatest common divisor of $q(L)$ ,

• ∘ the level N of L to be the reciprocal of the content of the dual lattice $L^\vee $ and

• ∘ the (unsigned) discriminant $\Delta $ of L to be the absolute value of the determinant of the Gram matrix of q on L. (The discriminant of a quadratic form is traditionally defined without taking absolute values, but the sign will not matter for us.)

To get acquainted with these quantities, consider for instance the case that L admits a basis $e_1, \dotsc , e_n$ with respect to which q is given by the diagonal quadratic form $q(\sum x_i e_i) = \frac {1}{2} \sum a_i x_i^2$ for some nonzero rational numbers $a_1,\dotsc ,a_n$ . Then

$$\begin{align*}C = \tfrac{1}{2} \gcd(a_1,\dotsc,a_n),\end{align*}$$

$$\begin{align*}N = 2/\gcd(1/a_1,\dotsc,1/a_n) = 2 \cdot \operatorname{lcm}(a_1,\dotsc,a_n),\end{align*}$$

$$\begin{align*}\Delta = |a_1 \dotsb a_n|.\end{align*}$$

We may relate the invariants attached to homothetic lattices: the effect of the substitution $L \mapsto m L$ for a nonzero rational scalar m is

$$\begin{align*}C \mapsto m^2 C, \quad \quad \quad N \mapsto m^2 N, \quad \quad \quad \Delta \mapsto m^{2 n} N.\end{align*}$$

8.2. Archimedean invariants

Next, write $V_{\mathbb {R}} := V \otimes _{\mathbb {Q}} \mathbb {R}$ and let $Q : V_{ \mathbb {R} } \rightarrow \mathbb {R}$ be a positive-definite quadratic form. We may find a basis $e_1,\dotsc ,e_n$ of $V _{ \mathbb {R} }$ so that, writing $x = \sum x_i e_i$ , we have $Q(x) = \frac {1}{2}\sum x_i^2$ and $q(x) = \frac {1}{2} \sum a_i x_i^2$ for some nonzero real numbers $a_1,\dotsc ,a_n$ . The dual $Q^\vee $ of Q with respect to q may be defined by $Q^{\vee }(x) = \frac {1}{2} \sum a_i^2 x_i^2$ . We note that in the scaled coordinates $y = \sum a_i^{-1} y_i e_i$ , we have $Q^{\vee }(y) = \frac {1}{2} \sum y_i^2$ and $q(y) = \frac {1}{2} \sum a_i^{-1} y_i^2$ . The Gram matrix relative to q of Q is defined to be the diagonal matrix with entries $(a_1,\dotsc ,a_n)$ .

Define

• ∘ the content C of Q to be the infimum of the ratio $Q/|q|$ over the set where $q \neq 0$ ,

• ∘ the level N of Q to be the reciprocal of the content of the dual $Q^{\vee }$ of Q and

• ∘ the discriminant $\Delta $ of Q to be the absolute value of the reciprocal of the determinant of the Gram matrix relative to q of Q.

The invariants of Q may be described in terms of coordinates as above by

$$\begin{align*}C = 1/\max(|a_1|,\dotsc,|a_n|) = \min(1/|a_1|,\dotsc,1/|a_n|),\end{align*}$$

$$\begin{align*}N = \max(1/|a_1|,\dotsc,1/|a_n|) = 1/\min(|a_1|,\dotsc,|a_n|).\end{align*}$$

$$\begin{align*}\Delta = 1/ |a_1 \dotsb a_n|.\end{align*}$$

These again behave predictably under homotheties: If we substitute ] for some nonzero real scalar m (which has the effect of multiplying the Q-unit ball by m), then the coefficients transform like $a_i \mapsto m^2 a_i$ and hence the invariants like

$$\begin{align*}C \mapsto m^{-2} C, \quad \quad \quad N \mapsto m^{-2} N, \quad \quad \quad \Delta \mapsto m^{-2 n} \Delta.\end{align*}$$

For later reference, it will be convenient to explicate the definition of ‘level’ in terms of matrices. To that end, we note first that for any basis $e_1,\dotsc ,e_n$ of V, we may find symmetric matrices S and P that represent q and Q in the sense that, for example, $q(\sum x_i e_i) = \frac {1}{2} \sum \sum x_i S_{i j} x_j$ . By singular value decomposition, we may find nonsingular matrices A and D, with D diagonal, so that

(8.1) $$ \begin{align} P = A^t A \ \ \text{ and } \ \, S = A^t D A. \end{align} $$

The level of Q is then the operator norm of the matrix $D^{-1}$ . For instance, if we choose our basis so that $Q(\sum x_i e_i) = \frac {1}{2} \sum x_i^2$ and $q(\sum x_i e_i) = \frac {1}{2} \sum d_i x_i^2$ , then the level of Q is $\max |d_i|^{-1}$ .

8.3. Duality

We note that replacing L (resp. Q) with its dual $L^\vee $ (resp. $Q^\vee $ ) has the following effect on the invariants:

(8.2) $$ \begin{align} (C, N, \Delta) \mapsto (1/N, 1/C, 1/\Delta). \end{align} $$

In what follows, this relation allows us to reduce slightly the number of computations required. For instance, we can read off the invariants of the dual of an Eichler order (or its traceless submodule) from those of the Eichler order itself.

8.4. Adelic invariants

Let $(Q,L)$ be a pair consisting of a positive-definite quadratic form Q and a lattice L as above. We define the content (resp. level, discriminant) of the pair to be the product of the corresponding invariants of Q and L.

We note that the invariants $C, N, \Delta $ assigned to the pair $(Q,L)$ are invariant by rational homotheties, that is, replacing L by $m L$ and Q by $Q_m$ for the same nonzero rational scalar m, and also under automorphisms of V that preserve q.

Furthermore, the discriminant of the pair $(Q,L)$ is the same as the determinant of the Gram matrix of Q with respect to a $\mathbb {Z}$ -basis of L, as the discriminant of Q is nothing but the inverse of the determinant of the matrix D in the singular value decomposition (8.1).

8.5. Statement of result

Proposition 8.4. Let V be an n-dimensional $\mathbb {Q}$ -vector space. Let $q : V \rightarrow \mathbb {Q}$ be an anisotropic quadratic form. Let $L \subset V$ be a lattice. Let $Q : V_{\mathbb {R}} \rightarrow \mathbb {R}$ be a positive-definite quadratic form. Let $\lambda _1 \le \dotsb \le \lambda _n$ denote the successive minima of Q on L (see Definition 6.1). Let $C, N, \Delta $ denote the content, level and discriminant of the pair $(Q,L)$ . Then,

1. (i) $\lambda _1 \ge C^{1/2}$ ,

2. (ii) $\lambda _1 \dotsb \lambda _n \asymp \Delta ^{1/2}$ and

3. (iii) $\lambda _1 \dotsb \lambda _{n-1} \gg (\Delta / N)^{1/2}$ .

In particular, for $n = 3$ , we have for all $X> 0$ that

$$\begin{align*}| \{ v \in L : Q(v) \le X^2 \}| \ll 1 + \frac{X}{\sqrt{C}} + \frac{X^2}{\sqrt{\Delta/N}} + \frac{X^3}{\sqrt{\Delta}}.\end{align*}$$

Proof. We follow the basic strategy of Blomer–Michel [Reference Blomer and MichelBM13, §3, §4], who established the corresponding result for q positive-definite and $Q=q$ .

Let v be a nonzero element of L. Since L is anisotropic, we have $q(v) \neq 0$ . Let $C_Q, C_L$ denote the content of Q, respectively L. By the definition of the content, we have $C_L|q(v)$ and $Q \ge C_Q |q|$ . Thus, $Q(v) \ge C_Q |q(v)| \ge C_QC_L = C$ , giving Claim (i).

By Lemma 6.3, we may find a basis $e_1,\dotsc ,e_n$ of L so that the submodules have covolume $\asymp \prod _{j \le m} \lambda _j$ in their real span, with volume defined using the restriction of Q. On the other hand, that covolume is the square root of the Gram determinant of Q on $L_m$ . Write $\det (Q,L_m)$ and $\det (q,L_m)$ for the respective Gram determinants of Q and q. Then

(8.3) $$ \begin{align} \prod_{j \le m} \lambda_j \asymp \det(Q,L_m)^{\frac{1}{2}}. \end{align} $$

Since $\det (Q,L_n) = \Delta $ by the remark in §8.4, the case $m=n$ of this estimate gives Claim (ii).

For the proof of Claim (iii), observe first that in view of Claim (ii), it is equivalent to check that $\lambda _n \ll (N_L N_Q)^{1/2}$ , where $N_L$ , respectively $N_Q$ , is the level of L, respectively Q. To that end, write $P = A^t A$ and $S = A^t D A$ for the matrices of Q and q, as in §8.2, and consider the final matrix entry $(P^{-1})_{n n}$ of the inverse of P. Cramer’s rule expresses $(P^{-1})_{n n} = \det (Q,L_{n-1}) / \det (Q, L_n)$ , so by the cases $m=n-1,n$ of Equation (8.3), we have $(P^{-1})_{n n} \asymp 1/\lambda _n^2$ . On the other hand, since $N_Q$ bounds the operator norm of $D^{-1}$ , we have

$$\begin{align*}|\,(S^{-1})_{n n}\,| = |\,\langle A^{-t} e_n, D^{-1} A^{-t} e_n \rangle\,| \le N_Q \langle A^{-t} e_n, A^{-t} e_n\rangle = N_Q (P^{-1})_{n n}.\end{align*}$$

Cramer’s rule likewise expresses $(S^{-1})_{n n} = \det (q,L_{n-1}) / \det (q,L_n)$ as a ratio of Gram determinants. Since q is anisotropic, both determinants are nonzero. Since $2N_L (S^{-1})_{n n} \in \mathbb {Z}$ , it follows that $1/(2N_L) \le |(S^{-1})_{n n}|$ . Thus, $1 / \lambda _n^2 \asymp (P^{-1})_{n n} \ge 1/(2N_L N_Q)$ , giving the required estimate.

10.1. Bounds for representation numbers of binary quadratic forms