Theta functions, fourth moments of eigenforms and the sup-norm problem II

Abstract

Let f be an $L^2$-normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate

$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

with absolute implied constant. For a cuspidal Maaß newform $\varphi $ of eigenvalue $\lambda $ on such a surface, we prove that

$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

We establish analogous estimates in the setting of definite quaternion algebras.

1. Introduction

Let $\Gamma \backslash \mathbb {H}$ be a finite volume hyperbolic surface. A basic problem in quantum chaos is to understand the limiting behavior of $L^2$ -normalized Laplace eigenfunctions $\varphi $ on $\Gamma \backslash \mathbb {H}$ . This behavior can be quantified through weak limits of $L^2$ -masses (‘quantum ergodicity’), bounds for $L^p$ -norms and so forth. We consider in this paper the sup-norm problem, which consists of bounding the supremum or $L^\infty $ -norm of an $L^2$ -normalized eigenfunction $\varphi $ with respect to the eigenvalue $\lambda _\varphi $ and/or the geometry of the underlying manifold $\Gamma \backslash \mathbb {H}$ . A general bound in this direction, due to Bérard [Bér77], asserts that

(1.1) $$ \begin{align} \|\varphi \|_{\infty} \ll_{\Gamma} (1+|\lambda_{\varphi}|)^{\frac{1}{4}} / \log(2 + |\lambda_\varphi|). \end{align} $$

Here and henceforth, $A \ll B$ means that there is a constant C such that $|A| \le CB$ ; we allow C to depend on any subscripts of $\ll $ and write $\varepsilon $ for an arbitrary, but sufficiently small, positive constant, which may change from line to line.

Stronger bounds have been established in the arithmetic case that

• ∘ $\Gamma \backslash \mathbb {H}$ is an arithmetic manifold, such as the modular surface $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ or a congruence cover, and

• ∘ $\varphi $ is a Hecke–Maaß form, that is, an eigenfunction not only of the Laplacian but also of the Hecke operators.

The pioneering result in that case is due to Iwaniec–Sarnak [IS95], who showed for congruence lattices $\Gamma $ that

(1.2) $$ \begin{align} \|\varphi\|_{\infty} \ll_{\Gamma,\varepsilon} (1+|\lambda_{\varphi}|)^{\frac{5}{24}+\epsilon}. \end{align} $$

The above estimates depend in an unspecified manner upon the underlying manifold. Consider, for instance, the case that $\Gamma $ is the Hecke congruence subgroup $\Gamma _0(N) = \operatorname {\mathrm {SL}}_2(\mathbb {Z}) \cap \left ( \begin {smallmatrix} \mathbb {Z} &\mathbb {Z} \\ N \mathbb {Z} & \mathbb {Z} \end {smallmatrix} \right )$ so that $\Gamma \backslash \mathbb {H}$ is an arithmetic manifold of volume $N^{1+o(1)}$ . We suppose that N is squarefree. A direct quantification of the Iwaniec–Sarnak argument (see [BH10, §10]) gives the estimate

(1.3) $$ \begin{align} \|\varphi\|_{\infty} \ll_{\varepsilon} N ^{\frac{1}{2} + \varepsilon} (1+|\lambda_{\varphi}|)^{\frac{5}{24}+\epsilon}, \end{align} $$

where we normalize $\varphi $ to have $L^2$ -norm one with respect to the hyperbolic probability measure, that is, the multiple of the hyperbolic measure having total volume one. The level aspect case of the sup-norm problem is to improve the dependence of the bound (1.3) upon N. The first improvement in the exponent was a major breakthrough of Blomer–Holowinsky [BH10], achieved 13 years after the work of Iwaniec–Sarnak. For a Hecke–Maaß newform $\varphi $ of eigenvalue $\lambda _{\varphi }$ , they managed to show

(1.4) $$ \begin{align} \|\varphi\|_{\infty} \ll_{\lambda_\varphi} N^{ \frac{1}{2}-\frac{1}{37}} \end{align} $$

(with explicit polynomial dependence upon $\lambda _\varphi $ ). Subsequently, Templier [Tem10] and Harcos–Templier [HT12, HT13] established several improved bounds, culminating in

(1.5) $$ \begin{align} \|\varphi \|_{\infty} \ll_{\lambda_\varphi,\epsilon} N^{\frac{1}{3}+\epsilon}. \end{align} $$

The estimate (1.5) is comparable in strength to the Weyl bound for the Riemann zeta function and has long been regarded as a natural limit for the sup-norm problem in the squarefree level aspect [HT13, Remarks (i)]. It has been extended to number fields [BHM16, BHMM20, Ass24] and to more general vectors than newforms [HNS19, Ass21]. For levels that are not squarefree (e.g., powers of a fixed prime), the flavor of the problem is quite different (see Remark 1.4), and stronger estimates have been achieved in [Sah17, Mar16, Sah20, Com21, HS20].

In this work, we bring new methodology to bear on the sup-norm problem in the squarefree level aspect. By obtaining optimal solutions to the technical problems that arise in applying that methodology, we deduce the following improvement of Equation (1.5).

Theorem 1.1. Let N be a squarefree natural number. Let $\varphi $ be a cuspidal Hecke–Maaß newform for $\Gamma _0(N)$ with trivial (central) character. Suppose that $\varphi $ is $L^2$ -normalized with respect to the hyperbolic probability measure on $\Gamma _0(N) \backslash \mathbb {H}$ . Then

$$ \begin{align*} \|\varphi\|_{\infty} \ll_{\lambda_{\varphi}, \epsilon} N^{\frac{1}{4}+\epsilon}. \end{align*} $$

Our main results apply not only to $\Gamma _0(N) \backslash \mathbb {H}$ but also to compact arithmetic quotients. In general, such a manifold is of the shape $\Gamma \backslash \mathbb {H}$ , where $\Gamma $ is commensurable with a lattice attached to a maximal order in a quaternion algebra B over a totally real field F, with B split at exactly one Archimedean place. We are content here to consider the case $F = \mathbb {Q}$ so that B is an indefinite quaternion algebra, characterized up to isomorphism by its reduced discriminant $d_B$ . For each natural number N coprime to $d_B$ , we denote by $\Gamma _0^B(N)$ the group of proper (i.e., norm one) units arising from an Eichler order of level N in B (see Section 2.1 for details). For example, if $B = \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Q})$ , then we could take $\Gamma _0^B(N) = \Gamma _0(N)$ . We prove the following theorem.

Theorem 1.2. Let $\Gamma =\Gamma ^B_0(N)$ be as above with the level N being squarefree. Let $\varphi $ be a cuspidal Hecke–Maaß newform for $\Gamma $ with trivial (central) character, $L^2$ -normalized with respect to the hyperbolic probability measure on $\Gamma \backslash \mathbb {H}$ . Then, with $V = (d_BN)^{1+o(1)}$ the covolume of $\Gamma $ ,

(1.6) $$ \begin{align} \|\varphi\|_{\infty} \ll_{\lambda_{\varphi}, \epsilon} V^{\frac{1}{4}+\epsilon}. \end{align} $$

Theorem 1.2 specializes to Theorem 1.1 upon taking $B = \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Q})$ . It improves upon (the $F = \mathbb {Q}$ case of) Templier’s result [Tem10], which gave the nontrivial bound $V^{\frac {1}{2} - \frac {1}{24} + \varepsilon }$ . We emphasize that the estimate (1.6) is uniform in the quaternion algebra B, hence gives a strong saving in the ‘discriminant aspect’; the first nontrivial results in that aspect (for B indefinite, as we have assumed) were established only very recently by Toma [Tom23], updating an earlier preprint, giving (among other things) the bound $V^{\frac {1}{2}-\frac {1}{30}+\varepsilon }$ . Our method applies equally in the setting of definite quaternion algebras, where we improve the exponent $\frac {1}{3}$ of Blomer–Michel [BM11, BM13] down to $\frac {1}{4}$ in analogy with Theorem 1.2 (see Section §2.3 for details).

Remark 1.3. The dependence on the eigenvalue in Equation (1.6) that follows from our proof is of exponential nature. With some finer Archimedean considerations, it seems likely that one could show $\|\varphi \|_{\infty } \ll _{\varepsilon } \lambda _\varphi ^{\frac {1}{4} +\varepsilon } V ^{\frac {1}{4} + \varepsilon }$ ; indeed, by comparison, we obtain such an estimate for the definite analogue of Equation (1.6) (see Corollary 2.3). Such a refinement of Equation (1.6) seems to require lengthy Archimedean calculations that we feel would distract from the primary novelties of this paper concerning the level aspect.

Remark 1.4. We have noted already that we focus in this paper on the case of squarefree levels. The opposite case is the depth aspect, where the level is a power $N = p^n$ of a fixed prime p. In that case, local arguments give the bound $\|\varphi \|_\infty \ll _{p,d_B, \varepsilon } (\lambda _\varphi N)^{1/4+\varepsilon }$ [Mar16], which has been improved to $\|\varphi \|_\infty \ll _{\lambda _\varphi ,p,d_B, \varepsilon } N^{5/24+\varepsilon }$ [HS20] via arithmetic amplification and refined local analysis.

Remark 1.5. In a function field setting analogous to that of Theorem 1.1, Sawin [Saw21] has used geometric techniques to establish (among other things) the sup-norm bound $\ll N^{\frac {1}{4} + \alpha _q}$ , where $\alpha _q> 0$ tends to zero as the cardinality q of the underlying finite field tends to $\infty $ . We do not see any obstruction to adapting the techniques of this paper to the function field setting, where we expect they would give the improved bound $\ll _{\varepsilon } N^{\frac {1}{4} + \varepsilon }$ .

By combining the arguments of this paper with those of the prequel [KS20] concerning the weight aspect for holomorphic forms, we obtain the following uniform hybrid bound in the weight and level aspects.

Theorem 1.6. Let $\Gamma = \Gamma ^B_0(N)$ be as in Theorem 1.2. Let f be a cuspidal holomorphic newform for $\Gamma $ with trivial (central) character and weight $k\ge 2$ . Suppose f is $L^2$ -normalized with respect to the hyperbolic probability measure on $\Gamma \backslash \mathbb {H}$ . Then

$$ \begin{align*} \|\Im(\cdot)^{\frac{k}{2}}f\|_{\infty} \ll_{\epsilon} (kV)^{\frac{1}{4}+\epsilon}, \end{align*} $$

where $V = (d_BN)^{1+o(1)}$ denotes the covolume of $\Gamma $ .

1.1. Selected applications

A straightforward application of these improved sup-norms is to $L^p$ -norms for $2\le p \le \infty $ by means of interpolation. We state here only the split holomorphic case, as in this case, strong $L^4$ -bounds were given by Buttcane–Khan [BK15] with subconvexity input from [You17].

Corollary 1.7. Let q denote an odd prime and f a cuspidal holomorphic newform for $\Gamma _0(q)$ with trivial (central) character and weight k. Suppose f is $L^2$ -normalized with respect to the hyperbolic probability measure on $\Gamma _0(q) \backslash \mathbb {H}$ . Then, for $2 \le p \le \infty $ and any $\eta>0$ , we have

$$ \begin{align*} \| \Im(\cdot)^{\frac{k}{2}} f \|_p \ll_{k, \eta} \begin{cases} q^{\frac{1}{6}-\frac{1}{3p}+\eta} , & 2 \le p\le 4, \\ q^{\frac{1}{4}-\frac{2}{3p}+\eta} , & 4 \le p \le \infty, \end{cases} \end{align*} $$

for k sufficiently large in terms of $\eta $ .

Further applications of sup-norm bounds include shifted convolution problems and subconvexity results for L-functions; see, for example, [Har03, HM06, HC19, HS20, Nor21]. Often, such applications would be obtained from a uniform version of Wilton’s estimate. By applying the arguments of [HM06, §2.7] with our improved sup-norm bound, we derive the following corollary.

Corollary 1.8. Let $\lambda (m)$ , $m \in \mathbb {N}$ , denote the Hecke eigenvalues, normalized so that the Ramanujan conjecture reads $|\lambda (m)| \ll _\varepsilon m^{\varepsilon }$ , of either a cuspidal Hecke–Maaß newform or a cuspidal holomorphic newform of weight k on $\Gamma _0(N)$ with trivial (central) character, where N is squarefree. Then, for any $\alpha \in \mathbb {R}$ , one has

where the implied constant in the Maaß case further depends on the eigenvalue of the form.

As a consequence, we may, for example, improve the main theorem in [HC19].

Corollary 1.9. Let $\varphi $ either be a cuspidal Hecke–Maaß newform or a cuspidal holomorphic newform on $\Gamma _0(q)$ , with q prime. Let $\chi $ be a primitive Dirichlet character of modulus m with $(m,q)=1$ . Suppose that $q = m^{\eta }$ with $0 < \eta < 2$ . Then, we have

$$ \begin{align*} L(\varphi \otimes \chi, \tfrac{1}{2}) \ll_{\epsilon} \mathcal{C}^{\frac{1}{4}+\epsilon} \left( \mathcal{C}^{-\frac{\eta}{4(2+\eta)}}+\mathcal{C}^{- \frac{2- \eta - 4 \vartheta}{8(2+\eta)}} \right), \end{align*} $$

where the implied constant depends on the eigenvalue respectively weight of $\varphi $ , $\mathcal {C}=qm^2$ is the conductor of the L-function and $\vartheta $ is the current best bound towards the generalized Ramanujan conjecture if $\varphi $ is a Maaß form and $0$ if $\varphi $ is holomorphic.

1.2. The fourth moment and further applications

The method underlying most previous works on this problem, including the work of Harcos–Templier giving the bound $\ll _{\epsilon } N^{1/3+\epsilon }$ , is based on the amplification method introduced in the original paper of Iwaniec–Sarnak. Recently, Steiner [Ste20] and Khayutin–Steiner [KS20] introduced a new method based on analysis of fourth moments over families. The key observation of these papers was that such a fourth moment naturally arises as the $L^2$ -norm of a theta kernel. Alternatively, Blomer et al. [BHMM22] have demonstrated that one may use Voronoï summation for Rankin–Selberg convolutions in place of a theta kernel. Prior to the application to fourth moments, theta kernels have played similar roles in the study of quantum variance [Nel16, Nel17, Nel19, Nel20], numerical computations [Nel15] and in the proof of Waldspurger’s formula [Wal85]. In each of these earlier works, theta kernels apparently served as a substitute for parabolic Fourier expansions, giving a tool for establishing analogues on compact quotients (where such expansions are not available) of results known already for noncompact quotients. The present work differs in that our main result is new even for the noncompact quotients $\Gamma _0(N) \backslash \mathbb {H}$ .

In this paper, we follow generally the theta kernel strategy of the prequel [KS20] and prove a fourth moment bound from which one may deduce the Theorems 1.1, 1.2 and 1.6 after some additional analysis near any cusps. In what follows, we let $\Gamma = \Gamma ^B_0(N)$ be a lattice as in Theorem 1.2 and denote by $V = (d_B N)^{1+o(1)}$ the volume of $\Gamma \backslash \mathbb {H}$ .

The formulation of our results requires some quantification of the closeness of a point $z \in \Gamma \backslash \mathbb {H}$ to the cusps. If $\Gamma \backslash \mathbb {H}$ is noncompact (i.e., $d_B = 1$ ), then we may assume that $\Gamma = \Gamma _0(N)$ , and we set

$$ \begin{align*} H(z) = \max_{\gamma \in A_0(N)} \Im(\gamma z), \end{align*} $$

where $A_0(N)$ denotes the lattice of Atkin–Lehner operators for $\Gamma _0(N)$ (see Section §2.2 for another formulation of the definition of H). If $\Gamma \backslash \mathbb {H}$ is compact, then we set $H(z) = 0$ .

Theorem 1.10. Let $\Gamma = \Gamma ^B_0(N)$ be as in Theorem 1.2. Fix $\Lambda> 0$ , and let $(\varphi _i)_i$ be an orthonormal set of cuspidal Hecke–Maaß newforms with trivial (central) character and Laplace-eigenvalue bounded by $\Lambda $ on the hyperbolic surface $\Gamma \backslash \mathbb {H}$ equipped with the hyperbolic probability measure. Then, for any two points $z,w \in \Gamma \backslash \mathbb {H}$ , we have

(1.7) $$ \begin{align} \sum_i \left( |\varphi_i(z)|^2-|\varphi_i(w)|^2 \right)^2 \ll_{\epsilon, \Lambda} V^{1+\epsilon} \left(1+ V[H(z)^2+H(w)^2] \right). \end{align} $$

Similarly, for an orthonormal set $(f_i)_i$ of cuspidal holomorphic newforms for $\Gamma $ of weight k and trivial (central) character with respect to the hyperbolic probability measure on $\Gamma \backslash \mathbb {H}$ , we have

$$ \begin{align*} & \sum_i \left( |\Im(z)^{\frac{k}{2}}f_i(z)|^2-|\Im(w)^{\frac{k}{2}}f_i(w)|^2 \right)^2 \\ & \qquad\qquad\qquad\qquad\qquad\quad\qquad\ll_{\epsilon} (Vk)^{1+\epsilon} \left(1+V^{\frac{1}{2}}[H(z)+H(w)]+Vk^{-\frac{1}{2}}[H(z)^2+H(w)^2] \right) \end{align*} $$

for any two points $z,w \in \Gamma \backslash \mathbb {H}$ .

In the case that the hyperbolic surface $\Gamma \backslash \mathbb {H}$ is compact, we may integrate z and w over the whole surface and get an essentially sharp bound on the fourth moment of fourth norms in the level aspect, thereby extending a result of Blomer [Blo13] to the case of cocompact lattices $\Gamma $ .

Corollary 1.11. With notation and assumptions as in Theorem 1.10 and assuming further that $\Gamma \backslash \mathbb {H}$ is compact, we have

$$ \begin{align*} \sum_i \|\varphi_i\|_4^4 \ll_{\epsilon, \Lambda} V^{1+\epsilon}, \end{align*} $$

$$ \begin{align*} \sum_i \|\Im(\cdot)^{\frac{k}{2}}f_i\|_4^4 \ll_{\epsilon} (Vk)^{1+\epsilon}. \end{align*} $$

This result may also be recast as a double average of triple L-functions by means of Watson’s formula [Wat08, Theorem 3].

The final application of Theorem 1.10 we mention is to the diameter of compact arithmetic hyperbolic surfaces $\Gamma \backslash \mathbb {H}$ [Ste23]. Here, one may use the sharp bound on the ‘fourth moment’ of exceptional eigenforms, together with a strong density estimate for the exceptional eigenvalues, to get an optimal estimate on the almost diameter and an estimate on the diameter of the same strength as if one were to assume the Selberg eigenvalue conjecture.

1.3. The added complexity of the level aspect

Compared to the weight aspect treated in the prequel, the level aspect requires many new ideas. Here, we tacitly restrict to the case of squarefree level; the general case would require a more nuanced discussion. In some sense, the level aspect may be understood as intermediate in difficulty between the holomorphic and eigenvalue aspects. Indeed, relative to known techniques, the difficulty in the sup-norm problem is reflected in the essential support of the matrix coefficient of the automorphic form being bounded. In the weight, (squarefree) level and eigenvalue aspects, the matrix coefficient concentrates on a space of dimension one, two and three, respectively.

We now briefly recall the main idea of the theta approach and discuss some of the new challenges that arise in the level aspect. We focus first on the case of Hecke–Maaß forms on $\Gamma _0(N) \backslash \mathbb {H}$ , as in Theorem 1.1. Take $R=\left (\begin {smallmatrix} \mathbb {Z} & \mathbb {Z} \\ N \mathbb {Z} & \mathbb {Z} \end {smallmatrix}\right )$ so that that the set of proper units of R is precisely $\Gamma _0(N)$ . For $\ell \mid N$ , let $R(\ell )= \left (\begin {smallmatrix} \mathbb {Z} & \mathbb {Z} / \ell \\ N \mathbb {Z} / \ell & \mathbb {Z} \end {smallmatrix}\right )$ denote the partially dualized lattices of the order R. Let $\sigma _z \in \operatorname {\mathrm {SL}}_2(\mathbb {R})$ be any matrix taking i to $z \in \mathbb {H}$ . Let $\varphi $ be an arithmetically normalized cuspidal Hecke–Maaß newform. The theta identity at the heart of the argument then reads

(1.8) $$ \begin{align} \langle \theta(z,w;\cdot), \varphi \rangle = \frac{1}{V} \varphi(z)\overline{\varphi(w)}, \end{align} $$

where V denotes the covolume of $\Gamma _0(N)$ and the theta function is given by

(1.9) $$ \begin{align} \theta(z,w;s) = \Im(s) \sum_{\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \sigma_z^{-1} R \sigma_w} e^{- \pi (a^2+b^2+c^2+d^2) \Im(s)} e^{2 \pi i (ad-bc) \Re(s)}. \end{align} $$

By Bessel’s inequality, the left-hand side of Equation (1.7) is in essence captured by the $L^2$ -norm of the difference of the theta kernels $\theta (z,z;\cdot )-\theta (w,w;\cdot )$ . From here, one may then proceed as in the prequel by covering a fundamental domain by Siegel sets and making use of the orthogonality relations in the unipotent direction. One ends up with a weighted sum over matrices $\gamma _1, \gamma _2 \in R(\ell )$ satisfying $\det (\gamma _1)=\det (\gamma _2)$ and for which the entries of $\sigma _z^{-1} \gamma _{i} \sigma _z$ , $i=1,2$ , satisfy certain bounds (and similarly for w). The bounds imposed on these entries depend crucially upon the precise choice of Siegel domains, so it is important that we make a good choice. Like in the prequel, we split the count according to whether $\operatorname {\mathrm {tr}}(\gamma _1) = \operatorname {\mathrm {tr}}(\gamma _2)$ or not.

In the case of nonequal trace, the naïve choice of Siegel domains consisting of $\Gamma _0(N) \backslash \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ -translates of the standard Siegel domain for $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ leads to a rather challenging counting problem. In order to get a sharp bound on Equation (1.7), one faces the challenge of counting, for each divisor $\ell $ of N and each T with $\ell ^{-1/2} \ll T \ll 1$ , the sextuples of integers $(a_1,b_1,c_1,a_2,b_2,c_2)$ satisfying

(1.10) $$ \begin{align} (c_i y N / \ell)^2 + 2 ( a_i - c_i xN/\ell)^2 + y^{-2} ( 2 a _i x + b / \ell - c_i x^2 N/ \ell)^2 \le T^2, \quad i=1,2, \end{align} $$

(1.11) $$ \begin{align} b_1 c_1 \equiv b_2 c_2\quad \pmod{\ell^2/N}. \end{align} $$

We would need to know that the number of such sextuples is roughly $O(\ell T^2)$ in the range ${N^{-1} \ll y \ll N^{-1/2}}$ and $|x| \le \frac {1}{2}$ . We do not know how to establish such a bound directly when, for instance, $\ell = N$ . On the other hand, when $\ell = 1$ , the congruence condition is void and, using arguments of Harcos–Templier, we can prove the required bound with some room to spare, namely, for T up to $N^{1/2}$ . Our solution to this dichotomy is thus to decrease the size of the Siegel domains associated to larger $\ell $ at the expense of increasing those associated to smaller $\ell $ . This solution may be implemented most simply by applying an Atkin–Lehner involution to the covering of $\Gamma _0(N) \backslash \mathbb {H}$ by $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ -translates of the standard fundamental domain for $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ . With this maneuver, we reduce to considering the range $T \ll N^{\frac {1}{2}} \ell ^{-1}$ . We are then able to prove the required bound by forgoing the congruence condition, reducing the problem to counting triples of integers $(a_i,b_i,c_i)$ satisfying Equation (1.10), which we carry out using geometry of numbers techniques. We refer subsequently to this type of counting problem, where we count traceless matrices $\gamma \in R(\ell )^0$ with a bound on the entries of $\sigma _{z}^{-1}\gamma \sigma _z$ , as ‘Type I’.

In the case of equal trace, we need to count sextuples of integers $(a_1,b_1,c_1,a_2,b_2,c_2)$ satisfying Equation (1.10) and

(1.12) $$ \begin{align} a_1^2+b_1c_1 \tfrac{N}{\ell^2} = a_2^2+b_2c_2 \tfrac{N}{\ell^2}. \end{align} $$

We need to bound this count by $O(\ell T)$ in the same ranges as before. We refer to this type of counting problem as ‘Type II’. The key observation is that $(a_1,b_1,c_1)$ turns out to determine $(a_2,b_2,c_2)$ up to a small number of possibilities. This allows us to reduce Type II estimates to Type I estimates.

The above arguments suffice for noncompact quotients, that is, for the proof of Theorem 1.1. They rely on the use of matrix coordinates $\left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right )$ with respect to which the lattices $\Gamma _0(N)$ are described by the simple congruence condition $c \equiv 0\ \ \pod {N}$ . We were unable to find an analogously straightforward way to separate the variables in the compact setting (e.g., using fixed quadratic subalgebras of B). In the case that B is definite, the Type I counts were treated in a coordinate-free way by Blomer–Michel [BM11, BM13], who controlled the successive minima of the ternary quadratic lattice underlying $\Gamma _0^B(N)$ in terms of only the content, level and discriminant of that lattice. We extend their arguments to the case that B is indefinite by defining analogous Archimedean quantities that control the disparity of the reduced norm and a majorant, such as the square of the Frobenius norm of $\sigma _z^{-1} \gamma \sigma _z$ for $\gamma \in R(\ell )^0$ .

Following the same strategy as in the noncompact case, it remains then only to reduce Type II estimates to Type I estimates. This reduction is perhaps the most subtle part of our counting arguments. It requires us to establish the analogue in the compact setting of the key observation noted following Equation (1.12). For example, in case that B is definite, writing R for an Eichler order of level N, we need to show that for each $n \ll V$ , the number of elements $\gamma \in R$ with trace $0$ and norm n is essentially $O(1)$ , uniformly in N and B. We eventually managed to do so through a delicate argument involving commutators and representations of binary quadratic forms.

1.4. Organization of the paper

The complete statements of our results may be found in Section §2. In Section §3, we reduce the proofs to those of two auxiliary collections of results:

• ∘ those concerning matrix counting, and

• ∘ those reducing the required estimates for theta functions to matrix counting.

The latter, including the appropriate splicing of a fundamental domain into Siegel sets, may be found in Section §4. In Section §5, we summarize the required properties of the theta functions. The proofs of said properties are deferred to Appendix A.

Sections §7 and §8 are dedicated to the anisotropic extension of the lattice counting argument of Blomer–Michel, which we subsequently apply to the Type I counting problem in Section §9.

The final section, §10, treats the crucial Type II counting problem.

2. Statement of results

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

Remark 2.2. In the indefinite holomorphic case (2.5), one may have the same bound for the fourth moment rather than the squared difference under the assumption that the weight satisfies $k \gg _{\eta } (d_BN)^{\eta }$ for some $\eta>0$ , in which case the implied constant also depends on $\eta $ and the implied constant in the assumed lower bound for the weight.

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, [Bum97, §3.2 & §3.6] for the indefinite case and [BM13] for the definite case.

2.4. Counting problems: setup

2.4.1. Lattices locally dual to R

Let $\ell $ be a divisor of the squarefree number $d_B N$ . We denote by $R(\ell )$ the lattice in B whose local components $R(\ell )_p$ are given

• ∘ for p dividing $\ell $ , by the lattice $R_p^\vee \subseteq B_p$ dual to $R_p$ , and

• ∘ otherwise, by $R_p$ .

2.4.2. Reduced trace and norm

We denote by $\operatorname {\mathrm {tr}}$ and $\det $ the reduced trace and reduced norm on B, and also on its completions. We use a superscripted $0$ , as in $R^0$ or $R(\ell )^0$ , to denote the kernel of the reduced trace.

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. Division and reduction of the proof

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) $H(h)=H(h')$ .

Remark 3.1. By considering a right translate of $\varphi \in \mathcal {F}$ and thereby moving the maximal compact $K_{\infty }$ and the Eichler order R around, one could reduce the statement of the main Corollary 2.3 to the case that g is the identity. However, in the split case, our counting arguments do depend on the particular order in the genus. Moreover, our method relies on a difference of theta kernels defined relative to different g. Such a reduction would thus be premature.

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 [Tem15, Prop. 3.1 & 3.2], for example, and the latter may be deduced from the Fourier expansion along the lines of Xia [Xia07]. 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 [Del71, Del74]. This implies $|a_n| \ll _{\varepsilon } n^{\frac {k-1}{2}+\varepsilon } |a_1|$ .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 [HL94] (cf. [HM06, Eq. (31)]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. Arithmetic quotients as real manifolds

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 [Bor63], $[G']/K_R^1$ 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 [Voi18, Thm 39.1.8] in the indefinite case and [Voi18, 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 $ [Iwa97, §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 $A_0(M) / \Gamma _0(M)$ [AL70, 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. Theta kernels and their $L^2$ -norms

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 [JL70, Cas73]. 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 restriction³ 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 [HL94] and Iwaniec [Iwa90] then imply the following bounds for the $L^2$ -norm of the arithmetically normalized Jacquet–Langlands lift (cf. [HM06, (30), (31)]⁴). One may also compare with the geometric normalization in [PY19, 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 [Ber31] 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. Preliminaries on the geometry of numbers

6.1. Bounds on successive minima

Definition 6.1. Let V be an n-dimensional real vector space. Let $L \subseteq V$ be a lattice (i.e., a cocompact discrete subgroup). Given a compact convex $0$ -symmetric subset $\mathcal {K}$ of V with nonempty interior, we define a function $N : V \rightarrow \mathbb {R}_{\ge 0}$ by $N(v) := \inf \{t> 0 : v \in t \mathcal {K} \}$ . Given a positive-definite quadratic form Q on V, we define such a function by $N(v) := Q(v)^{1/2}$ , or equivalently, by applying the previous definition with $\mathcal {K}$ the unit ball for Q. In either case, we define the successive minima $\lambda _1 \le \dotsb \le \lambda _n$ of $\mathcal {K}$ on L (or of Q on L) as: $\lambda _k$ is the smallest positive real for which there is a linearly independent subset $\{v_1,\dotsc ,v_k\}$ of L for which $N(v_j) \le \lambda _k$ for each $1 \le j \le k$ .

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 $(0,0)$ .

Proof. This is essentially [HT12, Lemma 1]. That reference gives the slightly weaker bound obtained by omitting the factor $(c,N)$ , but the stronger bound that we have stated follows from their proof, keeping track of $(c,N)$ at each step rather than bounding it from below by $1$ .

6.2. Lattice counting

Lemma 6.3. Let $f_{\mathcal {K}}$ be the distance function of a closed convex $0$ -symmetric set $\mathcal {K}\subseteq \mathbb {R}^n$ 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 (see Definition 6.1) of $\mathcal {K}$ on $\Lambda $ . Then, there is a basis $v_1,\dots ,v_n$ of $\Lambda $ such that $f_{\mathcal {K}}(v_i) \asymp _n \lambda _i$ .

Proof. This is [GL87, Thm. 2, p. 66].

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 [vdC36]. 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 [BHW93, 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

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

Proof. See [HT13, Lemma 2.1].

7. Local preliminaries on orders

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 $q(\mathfrak {a})$ .

Lemma 7.1. We have $\mathfrak {a} = \mathfrak {d} \mathfrak {n} \mathfrak {a}^\vee $ .

Proof. Let $\alpha $ be a generator of $\mathfrak {a}$ . Then $\mathfrak {n} = q(1)\operatorname {\mathrm {nr}}(\mathfrak {a}) = \mathfrak {o} q(1) \alpha \bar {\alpha }$ , $\mathfrak {a}^\vee = \{ q(1)^{-1} x: x \in E, \operatorname {\mathrm {tr}}(x \bar {\mathfrak {a}}) \subseteq \mathfrak {o} \} = q(1)^{-1} \bar {\alpha }^{-1} \mathfrak {d}^{-1}$ . Multiplying through, the conclusion follows.

Corollary 7.2. Suppose that $E/F$ is unramified and that q is integral on $\mathfrak {a}$ so that $\mathfrak {a} \subseteq \mathfrak {a}^\vee $ . Then the elementary divisors for the $\mathfrak {o}$ -module inclusion $\mathfrak {a} \hookrightarrow \mathfrak {a}^\vee $ are $(\mathfrak {n}, \mathfrak {n})$ .

Proof. Our hypotheses imply that $\mathfrak {d} = \mathfrak {o}$ and that $\mathfrak {n}$ is an integral ideal. The lemma implies that there is an isomorphism (first of $\mathfrak {o}_E$ -modules, then of $\mathfrak {o}$ -modules) $\mathfrak {a}^\vee / \mathfrak {a} \cong \mathfrak {o}_E / \mathfrak {n} \mathfrak {o}_E \cong (\mathfrak {o}/\mathfrak {n})^2$ , whence the conclusion.

Remark 7.3. Under the hypotheses of the corollary, the discriminant ideal of the binary quadratic form $(q,\mathfrak {a})$ is $\mathfrak {n}^2$ . More generally, under the hypotheses of the lemma, the discriminant ideal is $\mathfrak {D} \mathfrak {n}^2$ , with $\mathfrak {D} =\operatorname {\mathrm {nr}} (\mathfrak {d})$ . Conversely, given the discriminant ideal, we may compute $\mathfrak {n}$ as its square root.

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 [HNS19, HN18, Sah20].)

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. [Voi18, §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. [Voi18, §15.2.11, §23.4]), hence $\mathfrak {n} = \mathfrak {p}$ .

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

Lemma 7.4. Let $E / F$ unramified and R be an E-adapted order. Let $[\gamma _1,\gamma _2] = \gamma _1\gamma _2-\gamma _2\gamma _1$ denote the commutator for two elements $\gamma _1,\gamma _2$ in B. Then, we have

1. (i) $q([\gamma _1,\gamma _2]) \in \mathfrak {n}$ for all $\gamma _1,\gamma _2 \in R$ ,

2. (ii) $[\gamma _1,\gamma _2] \in \mathfrak {n}^{-1}R$ for all $\gamma _1,\gamma _2 \in R^{\vee }$ ,

3. (iii) $q([\gamma _1,\gamma _2]) \in \mathfrak {n}^{-2}$ for all $\gamma _1,\gamma _2 \in R^{\vee }$ .

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. Invariants of rational quadratic forms

Let V be an n-dimensional $\mathbb {Q}$ -vector space and $q : V \rightarrow \mathbb {Q}$ a nondegenerate quadratic form. We normalize the polarization $\langle \,, \rangle $ of q such that $\langle x,x \rangle = 2 q(x)$ .

Given a lattice $L \subseteq V$ and a positive-definite quadratic form Q on $V \otimes _{\mathbb {Q}} \mathbb {R}$ , one can define the successive minima of the pair $(L,Q)$ . Our aim in this section is to provide certain estimates for those successive minima in terms of other invariants of $(L,Q)$ . Our results may be understood as a generalization of those of Blomer–Michel [BM13, §3, §4], who treated the special case that q is definite and $Q = q$ . We mention also the work of Saha [Sah20, §2].

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.)

Remark 8.1. In general, the content of L does not agree with the content of the Gram matrix of q on L, that is, the greatest common divisor of the entries. However, if q is integral on L, then the level of L agrees with the level of the Gram matrix of q.

Remark 8.2. For our purposes, the content, respectively level, may be replaced by the first, respectively last, elementary divisor of the Gram matrix of q as these quantities differ by a bounded power of $2$ .

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 $\max |d_i|^{-1}$ .

Remark 8.3. We may relate the above definition of level to more standard notions. We recall that the form Q is a majorant of q if $P S^{-1} P = S$ , or equivalently, if $D^2 = 1$ . Suppose that $|q| \le Q$ . Then the level of Q is always at least $1$ , and it is equal to $1$ if and only if Q is a majorant of q. Indeed, the assumption $|q| \le Q$ implies that $|d_i| \le 1$ for each i, with equality precisely when $D^2 = 1$ .

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

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

Remark 8.5. This last estimate is scale-invariant in the sense that replacing $(Q,L)$ with $(Q_m, m L)$ for a positive rational scalar m has no effect on the right-hand side. This feature is not surprising in view of the multiplication-by-m bijection $\{ v \in L : Q(v) \le X^2 \} \cong \{ v \in m L : Q_m(v) \le X^2\}$ . The estimate is likewise invariant under replacing $(Q,X)$ by $(Q_m, X/m^2)$ for some nonzero real number m, as one might expect for similar reasons.

Proof. We follow the basic strategy of Blomer–Michel [BM13, §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.

9. Type I estimates

The local computations of Section §7, together with the behavior of invariants under duality recorded in §8.2, imply that the elementary divisors of the Gram matrix of the reduced trace form on $g^{-1}R(\ell )^0g$ are given by

(9.1) $$ \begin{align} \begin{cases} \left(\frac{1}{\ell},\frac{d_B N}{\ell^2},\frac{2d_B N}{\ell}\right), & 2 \nmid d_BN, \\ \left(\frac{2}{\ell},\frac{d_BN}{\ell^2},\frac{d_BN}{\ell}\right), & 2 | d_BN, 2 \nmid \ell, \\ \left(\frac{1}{\ell},\frac{d_BN}{\ell^2},\frac{d_BN}{2\ell}\right), & 2 | d_BN, 2 | \ell. \end{cases} \end{align} $$

Hence, the content, level and discriminant of $g^{-1}R(\ell )^0g$ with respect to the reduced norm are comparable to $1/\ell $ , $d_BN/\ell $ and $(d_BN)^2/\ell ^4$ respectively. Here, ‘comparable to’ means the ratios are bounded from above and below by positive constants. Suppose that the reduced norm on R is anisotropic. In this case, we wish to apply Proposition 8.4 to the lattice $g^{-1}R(\ell )^0g$ with q given by the reduced norm. Recall the notation P, u and X from Equation (5.1). As a first choice, we let $Q=P+\delta ^{-1}u$ , whose content, level and discriminant are comparable to $1,\delta ^{-1}$ and $\delta ^{-2}$ , respectively. This yields that the first successive minima of $g^{-1}R(\ell )^0g$ with respect to $P+\delta ^{-1}u$ is $\gg \ell ^{-\frac {1}{2}}$ , and hence also with respect to $\Omega (\delta ,1) \cap B_{\infty }^0$ . Furthermore,

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

This proves the first half of Theorem 2.5. Similarly, we have for the choice $Q=P+\delta ^{-1} |X|^2$ that the content, level and discriminant Q are comparable to $1,\delta ^{-1}$ and $\delta ^{-1}$ , respectively. Thus, the first successive minima of $g^{-1}R(\ell )^0g$ with respect to $\Psi (\delta ,1)\cap B_{\infty }^0$ is $\gg \ell ^{-\frac {1}{2}}$ and

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

which is the second half of Theorem 2.5.

We now turn to the case that the quaternion algebra B is split. Here, we proceed in a more ad hoc manner. First, we note that $R(\ell )^0$ is normalized by the Atkin–Lehner operators. Thus, we need only consider $g \in G(\mathbb {R})$ such that $g \cdot i =x+iy$ has maximal imaginary part under the action of the Atkin–Lehner operators. In particular, we have $H(g)=y$ . Let $\lambda _1\le \lambda _2 \le \lambda _3$ be the successive minima (Definition 6.1) of the closed convex $0$ -symmetric set $\Omega (\delta ,1)\cap B_{\infty }^0$ with respect to the lattice $g^{-1}R(\ell )^0g$ . Since $\Omega (\delta ,1)$ is both left and right $K_{\infty }$ -invariant, we may further assume that $g=\left (\begin {smallmatrix} 1 & x \\ & 1 \end {smallmatrix}\right ) \operatorname {\mathrm {diag}}(y^{\frac {1}{2}} , y^{-\frac {1}{2}})$ . By Lemma 6.4, we have

$$ \begin{align*} |g^{-1}R(\ell)^0g \cap \Omega(\delta,T) | = |g^{-1}R(\ell)^0g \cap T\Omega(\delta,1)| \asymp 1+\frac{T}{\lambda_1}+\frac{T^2}{\lambda_1\lambda_2}+\frac{T^3}{\lambda_1\lambda_2\lambda_3}. \end{align*} $$

Let $\beta _0 = \left (\begin {smallmatrix} a & b \\ c & -a \end {smallmatrix}\right ) \in R(\ell )^0$ , thus $a \in \mathbb {Z}, b \in \frac {1}{\ell }\mathbb {Z}, c \in \frac {N}{\ell }\mathbb {Z}$ . Let

$$ \begin{align*} \alpha_0= g^{-1} \beta_0 g = \begin{pmatrix} a-cx & \frac{1}{y}(2ax+b-cx^2) \\ cy & cx-a \end{pmatrix}. \end{align*} $$

Suppose $\beta _0 \neq 0$ . If $(a,c)\neq (0,0)$ , then by Lemma 6.2 we have $P(\alpha _0) \ge \frac {1}{2}|cz-a|^2 \ge \frac {1}{2\ell }$ . Otherwise, we have $u(\alpha _0)= (\frac {b}{y})^2 \ge \frac {1}{2(\ell y)^2}$ . Hence, we have $\lambda _1 \gg \min \{ \ell ^{-\frac {1}{2}} , \ell ^{-1} y^{-1} \delta ^{-\frac {1}{2}} \}$ . In order to get a lower bound on $\lambda _1\lambda _2$ and $\lambda _1\lambda _2\lambda _3$ , we shall give an upper bound on $|g^{-1}R(\ell )^0g \cap \Omega (\delta ,T)|$ along the lines of Harcos–Templier [HT13]. First, we bound the number of choices of c by $\ll 1+\frac {\ell T}{Ny}$ as $|cy| \le P(\alpha _0)^{\frac {1}{2}} \le T$ . For each such choice of c, the equation

$$ \begin{align*} \frac{1}{y^2} \left|-cz^2+2az+b\right|^2 = u(\alpha_0) \le \delta T^2 \end{align*} $$

defines a circle of radius $\delta ^{\frac {1}{2}}T y$ and center $cz^2$ in which we need to count lattice points of the lattice generated by $2z$ and $\frac {1}{\ell }$ . This lattice has covolume $2y/\ell $ and first successive minima $\ge (\ell N)^{-\frac {1}{2}}$ by Lemma 6.2. We may thus apply Lemma 6.5 to bound the number of $(a,b)$ by $\ll 1+\ell ^{\frac {1}{2}}N^{\frac {1}{2}}\delta ^{\frac {1}{2}}T y + \ell \delta T^2 y$ . We obtain

$$ \begin{align*} \frac{T^2}{\lambda_1\lambda_2} + \frac{T^{3}}{\lambda_1\lambda_2\lambda_3} \ll \left( 1+\frac{\ell T}{Ny} \right)\left( 1+\ell^{\frac{1}{2}}N^{\frac{1}{2}}\delta^{\frac{1}{2}}Ty + \ell\delta T^2 y \right). \end{align*} $$

By letting T tend to $\infty $ , it follows that $\lambda _1\lambda _2\lambda _3 \gg N \ell ^{-2}\delta ^{-1}$ . By taking $T=N^{\frac {1}{2}}/(\ell \delta )^{\frac {1}{2}}$ , we obtain

$$ \begin{align*} \frac{1}{\lambda_1\lambda_2} \ll \frac{\ell\delta}{N}+\frac{\ell^{\frac{3}{2}}\delta^{\frac{1}{2}}}{N^{\frac{3}{2}}y}+\ell\delta y + \frac{\ell^{\frac{3}{2}}\delta^{\frac{1}{2}}}{N^{\frac{1}{2}}} \ll \frac{\ell^{\frac{3}{2}}\delta^{\frac{1}{2}}}{N^{\frac{1}{2}}} + \ell \delta y. \end{align*} $$

We thereby conclude the proof of the final case of Theorem 2.5.

10. Type II estimates

10.1. Bounds for representation numbers of binary quadratic forms

Lemma 10.1. Let M be a free $\mathbb {Z}$ -module of rank $2$ . Let $q : M \rightarrow \mathbb {Z}$ be a nondegenerate integral binary quadratic form. Let Q be a positive-definite quadratic form on $M \otimes \mathbb {R}$ such that $|q| \le Q$ . Let n be a nonzero integer and let $X \ge 1$ . Set

Then,

$$\begin{align*}|S| \ll_{\varepsilon} (X |n|)^\varepsilon. \end{align*}$$

Proof. We begin with a preliminary reduction. Suppose we can find a lattice $M'$ in $M \otimes \mathbb {Q}$ that contains M and on which the $\mathbb {Q}$ -bilinear extension of q is integral. It suffices then to verify the lemma after replacing M with $M'$ . Indeed, doing so enlarges the set S. In particular, we may assume that the quadratic form M is primitive.

Suppose now that M is primitive and anisotropic. Without loss of generality, M is either positive-definite or indefinite. By the form-ideal correspondence, we may assume then that M is an invertible ideal of an order $\mathfrak {o}$ in a quadratic field, with q given by the element norm $\nu $ divided by the ideal norm $\nu (M)$ of M:

$$\begin{align*}q(\beta) = \nu(\beta)/ \nu(M). \end{align*}$$

We will establish the estimate

(10.1) $$ \begin{align} |S| \ll \log(2 + X^2/|n|) \tau(n), \end{align} $$

which suffices in view of the divisor bound. Let $\mathfrak {o}_{\max }$ denote the maximal order in the quadratic field containing $\mathfrak {o}$ . For each $\beta \in S$ , the $\mathfrak {o}$ -ideal $M^{-1} \beta $ has norm $|n|$ , as does the $\mathfrak {o}_{\max }$ -ideal $\mathfrak {o}_{\max } M^{-1} \beta $ . The number of $\mathfrak {o}_{\max }$ -ideals of norm $|n|$ is at most $\tau (n)$ . Suppose two elements $\beta _0, \beta \in S$ give rise to the same $\mathfrak {o}_{\max }$ -ideal. Then, $\beta /\beta _0$ is a norm one unit in $\mathfrak {o}_{\max }^{\times }$ . The required estimate follows in the imaginary quadratic case (without the logarithmic factor) because $| \mathfrak {o}_{\max }^{\times }| \le 6$ . In the real quadratic case, we fix a positive generator $\eta $ for the group $\cong \mathbb {Z}$ of non-root-of-unity norm one units in $\mathfrak {o}_{\max }$ and write $\beta = \pm \beta _0 \eta ^{\ell }$ for some $\ell \in \mathbb {Z}$ . It will suffice then to verify that $\ell \ll \log (2 + X^2/|n|)$ . To that end, we estimate $q(\beta _0 + \beta )$ in two ways. On the one hand, the triangle inequality for the Euclidean norm defined by Q gives the upper bound $|q(\beta _0 + \beta )| \le Q(\beta _0 + \beta ) \ll Q(\beta _0) + Q(\beta ) \ll X^2$ . On the other hand, the multiplicativity of $\nu $ gives the identity $q(\beta _0 + \beta ) = n \nu (1 \pm \eta ^{\ell })$ . The lower bound $\nu (1 \pm \eta ^{\ell }) \ge \frac {1}{4} \cdot 1.618^{\ell }$ for fundamental units now yields the required estimate for $\ell $ .

It remains to consider the case that M is isotropic and q nondegenerate. In that case, after applying our preliminary reduction to enlarge M if necessary, we may assume that $M = \mathbb {Z}^2$ and $q(x,y) = x y$ . Indeed, we may choose a basis $e_1, e_2$ for M with $e_1$ isotropic. Then, q is given with respect to the coordinates $x e_1 + y e_2$ by $q(x,y) = a x y + b y^2$ for some $a,b \in \mathbb {Z}$ , with $a \neq 0$ . Then $q (\tfrac {x - b y}{a}, y) = x y$ and $M \subseteq M' := \{ \tfrac {x - b y}{a} e_1 + y e_2 : x, y \in \mathbb {Z} \}$ , so $(M',q)$ gives the required enlargement. Now, since $n \neq 0$ , the divisor bound gives $|S| = 2 \tau (n) \ll _{\varepsilon } |n|^{\varepsilon }$ .

10.2. Local quaternionic preliminaries

Let B be a quaternion algebra over the rationals. We write $d_B$ for its reduced discriminant and q for its reduced norm.

10.2.1. Non-Archimedean preliminaries

Let $R \subset B$ be an Eichler order of level N, with N coprime to $d_B$ .

Lemma 10.2. For $x,y \in R(\ell )^0$ , we have $[x,y] \in \frac {1}{\ell } R^0$ and $q([x,y]) \in \frac {d_B N}{\ell ^3} \mathbb {Z}$ .

Proof. This follows from the local computations in Lemma 7.4 together with the fact that the trace of a commutator is zero.

10.2.2. Archimedean preliminaries

Recall the notation ‘ $\Omega $ ’ from §2.4.4 and ‘P’ from Equation (5.1).

Lemma 10.3. For $0 < \delta \le 1$ and $x, y \in \Omega (\delta ,T)$ , we have

$$\begin{align*}q([x,y]) \ll \delta T^4\end{align*}$$

and

$$\begin{align*}P([x,y]) \ll \delta T^4.\end{align*}$$

Proof. For $\delta = 1$ , the required estimates reduce via homogeneity to the compactness of the unit ball and the continuity of multiplication. We turn to the case $0 < \delta < 1$ . We embed $B_{\infty } \hookrightarrow \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {C})$ by ${\mathbf{i}} \mapsto \left (\begin {smallmatrix} i & 0 \\ 0 & -i \end {smallmatrix}\right )$ , ${\mathbf{j}} \mapsto \left (\begin {smallmatrix} 0 & 1 \\ \pm 1 & 0 \end {smallmatrix}\right )$ . Then, P is asymptotic to the restriction of the squared Euclidean norm on the matrix entries, while q is the restriction of the determinant. We may assume that $x=[a_1,b_1,c_1]$ and $y=[a_2,b_2,c_2]$ are nonzero. We may assume (by the known $\delta = 1$ case) that $\delta $ is sufficiently small. The required conclusion then reduces via homogeneity to the following assertion: The commutator of any two matrices of the form

$$ \begin{align*} \begin{pmatrix} a_1 i & O(\delta^{1/2}) \\ O(\delta^{1/2}) & - a_1 i \end{pmatrix} \quad \text{and}\quad \begin{pmatrix} a_2 i & O(\delta^{1/2}) \\ O(\delta^{1/2}) & - a_2 i \end{pmatrix} \end{align*} $$

is of the form $\left (\begin {smallmatrix} O(\delta ) & O(\delta ^{1/2}) \\ O(\delta ^{1/2}) & O(\delta ) \end {smallmatrix}\right )$ , and in particular has Euclidean norm $O(\delta ^{1/2})$ and determinant $O(\delta )$ . Indeed, the product of any two matrices of the indicated form is readily computed to be of the form $\left (\begin {smallmatrix} -a_1a_2 + O(\delta ) & O(\delta ^{1/2}) \\ O(\delta ^{1/2}) & -a_1a_2 + O(\delta ) \end {smallmatrix}\right )$ , hence the commutator of two such matrices, being a difference of such products, has the required form.

10.3. The nonsplit case

We retain the above setting and further assume that B is nonsplit and that the level N of the Eichler order R is squarefree.

The following estimate is, in some sense, the most intricate one in the paper. It requires us to bound certain matrix counts in the critical range (see Remark 10.5 below) by essentially $O(1)$ , uniformly in the discriminant and level. To achieve such uniformity seems to require the delicate argument involving commutators recorded below.

Proposition 10.4. Let n be a nonzero integer, let $0 < \delta \le 1$ and let $T \ge \ell ^{-\frac {1}{2}}$ . Then the set has cardinality

$$\begin{align*}|S| \ll_{\varepsilon} (\ell T)^\varepsilon \tau(d_B N) \left( 1 + \frac{\ell^2}{d_B N} \min\left\{\delta^{\frac{1}{2}} T^2 , \frac{\delta T^4}{|n|} \right\} \right).\end{align*}$$

Remark 10.5. The critical range is when $n \asymp T^2 \asymp d_B N(1+k) / \ell ^2$ and $\delta \asymp (1+k)^{-1}$ ; in that range, we obtain $|S| \ll _{\varepsilon } (d_B N (1+k) )^\varepsilon $ .

Proof. Suppose S is not empty, and let $\gamma _1, \gamma _2 \in S$ . Our strategy will be to bound for each $\gamma _1$ the number of possibilities for $\gamma _2$ .

Set and . Then, $2 \gamma _1 \gamma _2 = a + \beta $ , $4 n^2 = a^2 + q(\beta )$ . In particular, $\gamma _2 = (a + \beta ) / 2 \gamma _1$ , so it suffices to bound the number of possibilities for a and $\beta $ .

Lemmas 10.2 and 10.3 give $q(\beta ) \ll \delta T^4$ and $q(\beta ) \in \frac {d_B N}{\ell ^3} \mathbb {Z}$ , that is,

$$\begin{align*}a^2 = 4 n^2 + O(\delta T^4),\end{align*}$$

$$\begin{align*}a^2 \equiv 4 n^2 \quad\pod{\tfrac{d_B N}{\ell^3}},\end{align*}$$

thus

$$\begin{align*}a = \pm 2 n + O\left(\min\left\{\delta^{\frac{1}{2}} T^2 , \frac{\delta T^4}{|n|} \right\} \right),\end{align*}$$

$$\begin{align*}a \equiv a_0 \quad\pod{\tfrac{d_B N}{\ell^2}}\end{align*}$$

for some sign $\pm $ and some residue class $a_0$ modulo $d_B N/ \ell ^2$ with $a_0^2 \equiv 4 n^2\ \, \pod{\tfrac {d_B N}{\ell ^2}}$ . Since $d_B N$ is squarefree, there are at most $\tau (d_B N)$ such classes. For each $a_0$ , the number of possibilities for a is $O(1 + (\ell ^2 / d_B N) \cdot \min \{\delta ^{\frac {1}{2}} T^2, \delta T^4 /|n|\})$ .

We now bound for each a the number of possibilities for $\beta $ . Let M denote the orthogonal complement in $\frac {1}{\ell } R^0$ of $\gamma _1$ , thus $M = \{ \gamma \in \frac {1}{\ell }R^0 : \operatorname {\mathrm {tr}}(\gamma \gamma _1) = 0 \}$ . By restricting q to M, we obtain an integral binary quadratic form. Since B is nonsplit, M is anisotropic. Since $\operatorname {\mathrm {tr}}(\gamma _1 \gamma _2 \gamma _1) = \operatorname {\mathrm {tr}}(\gamma _2 \gamma _1 \gamma _1)$ , we have $\beta \in M$ . From Lemma 10.3, we obtain $P(\beta ) \ll T^4$ . Thus, $\beta $ satisfies the system

$$\begin{align*}\beta \in M, \quad q(\beta) = 4n^2 - a^2, \quad P(\beta) \ll T^4. \end{align*}$$

Since $q(M) \subseteq \frac {1}{\ell ^2} \mathbb {Z}$ and $4 n ^2 - a^2 \ll \delta T^4$ , we see by Lemma 10.1 that the number of possibilities for $\beta $ is $\ll _{\varepsilon } ( \ell T)^\varepsilon $ .

By multiplying together the number of possibilities for $\pm , a_0, a$ and $\beta $ , we achieve the required bound.

10.4. Extension to the split case

Recall from §2.2, that we may assume our Eichler order of level N in the split quaternion algebra $B = \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Q})$ to be of the shape $g^{-1}Rg$ , where $R=\left (\begin {smallmatrix} \mathbb {Z} & \mathbb {Z}\\ N\mathbb {Z} & \mathbb {Z} \end {smallmatrix}\right )$ and $g \in G(\mathbb {R})$ . Our aim is to bound the cardinality of the set

In the nonsplit case, we had verified that

$$\begin{align*}|S| \ll_{\varepsilon} C,\end{align*}$$

where

We extend this to the split case as follows.

Proposition 10.6. Let n be an integer. If $-n$ is not a square, then $|S| \ll _{\varepsilon } C$ . If $-n$ is a square, then $|S| \ll _{\varepsilon } C + \delta ^{1/2} T \ell H(g)$ , where H denotes the normalized height function defined in §2.2.

Proof. We proceed as in the original argument, aiming to bound for fixed $\gamma _1 \in gSg^{-1}$ the number of possible $\gamma _2 \in gSg^{-1}$ . As before, we write

$$\begin{align*}\alpha = \operatorname{\mathrm{tr}}(\gamma_1 \gamma_2) = \gamma_1 \gamma_2 + \gamma_2 \gamma_1 \in \tfrac{1}{\ell}\mathbb{Z},\end{align*}$$

(10.2) $$ \begin{align} \beta = [\gamma_1, \gamma_2] = \gamma_1 \gamma_2 - \gamma_2 \gamma_1 \in \tfrac{1}{\ell} R^0 \end{align} $$

so that

$$\begin{align*}2 \gamma_1 \gamma_2 = \alpha + \beta,\end{align*}$$

$$\begin{align*}4n^2 = q(\gamma_1 \gamma_2) = \alpha^2 + q(\beta).\end{align*}$$

Since $\gamma _1 = (\alpha + \beta )/2 \gamma _2$ , it suffices to count the number of possible pairs $(\alpha ,\beta )$ . The pairs with $q(\beta ) \neq 0$ may be counted as before after noting that the restriction of q to the orthogonal complement of $\gamma _1$ is nondegenerate. There are at most two pairs with $\beta = 0$ , since then $\alpha = \pm 2n$ . We thereby reduce to counting the number of pairs for which

$$\begin{align*}q(\beta) = 0, \quad \beta \neq 0.\end{align*}$$

Recall the notation of §4.3.1. We note that for each cusp $\mathfrak {a}$ , we may and shall choose $\sigma _{\mathfrak {a}} \in \Gamma _0(N) \tau _{t}$ for some $t | N$ . With this choice, we have

(10.3) $$ \begin{align} \sigma_{\mathfrak{a}}^{-1} R \sigma_{\mathfrak{a}} = \begin{pmatrix} \mathbb{Z} & w_{\mathfrak{a}}\mathbb{Z} \\ \frac{N}{w_{\mathfrak{a}}}\mathbb{Z} & \mathbb{Z} \end{pmatrix}, \end{align} $$

where $w_{\mathfrak {a}}$ is the cusp width of the cusp $\mathfrak {a}$ . This is easily verified locally. We further introduce the following notation: For $\mathfrak {a} \in \mathbb {P}^1(\mathbb {Z})$ and $\kappa \in B^0$ , we set .

We observe, by Equation (10.2), that $\ell \beta $ is a nonzero element of $R^0 \subseteq \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Z})^0$ . There is thus a unique (up to sign) primitive element $\beta _0$ of $\operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Z})^0$ that generates $\mathbb {Q} \ell \beta \cap \operatorname {\mathrm {Mat}}_{2 \times 2}(\mathbb {Z})^0$ . We then have $\beta \in \frac {1}{\ell } \mathbb {Z} \beta _0$ .

We may and shall choose $\mathfrak {a}$ so that $\beta _0^{\mathfrak {a}} = \left (\begin {smallmatrix} 0 & \pm 1 \\ 0 & 0 \end {smallmatrix}\right )$ . Note that $\beta $ (equivalently, $\beta _0$ ) is orthogonal not only to $\gamma _1$ (as was used in the original argument), but also to $\gamma _2$ . From this, we deduce that

$$\begin{align*}\gamma_2^{\mathfrak{a}} = \begin{pmatrix} a & b \\ 0 & -a \end{pmatrix}\end{align*}$$

for some $a \in \mathbb {Z}$ and $b \in \frac {1}{\ell } \mathbb {Z}$ . We have $n = q(\gamma _2) = -a^2$ , which shows that $-n$ must be a square and also that there are at most two possibilities for a. It remains to verify that the number of possibilities for b is $O(1+\delta ^{1/2} T \ell H(g))$ . We will show in fact that the number of b is

$$\begin{align*}O(1 + \delta^{1/2} T \ell y_{\mathfrak{a}} / w_{\mathfrak{a}}),\end{align*}$$

where $\sigma _{\mathfrak {a}}^{-1}z=z_{\mathfrak {a}}=x_{\mathfrak {a}}+iy_{\mathfrak {a}}$ . To see this, observe first that the condition $\gamma _2 \in R(\ell )^0 \subseteq \frac {1}{\ell } R^0 \Leftrightarrow \gamma _2^{\mathfrak {a}} \in \sigma _{\mathfrak {a}}^{-1}R(\ell )^0\sigma _{\mathfrak {a}} \subseteq \frac {1}{\ell } \sigma _{\mathfrak {a}}^{-1}R^0\sigma _{\mathfrak {a}}$ yields the congruence $b \equiv 0\ \,\pod {\tfrac {w_{\mathfrak {a}}}{\ell }}$ ; see Equation (10.3). The condition $\gamma _2 \in g\Omega (\delta ,T)g^{-1}$ may be restated as

$$\begin{align*}\gamma_2^{\mathfrak{a}} \in \sigma_{\mathfrak{a}}^{-1}g \Omega(\delta, T) (\sigma_{\mathfrak{a}}^{-1}g)^{-1}.\end{align*}$$

Let $g'$ be an upper-triangular element of $G(\mathbb {R})$ for which $g' \cdot i = \sigma _{\mathfrak {a}}^{-1}g i=\sigma _{\mathfrak {a}}^{-1} z = z_{\mathfrak {a}}$ . Then, by the $K_{\infty }$ invariance of $\Omega (\delta ,T)$ on the left and right, we have $\gamma _2^{\mathfrak {a}} \in g' \Omega (\delta , T) (g')^{-1}$ . We compute

$$\begin{align*}(g')^{-1} \gamma_2^{\mathfrak{a}} g' = \begin{pmatrix} a & y_{\mathfrak{a}}^{-1} (b + 2 a x_{\mathfrak{a}}) \\ 0 & -a \end{pmatrix} \in \Omega(\delta,T).\end{align*}$$

This last condition forces b to lie in an interval of length $O(\delta ^{1/2} T y_{\mathfrak {a}})$ . We thereby obtain the required bound for the number of possible b’s.

10.5. Proof of Theorem 2.6

We may split the set $\Omega (\delta , T)$ into $\Omega (1/16, 4\delta ^{\frac {1}{2}} T)$ and the dyadic sets which are comprised of the elements $[a,b,c] +d\in B_{\infty }$ for which

$$\begin{align*}\tfrac{1}{2} \delta_j T^{2} \le a^2 + b^2 + c^2 + d^2 \le \delta_j T^2, \quad b^2 + c^2 \le \delta T^2, \end{align*}$$

for some $16 \delta \le \delta _j \le 1$ . We note that these are contained in $\Omega (\delta \delta _j^{-1} , \delta _j^{\frac {1}{2}} T)$ . In order for the dyadic sets to contain an element of trace $0$ and norm n, one must have $|n| \asymp a^2 \asymp \delta _j T^2$ . Hence, if we apply Propositions 10.4 and 10.6, we get

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

where we used $|n|\asymp \delta _jT^2$ , and

$$ \begin{align*} |g^{-1} R(\ell)^0 g \cap \Omega(\tfrac{1}{16}, 4 \delta^{\frac{1}{2}}T) \cap \det{}^{-1}(\{n\})| \prec 1+ \ell \delta^{\frac{1}{2}}H(g)T +\frac{\ell^2}{d_BN} \delta T^2. \end{align*} $$

If $\delta $ happens to be very small, say, if $\delta \le (16 d_BNT^2)^{-1}$ , then it suffices to consider only $\delta _j \ge (d_BNT^2)^{-1}$ and the final set $\Omega (\delta d_B N T^2,(d_BN)^{-\frac {1}{2}})$ . This avoids a factor $\delta ^{-o(1)}$ for a too small $\delta $ . We conclude Theorem 2.6.