Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-manifolds

Fix integers $r,s\geq 0$ so that $r+s\geq 1$ and consider the symmetric space

where $\mathbb{H}^{(n)}$ denotes hyperbolic $n$-space. Let $\Gamma$ be a lattice in the isometry group $G=\mathrm{SL}_{2}(\mathbb{R})^{r}\times\mathrm{SL}_{2}(\mathbb{C})^{s}$. Then the finite-volume manifold $Y=\Gamma\backslash S$ is equipped with a family of $r+s$ commuting differential operators coming from the Laplace–Beltrami operator in each factor (the sum of which is the Laplace–Beltrami operator of $Y$). When $\Gamma$ is, in addition, a congruence lattice the manifold $Y$ is also equipped with a ring of discrete averaging operators, the Hecke algebra, which further commute with the differnential operators noted above. Write $dy=\mathop{}\!\mathrm{d}\!\operatorname{vol}_{Y}$ for the Riemannian volume element normalized to have total volume 1. In this paper we establish the following result:

The new ingredient in the proof is a method for ruling out certain measures as components of weak-$*$ limits of lifts of this setup to $X=\Gamma\backslash G$. To state its realization here let $G_{i}$ be one of the factors isomorphic to $\mathrm{SL}_{2}(\mathbb{C})$ in the product defining $G$. Identifying $G_{i}$ with $\mathrm{SL}_{2}(\mathbb{C})$ for the moment let $H_{i}=\mathrm{SL}_{2}(\mathbb{R})$, and let $M_{i}=\{\operatorname{diag}(e^{i\theta},e^{-i\theta})\}$ be the group of diagonal matrices with entries of modulus $1$. Finally set $H=H_{i}\times\prod_{j\neq i}G_{j}$ (i.e. multiply $H_{i}$ at the $i$th factor with the full isometry groups $G_{j}$ at the other factors). Let $g=(g_{j})_{j}\in G$ be such that the entries of the matrix $g_{i}$ are algebraic numbers.

More generally, let $G$ be a semisimple Lie group, $K\subset G$ a maximal compact subgroup, $\Gamma<G$ a lattice, and consider the locally symmetric space

with its uniform probability measure $dy$. The ring of $G$-invariant differential operators on $G/K$ is commutative and commutes with the action of $\Gamma$, hence acts on functions on $Y$. Let $\{\psi_{j}\}_{j\geq 1}\subset L^{2}(Y)$ be an orthonormal sequence of joint eigenfunctions of this ring. We remark that the Laplace–Beltrami operator of $Y$ belongs to the ring and that writing $\lambda_{j}$ for the corresponding eigenvalue for $\psi_{j}$ we necessarily have $\left|{\lambda_{j}}\right|\to\infty$. The Quantum Unique Ergodicity (QUE) Conjecture for locally symmetric spaces asserts the sequence $\{\psi_{j}\}$ becomes equidistributed on $Y$, in the sense that

for all test functions $f\in C_{c}(Y)$. In other words, the sequence $|\psi_{j}|^{2}dy$ of probability measures converges in the weak-$*$ topology to the uniform measure $dy$. Next, when $\Gamma$ is a congruence lattice the space $Y$ also affords a commutative family of discrete averaging operators, the Hecke operators, which commute with the invariant differential operators. The Arithmetic QUE Conjecture (AQUE) is the restricted form of the QUE Conjecture for sequences of Hecke eigenfunctions, that is for seuqences where each $f_{j}$ is simultaneously an eigenfunction of the ring of invariant differential operators and of the ring of Hecke operators.

Investigation of Arithmetic QUE goes back to the work [11] of Rudnick–Sarnak, who formulated the QUE conjecture for hyperbolic surfaces and, more generally, for compact manifolds of negative sectional curvature. A major breakthrough on this problem was due to Lindenstrauss, who in [8] established AQUE for congruence hyperbolic surfaces, that is congruence quotients $Y=\Gamma\backslash\mathbb{H}^{(2)}$ of the hyperbolic plane. More precisely, Lindenstrauss proved that weak-$*$ limits as above were proportional to the uniform measure. This fully resolved the Conjecture in the case of uniform lattices, i.e. when the quotient is compact. For non-uniform lattices, however, the possibility remained that the constant of proportionality was strictly less than one ("escape of mass"), an alternative ruled out later by Soundararajan [16]. Silberman–Venkatesh [13, 14] obtains generalizations of some of this work to the case of general semisimple groups $G$, formulating the QUE and AQUE conjectures in the context of locally symmetric spaces and obtaining further cases of AQUE.

As stated above, we establish AQUE for congruence quotients of products of hyperbolic 2- and 3-spaces. The case $Y=\mathrm{SL}_{2}(\mathbb{Z}[i])\backslash\mathbb{H}^{(3)}$ is already new and contains most of the the novel ideas of this paper; reading the paper with this assumption in mind will give the reader most of the insight but avoid the technical legedermain needed to handle number fields with multiple infinite places. For the rest of the introduction we concentrate on this case.

Thus let $G=\mathrm{SL}_{2}(\mathbb{C})$ acts transitively by isometries on the space $S=\mathbb{H}^{(3)}$ with point stablizer the maximal compact subgroup $K=\mathrm{SU}(2)$. Let $\Gamma=\mathrm{SL}_{2}(\mathbb{Z}[i])$, the group of unimodular matrices with Gaussian integer entries, which is indeed a lattice in $G$. Then $Y=\Gamma\backslash S$ is a finite-volume hyperbolic $3$-manifold and $X=\Gamma\backslash G$ is its frame bundle. The action of $A=\{\operatorname{diag}(a,a^{-1})\mid a>0\}$ on $X$ from the right corresponds to the geodesic flow on the frame bundle, with the commuting subgroup $M=\{\operatorname{diag}(e^{i\theta},e^{-i\theta})\}$ acting by rotating the frame around the tangent vector.

Let $H=\mathrm{SL}_{2}(\mathbb{R})\subset G$. Then $Y_{H}=\mathrm{SL}_{2}(\mathbb{Z})\backslash H/\mathrm{SO}(2)\subset Y$ is a finite-volume hyperbolic surface embedded in $Y$; its unit tangent bundle is the finite-volume $H$-orbit $X_{H}=\mathrm{SL}_{2}(\mathbb{Z})\backslash H\subset X$. Very relevant to us will be the subset $\mathrm{SL}_{2}(\mathbb{Z})\backslash HM$, correspoding to the pullback of the frame bundle of the hyperbolic $3$-fold to the surface. Geometrically the unit tangent bundle of $Y_{H}$ embeds in the frame bundle of $Y$ in multiple ways invariant by the geodesic flow, parametrized by choosing a normal vector to the tangent vector at one point. The set of choices is thus parametrized by the group $M$ rotating the frame at the chosen point around the tangent vector.

The Casimir element in the universal enveloping algebra of the Lie algebra $G$ acts on functions on $X$ and on $Y$, where it is propotional to the Laplace–Beltrami operator. In addition there is a family of discrete averaing operators acting on functions on $X$ and commuting with the right $G$ action, hence also on functions on $Y$ (thought of as $K$-invariant functions on $X$). These Hecke operators can be constructed as follows: for each $g\in\mathrm{SL}_{2}(\mathbb{Q}(i))$ the set $[g]=\Gamma\backslash\Gamma g\Gamma\subset\Gamma\backslash\mathrm{SL}_{2}(\mathbb{Q}(i))$ is finite, so for a function $f$ on $X$ we can set

Since these operators are $G$-equivariant they also commute with the Casimir element, and hence with the Laplace–Beltrami operator on $Y$. They are bounded (the $L^{2}$ operator norm of $T_{g}$ is at at most the cardinality of $[g]$) and it is not hard to check that the adjoint of $T_{g}$ is $T_{g^{-1}}$. It is a non-trivial fact that the $T_{g}$ commute with each other, and hence are a commuting family of bounded normal operators. For the sequel we will take a different point of view that gives better control of these operators; see the construction in Section 2.5.

The proof of 1 is given in Section 3. Our strategy is the one laid down by Lindenstrauss and followed by later work on AQUE.

1. (1)

   The microlocal lift of [13] (see also [7] for the case $s=0$; the original such constructions in much greater generality are due to Shnirelman, Zelditch and Colin de Verdière [15, 18, 4]) shows that any weak-* limit on $Y$ as in 1 is the projection from $X$ of a limit $\mu$ as in 2 which, in addition, is an $AM$-invariant measure.

2. (2)

   We show that almost every $A$-ergodic component of $\mu$ has positive entropy under the $A$-action. These arguments are standard (see [14] generalizing [1]) but we need to adjust them to handle the fact that $M$ isn’t finite and that the group is essentially defined over a number field. What is shown is that the measure of an $\epsilon$-neighbourhood of a (compact piece) of an orbit $xAM$ in $X$ decays at least as fast as $\epsilon^{h}$ for some $h>0$.

3. (3)

   The classification of $A$-invariant measures in [5] implies that all the ergodic components of $\mu$ other than the $G$-invariant measure are contained in sets of the form $xHM$ where $xH$ is a finite-volume $H$-orbit in $X$.

4. (4)

   In a closed $H$-orbit $\Gamma gH\subset X$ the entries of $g$ are algebraic numbers, so by 2 the exceptional possibilies are contained in a countable family of sets each of which has measure zero. It follows that $\mu$ is $G$-invariant.

5. (5)

   It was shown in Zaman’s M.Sc. Thesis [17] that the measure $\mu$ is a probability measure (a generalization of Soundararajan’s proof [16] for the case of $\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathrm{SL}_{2}(\mathbb{R})$).

The bulk of the paper is then devoted to realizing step 4. Before discussing how that is achieved, let us remark on its necessity. The group $G=\mathrm{SL}_{2}(\mathbb{R})$ has no proper semisimple subgroups, so the issue of ruling out components supported on orbits of such subgroups did not arise in the original work of Lindenstrauss. In followup generalizations to higher-rank groups the difficulty was addressed by choosing the lattice $\Gamma$ so that the subgroups $H$ that could arise did not have finite-volume orbits on $X$ and $G$ itself was chosen $\mathbb{R}$-split so that $M$ was finite and could be ignored.

We now go further. The group $G=\mathrm{SL}_{2}(\mathbb{C})$ has $H=\mathrm{SL}_{2}(\mathbb{R})$ as a relevant subgroup, forcing us to confront the problem of ruling out components supported on finite-volume $H$-orbits head-on. It is also not $\mathbb{R}$-split, forcing us to contend with the infinitude of $M$. Indeed, equip a finite-volume $H$-orbit $xH$ with the $H$-invariant probability measure $\nu$. Translating $\nu$ by $m\in M$ gives further $A$-invariant probability measures supported on the subsets $xHm$, and averaging these measures to gives an $A$-invariant probability measure supported on $xHM$. The construction in step 1 is such that if $\nu$ occurs in the ergodic decomposition of $\mu$ then so do its $M$-translates, so we need to rule out the averaged measure on $xHM$ as occuring in $\mu$ (fortunately there are only countable many such subsets). Now previous technology as mentioned in step 2 could show that ($\epsilon$-neighborhoods of pieces of) subgroup orbits such as $xH$ have small measure¹¹1This technology does work in our context too if one uses the new amplifier of Section 5. However, that $\mu(xH)=0$ also automatically follows from the $M$-invariance on $\mu$, much in the same way that the $\mathbb{R}$-invariant of Lebesgue measure shows that it has no atoms. While showing that $\mu(xHm)=0$ is insufficient (there are uncountably many such subsets), still showing $\mu(xH)=0$ for limits of Hecke eigenfunctions was an important conceptual step in this work. but $xHM$ is not of this form. Instead the subgroup generated by $HM$ is all of $G$, so the only orbit of a closed subgroup containing $xHM$ is all of $X$.

Let $U\subset xHM$ be a bounded open set. Then showing that $\mu(xHM)=0$ amounts to bounding the mass Hecke eigenfunctions can give to $\epsilon$-neighbourhoods $U_{\epsilon}$. In other words, our goal is to prove

for any Hecke-eigenfunction $\phi$, uniformly in $\phi$.

We find a Hecke operator (“amplifier”) $\tau$ which acts on $\phi$ with a large eigenvalue $\Lambda$ yet geometrically “smears” ${U_{\epsilon}}$ around the space. The operator $\tau$ is a linear combination of operators $T_{g}$ so there is a subset $\operatorname{supp}(\tau)\subset\mathrm{SL}_{2}(\mathbb{Z}[i])\backslash\mathrm{SL}_{2}(\mathbb{Q}(i))$ so that $\Lambda\phi(x)=(\tau\star\phi)(x)=\sum_{s\in\operatorname{supp}(\tau)}\tau(s).\phi(sx)$. Choosing $\tau$ so that $\left|{\tau(s)}\right|\leq 1$ for each $s\in\operatorname{supp}(\tau)$, squaring, applying Cauchy–Schwartz and integrating over ${U_{\epsilon}}$ gives

The best kind of smearing is thus if there were very few non-empty intersections between the translates $s.{U_{\epsilon}}$, where a bound would follow as long as we could arrage for $\Lambda^{2}$ to be large compared with the size of the support of $\tau$, but unfortunately that is too got to hope fore. Thankfully with better geometric and spectral arguments than the naive one given above one requires less stringent control over the intersections. For increasingly sophisticated implementations of this strategy see [11, 1, 14, 2, 12], all in settings where $U$ is a piece of an orbit $xL$ for a subgroup $L$. Then an intersection between $s.{U_{\epsilon}}$ and ${U_{\epsilon}}$ (say) implies that $s$ is $O(\epsilon)$-close to the bounded neighbourhood $xUU^{-1}x^{-1}$ in $xLx^{-1}$; choosing the $s$ in the support of of $\tau$ to have sufficiently small denominators relative to $\epsilon$ then forces all the $s\in\mathrm{SL}_{2}(\mathbb{Q}(i))$ causing the intersections to jointly be contained in a single conjugate of $L$. If $L$ is very small (e.g. is a torus) or if the lattice $\Gamma$ is chosen so that the rational points must generate a torus, there will be very few intersections; it is sometimes even possible to choose the support of $\tau$ to avoid them entirely.

Here $UU^{-1}$ (let alone $U_{\epsilon}U_{\epsilon}^{-1}$) is an open subset of $G$ (a reflection of the fact that $HM$ generates $G$), so the number of intersections is large: $s\in xUU^{-1}x^{-1}$ holds generically, and no good bound on the number of intersections is possible. Instead for the first time we go beyond estimating the number of intersections and instead bound the measure of the pieces through an induction argument on their dimension. For this lift the picture to $G$, replacing $x\in X$ with a representative $g\in G$ and $L$ with the submanifold $L=gHM$ which is, in fact, an irreducible real algebraic subvariety of $G$. An intersection $sL\cap L$ with $s\in\mathrm{SL}_{2}(\mathbb{Q}(i))$ is then also a subvariety, and hence one of two possibilities must hold: either the intersection has strictly smaller dimension, in which case we call the intersection (and, by abuse of language, the element $s$) transverse, or it is not, in which case we call $sL$ (and $s$) parallel to $L$ and must actually have $sL=L$ by the irreducibility of $L$. By induction we may assume that the measures of all transverse intersections $s.{U_{\epsilon}}\cap{U_{\epsilon}}$ are small (one needs to show that $s.{U_{\epsilon}}\cap{U_{\epsilon}}$ is contained in a decreasing neighbourhood of $s.U\cap U$). On the other hand the parallel elements stabilize the subvariety $L$ and this forces them to lie in a proper subgroup $S\subset\mathrm{SL}_{2}(\mathbb{Q}(i))$ (here we gained over the naive attempt which considered the much larger subgroup generated by $\mathrm{SL}_{2}(\mathbb{Q}(i))\cap LL^{-1}$).

For example, the stabilizer of $HM$ in under the left action of $G$ on itself is exactly $H$, so parallel intersections can only arise from $s\in\mathrm{SL}_{2}(\mathbb{Q})$. We now introduce a final idea, allowing us to deal with a situation where this group is not a torus. Let $p\equiv 1\,(4)$ be a prime number so that $p=(a+bi)(a-bi)$ for integers $a,b$ and set

It turns out that for any Hecke eigenfunction $\phi$, the eigenvalue of $\lambda_{p}$ of one of the two operators $T_{g_{p}}$, $T_{g_{p}^{2}}$ is at least comparable to the square root of the size of its support. In addition, the supports $[g_{p}],[g_{p}^{2}]$ of these operators do not intersect $\mathrm{SL}_{2}(\mathbb{Q})$: complex conjguation exchanges the Gaussian prime $a+bi$ with the distinct Gaussian prime $a-bi$, so the ratio of matrices from the two double cosets must contain both primes and cannot be real), so these operators themselves do not cause intersections. Making use of these as building blocks and combining the contribution from many primes we then construct an amplifier which has good spectral properties and at the same time avoids intersections caused by $\mathrm{SL}_{2}(\mathbb{Q})$.

Each closed orbit $xHM$ is the projection of $L=gHM$ where the entries of $g$ are algebraic but need not be rational, so greater care must be taken to define the complex conjugation which limits the intersection and correspondgly the arithmetic progression from which we select the primes needs to be smaller. Also, we need to consider the stabilizers of the subvarieties of $L$ that will appear in the recursive argument. We show these stabilizers are either contained in a conjguate of $H$ (so that a corresponding complex conjugation exists) or are tori (which are already known to cause few intersections).

In Section 2 we fix our notation and examine the algebraic algebraic structure of forms of $\mathrm{SL}_{2}$ over number fields, constructing the complex conjugations which control returns of Hecke translates to real submanifolds. We take the adelic point view of the Hecke operators, which is more convenient than the one used for the introduction. In Section 3 we then reduce 1 to 2. Before giving the proof of 2 in Section 6 we devote Section 4 to classifying the stabilizers of the subvarieties that can occur in the recursion and Section 5 to constructing the amplifier.

Varieties (including algebraic groups) will be named in blackboard font,

Fix a number field $F$, and let $V=\left|{F}\right|$ be the set of places of $F$, $V_{\infty}\subset V$ the set of archimedian places, divided into real and complex places as $V_{\infty}=V_{\mathbb{R}}\sqcup V_{\mathbb{C}}$. For a place $v\in V$ write $F_{v}$ for the completion of $F$ at $v$. If the place is finite write $\mathcal{O}_{v}\subset F_{v}$ for the maximal compact subring $\{x\in F_{v}\mid\left|{x}\right|_{v}\leq 1\}$, and let $\varpi_{v}\in\mathcal{O}_{v}$ be a uniformizer. We normalize the associated absolute value so that $q_{v}=\left|{\varpi^{-1}}\right|_{v}$ is the cardinality of the residue field $\mathcal{O}_{v}/\varpi\mathcal{O}_{v}$. For a rational prime $p$ let $F_{p}=\prod_{v|p}F_{v}$ and similarly set $F_{\infty}=\prod_{v\mid\infty}F_{v}$. We write $\mathbb{A}_{\mathrm{f}}$ for the ring of finite adeles $\prod^{\prime}_{v<\infty}(F_{v}:\mathcal{O}_{v})$ and $\mathbb{A}_{F}$ for the ring of adeles $F_{\infty}\times\mathbb{A}_{\mathrm{f}}$.

If $\mathbb{G}$ is an $F$-variety and $E$ is an $F$-algebra write $\mathbb{G}(E)$ for the $E$-points of $\mathbb{G}$, equipped with the analytic topology if $E$ is a local field extending $F$. For a place $v$ of $F$ we also write $G_{v}=\mathbb{G}(F_{v})$. We further write $G=G_{\infty}$ for the manifold (Lie group) $\prod_{v\mid\infty}G_{v}$.

Let $D$ be a quaternion algebra over $F$, that is either the matrix algebra $M_{2}(F)$ or a division algebra of dimension $4$ over $F$ (for a first reading let $F=\mathbb{Q}(i)$, $D=M_{2}(F)$. Let $\mathbb{D}$ be the variety such that $\mathbb{D}(E)=D\otimes_{F}E$, and let $\det\colon\mathbb{D}\to\mathbb{A}^{1}$ be the reduced norm. Let $\mathbb{G}=\mathbb{D}^{1}$ be the algebraic group of elements of reduced norm $1$:

For each place $v\in V$ we have the algebra $D_{v}=D\otimes_{F}F_{v}$. When $v$ is complex necessarily $D_{v}\simeq M_{2}(\mathbb{C})$ and thus $G_{v}\simeq SL_{2}(\mathbb{C})$. When $v$ is real $D_{v}$ is either the split algebra $M_{2}(\mathbb{R})$ or Hamilton’s quaternions $\mathbb{H}$, and correspondingly $G_{v}$ is one of the groups $\mathrm{SL}_{2}(\mathbb{R})$ and $\mathbb{H}^{1}\simeq\mathrm{SU}(2)$. We suppose there are $s$ complex places and $r+t$ real places divided into $r$ of the first form and $t$ of the second so that

.

We have the factorizations $G_{\infty}=\prod_{v\mid\infty}G_{v}$ (and similarly we can also write $G_{p}=\prod_{v\mid p}G_{v}$ where $p$ is a finite place of $\mathbb{Q}$). Let $g\in\mathbb{G}(F)$ and suppose we have some information on in the image of $g$ in $G_{v}$ for an archimedean place $v\in V_{\mathbb{C}}$. A-priori there seems to be no way to translate this to information about $g$ at a particular finite place over $p$. However, it turns out that this is possible if we restrict our attention to a positive density subset of the primes: for such primes we will enumerate the respective places of $F$ over the two places $\infty,p$ of $\mathbb{Q}$ in such a way that the factors in some sense correspond.

Moreover, for a complex place $v\in V_{\mathbb{C}}$, the usual extension of scalars realizes $G_{v}$ as the group of complex points of the $\mathbb{C}$-group $\mathbb{G}\times_{F}\mathbb{C}$. However, when thought of as a Lie group this group has closed subgroups which are not complex – the key example for us being the subgroup $\mathrm{SL}_{2}(\mathbb{R})$ of $\mathrm{SL}_{2}(\mathbb{C})$. We thus would like to think of the group $G_{v}$ as the group of real points of an algebraic group defined over an extension of $F$.

In this section we address both concerns simultaneously: given the complex place $v$ we construct a number field $E$ equipped with a real place $w$ and an algebraic group $\tilde{\mathbb{G}}$ defined over $E$ factoring as an $E$-group into a product $\tilde{\mathbb{G}}=\prod_{i=1}^{r+t+2s}\tilde{\mathbb{G}}_{i}$ such that:

1. (1)

   The group $\mathbb{G}(F)$ embeds in $\tilde{\mathbb{G}}(E)$.

2. (2)

   We have an isomorphism

   $$\tilde{\mathbb{G}}(E_{w})=\prod_{i=1}^{r+t+s}\tilde{\mathbb{G}}_{i}(E_{w})\to G_{\infty}$$

   where the factors correspond.

3. (3)

   For a rational prime $p$ splitting completely in a particular quaratic extension $N$ of $E$ we have a place $E\hookrightarrow\mathbb{Q}_{p}$ such that For $1\leq i\leq s$ we have $\tilde{\mathbb{G}}_{i}(\mathbb{Q}_{p})\simeq(\mathrm{SL}_{2}(\mathbb{Q}_{p}))^{2}$, for $s<i\leq r+t+s$ we have $\tilde{\mathbb{G}}_{i}(\mathbb{Q}_{p})\simeq\mathrm{SL}_{2}(\mathbb{Q}_{p})$ and the resulting factorization of $\tilde{\mathbb{G}}(\mathbb{Q}_{p})$ is isomorphic to the factorization of $G_{p}$ into $2s+r+t$ copies of $\mathrm{SL}_{2}(\mathbb{Q}_{p})$ coming from the $2s+r+t$ places over $F$ over $\mathbb{Q}_{p}$.

4. (4)

   The latter two identifications are compatible with first embedding, in such a way that if an element $\gamma\in\mathbb{G}(F)$ embeds to a real-valued matrix in $\tilde{\mathbb{G}}_{1}(E_{v})$ (say) then its image in $\tilde{\mathbb{G}}_{1}(\mathbb{Q}_{p})$ lies in the diagonal subgroup of $\tilde{\mathbb{G}}_{1}(\mathbb{Q}_{p})\simeq(\mathrm{SL}_{2}(\mathbb{Q}_{p}))^{2}$. This will allow us to choose Hecke operators which "avoid" a real subgroup in a particular complex place.

Thus let $v$ be a complex place of $F$ and let $N$ be any finite Galois extension of $\mathbb{Q}$ containing $F$ (in Section 3 $N$ will the Galois closure of a splitting field of $D$; in Section 5 we make a different choice depending on the subvariety we are trying to avoid). Let $w$ be a place of $N$ extending $v$, let $c_{w}\in\operatorname{Gal}(N/\mathbb{Q})$ the element acting as complex conjugation in the completion of $N$ at $v$, let $E=N^{c_{w}}$ be the fixed field of $c_{w}$, and also write $w$ for the restriction of $w$ to $E$ – a real place of that field. The situation is simpler when $E$ contains $F$ (e.g. our running example of $F=N=\mathbb{Q}(i)$ where $E=\mathbb{Q}$) but we will not assume this is the case.

Writing $F=\mathbb{Q}[x]/(f)$ for an irreducible polynomial $f\in\mathbb{Q}[x]$ and factoring $f$ in the extensions $N$ and $E$ of $\mathbb{Q}$ we see that the $E$-algebra $A=F\otimes_{\mathbb{Q}}E$ is étale, we have $A\simeq\prod_{i}E_{i}$ which we can interpret both as an isomorphism of $F$-algebras and as an isomorphism of $E$-algebras. From the first point view composing each embedding $F\hookrightarrow E_{i}$ with the embedding $w\colon E_{i}\to\mathbb{C}$ we obtain an enumeration of the archimedean places of $F$, with $E_{i}\simeq N$ for complex places (say those are indexed by $1\leq i\leq s$) and $E_{i}\simeq E$ for real places (say for $s<i\leq s+r+t$). Without loss of generality we assume the inclusion $F\hookrightarrow E_{i}\hookrightarrow\mathbb{C}$ is the place $v$ fixed above.

Now thinking of $A$ as a $E$-algebra let $\tilde{\mathbb{G}}=\mathbb{G}\times_{F}A$ thought of as an algebraic group over $E$. Equivalently $\tilde{\mathbb{G}}=\left(\operatorname{Res}^{F}_{\mathbb{Q}}\mathbb{G}\right)\times_{\mathbb{Q}}E$ (Weil restriction of scalars). The factorization of $A$ then gives a factorization $\tilde{\mathbb{G}}=\prod_{i}\tilde{\mathbb{G}}_{i}$ as groups over $E$, where $\tilde{\mathbb{G}}_{i}(E_{w})\simeq\mathbb{G}(F_{v})$ if $v$ is the $i$th archimedean place of $F$ under the identification above. Having done this we also write $G_{i}$ for $G_{v}$, especially in situations where we would like $v$ to vary over finite places.

The embedding $F\hookrightarrow A$ gives an inclusion $\mathbb{G}(F)\to\tilde{\mathbb{G}}(E)$.

Finally let $p$ be a rational prime which splits completely in $N$ (hence also in $F\subset N$). Choosing any place $w_{p}\colon N\to\mathbb{Q}_{p}$ lying over $p$ all embeddings of $F$ into $\mathbb{Q}_{p}$ factor through $w_{p}$ (as above, with different embeddings of $F$ into $N$). Restricting $w_{p}$ to $E$ we then have

since when $E_{i}=N$ there are two places of $N$ lying over the place $w_{p}$ of $E$. Furthermore, $\operatorname{Gal}(N/E)=\{c_{w},\operatorname{id}\}$ acts transitively on these places so that $c_{w}$ swaps them.

Finally let $\gamma\in\mathbb{G}(F)$ and suppose that its image in $\tilde{\mathbb{G}}_{1}(E_{w})$ lies in the subgroup $g_{1}\tilde{\mathbb{G}}_{1}(E_{w})^{c_{w}}g_{1}^{-1}$ for some $g_{1}\in\tilde{\mathbb{G}}_{1}(E)$ (we think of $\tilde{\mathbb{G}}_{1}(E_{w})^{c_{w}}$ as the "standard" copy of $\mathrm{SL}_{2}(\mathbb{R})$ in $\tilde{\mathbb{G}}_{1}(E_{w})\simeq\mathrm{SL}_{2}(\mathbb{C})$). Since the image of $g_{1}^{-1}\gamma g_{1}in\tilde{\mathbb{G}}_{1}(E)$ is fixed by $c_{w}$, this persists when we embed our group in $\mathbb{Q}_{p}$ using the place $w_{p}$, and it follows that the image of $\gamma$ in $\tilde{\mathbb{G}}_{1}(E_{w_{p}})$ lies in $g_{1}\tilde{\mathbb{G}}_{1}(E_{w_{p}})^{c_{w}}g_{1}^{-1}$, in other words in a particular conjugate of the diagonal subgroup $\tilde{\mathbb{G}}_{1}(E_{w_{p}})^{c_{w}}$ (so-called because when we identified $\tilde{\mathbb{G}}_{1}(E_{w_{p}})^{c_{w}}\simeq\left(\mathrm{SL}_{2}(\mathbb{Q}_{p})\right)^{2}$ the Galois automorphism $c_{w}$ acts by exchanging the factors).

Before going any further, it may be helpful to have two concrete examples of the setup so far.

We return to the isomorphism $G_{\infty}\simeq\left(\mathrm{SL}_{2}(\mathbb{C})\right)^{s}\left(\mathrm{SL}_{2}(\mathbb{R})\right)^{r}\left(\mathrm{SU}(2)\right)^{t}$ and fix some subgroups of this group.

At each infinite $v$ where $D_{v}$ splits, let $A_{v}\subset G_{v}$ be the group corresponding to the group of diagonal matrices with positive real entries under the isomorphism above. Also let $K_{v}$ be a compatible maximal compact subgroup (corresponding to the subgroup $\mathrm{SU}(2)$ at a complex place, $\mathrm{SO}(2)$ at a real place). From these let $M_{v}=Z_{K_{v}}(A_{v})$ be the centralizer of $A_{v}$ in $K_{v}$, so that $M_{v}$ is the group $\{\pm I\}\subset\mathrm{SL}_{2}(\mathbb{R})$ at a real place or the group $U(1)\subset\mathrm{SL}_{2}(\mathbb{C})$ of diagonal matrices with inverse entries both of modulus $1$. Observe that in either case the group $A_{v}M_{v}$ consists of the $F_{v}$-points of a maximal $F_{v}$-split algebraic torus of $\mathbb{G}\times_{F}F_{v}$.

At the real places $v$ where $D_{v}$ remains a division algebra $K_{v}=G_{v}\simeq\mathrm{SU}(2)$ is a maximal subgroup. With this choice $K=K_{\infty}=\prod_{v\mid\infty}K_{v}$ is a maximal compact subgroup of $G$ and $G/K\simeq\left(\mathbb{H}^{(3)}\right)^{s}\left(\mathbb{H}^{(2)}\right)^{r}$.

We fix (arbitrarily) once and for all a left-invariant Riemannian metric on $G$ (equivalently, a positive definite quadratic form on $\operatorname{Lie}G$). This induces a left-invariant metric on $X=\Gamma\backslash G$ where the quotient map is non-expansive. For a subset $U\subset G$ we write ${U_{\epsilon}}$ for its $\epsilon$-neighborhood with respect to this metric. This metric is used in Sections 3 and 6, but in both cases we can first restrict our attention to compact subsets of $G$. The choice of metric thus affects some overall constants but not the bottom line. For example in Section 3 we establish bounds of the form $\mu({U_{\epsilon}})\leq C\epsilon^{h}$; with a different metric ${U_{\epsilon}}$ would be contained in $U_{c\epsilon}$ with respect to the new one and the bound would be identical except for the value of the constant $C$.

Let $R\subset D$ be an order, that is a subring which an $\mathcal{O}_{F}$-lattice in $D$. Then for every finite place $v$ of $F$, $R_{v}=R\otimes_{\mathcal{O}_{F}}\mathcal{O}_{v}$ is an order in $D_{v}$, that is a compact open $\mathcal{O}_{v}$-subalgebra of $D_{v}$. Its group of units $R_{v}^{1}$ is then a compact subgroup of $G_{v}$ which is a maximal compact subgroup $K_{v}$ at almost all places.

For all but finitely many places, $D_{v}\simeq M_{2}(F_{v})$ and then $G_{v}\simeq\mathrm{SL}_{2}(F_{v})$ and $K_{v}\simeq\mathrm{SL}_{2}(\mathcal{O}_{v})$ (we don’t choose $K_{v}$ at the finitely many places where $D_{v}$ is a division algebra or where $R_{v}$ is not a maximal order).

Let $\mathbb{G}(\mathbb{A}_{\mathrm{f}})=\prod^{\prime}_{v<\infty}(G_{v}:K_{v})$ be the restricted direct product of the $G_{v}$ and let $\mathbb{G}(\mathbb{A})=G_{\infty}\times\mathbb{G}(\mathbb{A}_{\mathrm{f}})$. Then the diagonal embedding $\mathbb{G}(F)\hookrightarrow\mathbb{G}(\mathbb{A})$ realizes $\mathbb{G}(F)$ as a lattice there. Fix an open compact subgroup $K_{\mathrm{f}}\subset\mathbb{G}(\mathbb{A}_{\mathrm{f}})$. Then there exists a finite set $S$ of finite places (including all places where $K_{v}$ was left undefined above) such that $K_{\mathrm{f}}=H\times\prod_{S\not\ni v<\infty}K_{v}$ for some open compact subgroup $H<\prod_{v\in S}G_{v}$. We will generally ignore all places in $S$.

Let $\Gamma=\mathbb{G}(F)\cap K_{\mathrm{f}}$. Its image in $G_{\infty}$ is a lattice, and we have the identification³³3We use here that $\mathbb{G}$ is a form of the simply connected algebraic group $\mathrm{SL}_{2}$; in general the adelic quotient would correspond to a disjoint union of quotients $\Gamma\backslash G$

given by mapping the coset $\Gamma g_{\infty}$ with the double coset $[g_{\infty},1]=\mathbb{G}(F)(g_{\infty},1)K_{\mathrm{f}}$.

Conversely every congruence subgroup of $G$ (with the respect to the $F$-rational structure $\mathbb{G}$) contains a lattice $\Gamma$ as above. Since equidistribution modulo $\Gamma$ implies equidistribution modulo any overgroup (assuming the eigenfunctions are properly invariant) it suffices to consider the case above.