The unipotent mixing conjecture

We show that shifted pairs of discrete or continuous low-lying horocycles equidistribute in the product space of two modular curves.

Similarly, one can consider for a prime q, say, "discrete" low-lying horocycles Except for the point iq ∈ H, this is the image of SL 2 (Z)i by the Hecke correspondence T q (which has degree q + 1), and non-trivial bounds for eigenvalues of Hecke operators imply that these discrete low-lying horocycles also equidistribute on X as q → ∞.
It is natural to investigate whether equidistribution persists when some additional constraints are imposed on the entries a. Identifying the set of a's with integers contained in the interval [0, q − 1], one can restrict a to belong to a subinterval I ⊂ [0, q − 1] (see [Str04]) or vary a along a subsequence of integers of a special shape like a = [n c ], c = 1.1 [Ven10] or ask for a to vary along the primes.
This last case was studied by Sarnak and Ubis [SU15] who proved, assuming the Ramanujan-Petersson-Selberg conjecture that, as q → ∞, the "prime" q-Hecke points H p q := p + i q , 1 p q − 1, p prime .become dense in X; more precisely any weak limit, µ p say, of the uniform probability measures supported by the H p q satisfies 1 5 µ X µ p 9 5 µ X .
As for the proof, the combinatorial decomposition of the characteristic function of the primes lead naturally to handling sums of "Type I" and "Type II".The treatment of type I sums is closely related to restricting a to a sub-interval while the treatment of type II leads a new equidistribution problem: the joint equidistribution of sets of the shape on the product X × X; here 0 < b 1 < b 2 are distinct integers bounded by a fixed power of q and with N satisfying b 1 N, b 2 N q.
In the present paper we investigate the distribution properties of the above sets when we let a vary over the complete set of congruence classes modulo q and look at how distributes in the product X × X as q → ∞ as we allow b to possibly vary with q.Clearly if b = 1 we cannot expect full equidistribution to hold: H q,b is simply trapped in the diagonally embedded copy of X in X × X (and equidistributes along it).A similar phenomenon will occur for any fixed (i.e.independent of q) residue class b.One might optimistically conjecture that something like q b/q → ∞ (where .denotes the distance in R/Z) may suffice to ensure equidistribution, but the situation is more subtle as we shall see in a moment.
This lattice has covolume q and we denote by s(q; b) = min 0 =n∈Λ q,b n its minimum (i.e. the minimal euclidean norm of a non-zero element).It is well-known from the geometry of numbers that (1.2) s(q; b) ≪ q 1/2 .
In particular s(q; b) → ∞ implies q → ∞.We are now ready to state our first main result.
Theorem 1.1.Let q be a large prime and b ∈ (Z/qZ) × .Then for pairs (q, b) such that s(q; b) → ∞ the set becomes equidistributed with respect to the product of hyperbolic measures µ X×X = µ X ⊗µ X .
Once the case of two factors is available, it is possible to obtain equidistribution for more ([EL19, Corollary 1.5]): Then for pairs (q, b) such that s(q, b) → ∞ the set becomes equidistributed with respect to the product of hyperbolic measures µ X d = µ ⊗d X .Remark 1.1.Theorem 1.1 and Corollary 1.2 hold true with X replaced by any fixed modular curve X 0 (D) = Γ 0 (D)\H for D 1 and In fact, for the proof we will need such an extension to a modular curve of higher level.
Remark 1.2.If we denote for a continuous, compactly supported function ϕ : X × X → C by (1.4) W ϕ (b; q) := 1 q a (mod q) ϕ a + i q , ab + i q the corresponding Weyl sum, then one approach to Theorem 1.1 is to show that If we assume the Ramanujan-Petersson conjecture for the Hecke eigenvalues of Maaß cusp forms for the group SL 2 (Z) and ϕ is cuspidal (in both components), we prove (1.5) with a polylogarithmic rate in the form for some fixed δ > 0 and any ε > 0. See Section 1.2 for more details.
Remark 1.3.The condition s(q; b) → ∞ means that b/q must not be too close to a rational number with fixed denominator (like 2/5), so b/q must be, in a rather weak sense, "badly approximable".We indicate in Remark 4.2 below why this condition is probably sharp.
Remark 1.4.Theorem 1.1 can probably be generalized to composite numbers q at the cost of slightly more work.For instance, some care must be taken if q has many small prime factors which might collide with the auxiliary primes to be chosen later.
We prove a similar theorem for "continuous" low-lying pairs of horocycles.
Theorem 1.3.Let T > 1, y ∈ [1, 2] and write y = a/q + O(1/qQ) for positive coprime integers a, q with q Q := T 0.99 .Let I ⊆ R be a fixed non-empty interval.Then The condition on y is similar: y must be in some weak sense badly approximable by rational numbers with small denominator.There is some flexibility in the definition of Q as the proof shows.Also Theorem 1.3 can be generalized to d factors in a similar way.
These results are new examples of a growing family of equidistribution results inspired by Sarnak's classical theorem [Sar81] mentioned above, as well as by Duke's celebrated equidistribution theorems for Heegner points and closed geodesics with increasing discriminants on modular curves [Duk88]; the latter needs crucially subconvexity for twisted L-functions [DFI93] as an input.
A new generation of equidistribution problems was posed by the second author and Venkatesh in their ICM address [MV06] in the context of Duke's theorem.Let H(D) be the set of Heegner points in the imaginary quadratic field K = Q( √ −D).Let [b] be an element in the class group of K (which acts on H(D)), and consider the set of pairs As D → ∞, does this equidistribute in the product space X × X? Similarly to the assumption s(q; b) → ∞ in Theorem 1.1, a necessary condition to escape from the diagonal is that the norm of the smallest ideal in [b] tends to infinity, and it is reasonable to conjecture that this is sufficient.This is (a special case of) the mixing conjecture.This conjecture is still open but two different conditional proofs were given by Khayutin [Kha19] and Blomer-Brumley-Khayutin [BBK].One of the key ingredients in the former is a deep result in homogeneous dynamics due to Einsiedler and Lindenstrauss [EL19] classifying under suitable condition joinings in products of locally homogeneous spaces.As we will see below this result also play a crucial role in the present paper.Further variations on this theme can be found in [AES16, AEW22, ALMW22, BB], and we recommend in particular the Bourbaki seminar by M. Aka [Aka21].
Remark 1.5.The astute reader might have noticed that H q is essentially the set H(−4q 2 ) of Heegner points of the highly non-maximal order of conductor −4q 2 .
More precisely, if q ≡ 3 (mod 4), we have #H(−4q 2 ) = 1 2 (q + 1), and H q corresponds to the q primitive quadratic forms F q,a := (q 2 , 2qa, a 2 + 1) for a (mod q).The forms F q,a 1 and F q,a 2 are equivalent if and only if a 1 a 2 ≡ −1 (mod q) or a 1 ≡ a 2 (mod q), so that H q covers H(−4q 2 ) twice except for the form F q,0 which corresponds uniquely to the identity in H(−4q 2 ).
However the group action in H(−4q 2 ) is quite different from the multiplication in H q .
We recall that Khayutin's work [Kha19] is tailored for fundamental discriminants and uses quite crucially the fact that the underlying order is maximal (or at least not far away from being maximal, cf.[Kha19, Section 1.8.2]).While our proof also builds crucially on the work of Einsiedler-Lindenstrauss, our approach is different from Khayutin's: instead of bounding a correlation function between two measures, we prove, by direct arguments, that the Weyl sums attached to a well chosen cuspidal test function ϕ converge to the correct limit (i.e.0); the classification theorem of Einsiedler-Lindenstrauss then allows us to bootstrap this partial information to full equidistribution.
Remark 1.6.Unlike other examples, our result does not require any splitting conditions: in our case such conditions are in fact automatically satisfied.As such, this seems to be one of the first unconditional instances of a mixing type conjecture (see also [Saw21] as well as [ST17] for proofs of function field versions of this conjecture).
Remark 1.7.Lindenstrauss, Mohammadi and Wang [LMW] have obtained very general and effective forms of equidistribution with polynomial decay rates for certain types of one parameter unipotent flows on products of modular curves.It is not clear to us whether this would cover all cases of Theorem 1.3 (as the analysis in [LMW] depends on the injectivity radius of the base point x 0 ) and then whether this could be transferred to the discrete case treated in Theorem 1.1.1.2.Principle of the proof.By the spectral decomposition and Weyl's equidistribution criterion, it is sufficient to bound non-trivially the Weyl sum W ϕ (b; q) for ϕ = f 1 ⊗ f 2 when f 1 and f 2 are either constant functions or (non-constant) Hecke eigenforms.The case where f 1 or f 2 is constant follows from the equidistribution of (1.3) when projected to each factor X. To handle the remaining sums and prove Theorem 1.1 we use different techniques depending on how fast s(q; b) approaches ∞ within the range (1.2), a principle that is familiar from [EMV13], [Kha19] or [BBK].
More precisely, let θ = 7/64 be the best known approximation towards the Ramanujan-Petersson conjecture [Kim03] and suppose we are in the range s(q; b) q 1/2−2θ−η for some fixed η > 0. Bounds for the Weyl sums can be achieved by resolving some versions of the shifted convolution problem (see Proposition 2.1).In this regime, the Ramanujan-Petersson conjecture is not necessary and one can show that equidistribution holds with a polynomial decay rate o ϕ (1) ≪ q −δ(η) in the notation of (1.5).
In the remaining range q 1/2−2θ−η s(q; b) ≪ q 1/2 , methods from harmonic analysis do not work so well at the moment.Using sieve methods instead together with the analytic theory of multiplicative functions, we are able to bound non-trivially the Weyl sums (1.5) when f 1 and f 2 are cuspidal and both satisfy the Ramanujan-Petersson conjecture.The Ramanujan-Petersson conjecture is needed to implement the sieving argument as it insures that the (multiplicative) Hecke eigenvalue functions n → λ f i (n) are bounded in absolute value by the divisor function.This reduces the problem to that of obtaining a sufficiently good bound for sums of the Hecke eigenvalues along the primes, namely (1.6) for some fixed δ > 0 and sufficiently large z.Such a bound, which follows from suitable approximations to the Sato-Tate conjecture for cusp forms, yields a power saving in log p on the size of the Weyl sum (in a way similar to [Hol10]).Unfortunately, if f 1 or f 2 is an Eisenstein series, the distribution properties for their Hecke eigenvalues do not allow to obtain (1.6).Nevertheless it will be useful to remember that if f 1 and f 2 are both CM forms (cusp forms attached to Hecke characters of quadratic fields) the Ramanujan-Petersson conjecture holds and (1.6) is unconditional.
To deal with the rest of the spectrum we take a very different route and make use of a powerful measure classification theorem of Einsiedler-Lindenstrauss [EL19, Thm 1.4].Let µ q,b denote the uniform probability measure on X × X supported by H q,b .As we explain below in Section 5, µ q,b is the projection to X×X of a measure µ S q,b on a suitable S-arithmetic quotient Γ S × Γ S \G(Q S ) × G(Q S ), G = PGL 2 and (by non-trivial bounds for Hecke eigenvalues in the discrete case or Sarnak's theorem in the continuous case) any weak-⋆ limit of the µ S q,b projects to the Haar measure on each factor, i.e. it defines a joining of the Haar measures.Moreover, these limits are invariant under the action of a rank 2 diagonalisable subgroup, namely the diagonal subgroup of G(Q S ) × G(Q S ) generated by (t 1 , t 1 ) and (t 2 , t 2 ) where for q 1 , q 2 two primes distinct from q.This is an immediate consequence of the invariance of H q,b under multiplicative shifts that we have already observed: By the measure classification theorem, any such joining is a convex combination of algebraic measures which in the present situation are either the (image of the) Haar measure on the full product space or the Haar measure along diagonal G-orbits in the product.We will have to exclude the latter possibility.This will follow -in a rather subtle way -from what we have already proved.Testing the measure against a carefully selected test function ϕ = f 1 ⊗ f 2 , where both f 1 and f 2 are suitable vectors in a certain CM automorphic representation (which satisfies the Ramanujan conjecture), we show that the corresponding Weyl sum would have a positive limit if the measure were not the full product measure.
Remark 1.8.This principle of using measure rigidity to bootstrap the evaluation of Weyl sums for a small portion of the spectrum to full a equidistribution statement has occured in the past notably in [ELMV11] (for the Siegel Eisenstein series for SL 3 ).This principle is likely to be used again in higher rank situations in connection to functoriality: for instance we hope that the results of this paper will be useful for equidistribution problems associated with GSp 4 (on using Saito-Kurokawa lifts).
Remark 1.9.Notice that the two entries (a (mod q), ab (mod q)) in Theorem 1.1 are related by a linear equation.Another interesting question occurring in classical problems from analytic number theory is whether joint equidistribution holds for tuples of discrete horocycles whose entries are related by more general algebraic equations.For instance, the second named author together with Einsiedler and Lindenstrauss ([ELM18]) established joint equidistribution in the case of monomial relations: given two fixed integers 1 k < l, then as q → ∞ amongst the primes and for any b ∈ (Z/qZ) × , the set becomes equidistributed on X ×X.The proof uses again crucially the measure classification results of [EL19], but the application is simpler: the sets H k,l,q,b are then invariant under the rank 2 subgroup generated by (t k 1 , t l 1 ) and (t k 2 , t l 2 ), and the fact that 1 k < l excludes the possibility of having measures supported along diagonal G-orbits.The combination of the present results with those of [ELM18] will be the topic of a future work, joint with Einsiedler and Lindenstrauss.

Shifted convolution problems
In this section we prepare the stage for a proof of Theorem 1.1 in the "low regime" case, i.e. when s(q; b) q 1/2−2θ−δ for some δ > 0.
We start with a variation of [Blo04, Theorem 1.3] with an additional summation over h.
with an implied constant depending on ε and polynomially on P 1 , P 2 and the conductors of f and g.
The proof depends on the following bound for averages of twisted Kloosterman sums.
Here we work unconditionally and denote by θ 7/64 an admissible exponent towards the Ramanujan-Petersson conjecture.
with an implied constant that depends on ε and polynomially on P 0 , P 1 , P 2 .
Proof.We follow the proof of [Blo04, Proposition 3.5].Instead of U h (t, q) we work with As in [Blo04, Proposition 3.5] we now apply the Kuznetsov formula, Cauchy-Schwarz and the spectral large sieve for cusp forms of level N and character χ.Not that, as (d, N ) = (d, h/d) = 1, we can extract the divisibility condition d | h at the cost of a factor d θ in the h-sum and reduce the length of the h-sum to H/d.We then arrive in the same way at [Blo04,(3.19)]except that the factor T h θ is replaced with and the result follows with the same choice of T as in [Blo04,(3.20)].
For the proof of Proposition 2.1 we follow literally the proof of [Blo04, Theorem 1.3] in [Blo04, Section 4] with the only minor modification that f and g may be two different cusp forms of potentially levels, they may have non-trivial central characters, and that the summation condition can have either sign.We keep the extra sum over h outside until the very end when we apply our Lemma 2.1 with S = Q 2 /l 2 M 2 and N = N 1 N 2 l 1 l 2 as a replacement for [Blo04, Proposition 3.5].This replaces the factor h θ in [Blo04, (4.19)] with as desired.

Application of a sieve
For X, Y ≫ 1 and q ∈ N we denote by C q (X, Y ) the set of subsets S ⊆ N 2 contained in a ball of radius X 100 about the origin and satisfying is a sublattice of index q and shortest nonzero vector of length s, R ⊆ R 2 >0 a non-empty, bounded, connected and simply-connected set with piecewise smooth boundary ∂R, with Note that for (d 1 d 2 , q) = 1, the pairs (n 1 , n 2 ) ∈ Λ with d j | n j for j = 1, 2 form a sublattice of index d 1 d 2 .
For z 1 and q ∈ N let P q,z be the set of primes p z, p ∤ q.For S ∈ C q (X, Y ), a 1 , a 2 ∈ N and y 1 , y 2 1 define 1.
As usual we write P + (n) for the largest prime factor of n (with the convention P + (1) = 1).From a standard sieve, e.g.Selberg's sieve, we obtain For the proof of Proposition 3.1 we decompose a general n 1 appearing in the sum as . ., p j ∤ q and similarly for n 2 = a 2 b 2 where k j is maximal with a j z = X γ .Let ξ := (log z)(log log z).
We distinguish the following four cases for n 1 : and similarly for n 2 .This set-up goes back originally to Erdös [Erd52] and was refined by Wolke [Wol71], Nair-Tenenbaum [NT98], Khayutin [Kha19] and others.
1) Consider first pairs (n 1 , n 2 ) such that n 1 is in case (II).Then p z 1/2 , so n 1 is divisible by a prime power p e z 1/2 with p < z 1/2 .Let e 0 = e 0 (p, z) 2 be the smallest positive integer with p e 0 z 1/2 .Estimating λ 1 (n 1 )λ 2 (n 2 ) trivially, by Lemma 3.1 the contribution of such pairs is at most The same argument works with exchanged indices if n 2 is in case (II).
2) Next consider pairs (n 1 , n 2 ) such that n 1 is in case (III).Then n 1 is divisible by a number z 1/2 < a 1 z such that P + (a 1 ) < ξ.Since estimating λ 1 (n 1 )λ 2 (n 2 ) trivially, the contribution of such pairs is by Lemma 3.1 and partial summation at most The same argument works with exchanged indices if n 2 is in case (III).
We conclude that in either case the contribution of such pairs is where we used once again Lemma 3.1.By [Wol71, Lemma 3] we have for some constant c > 0, r ≪ log z/ log log z (this condition is needed in the proof and requires us to treat case (III) separately) and j = 1, 2, so that the total contribution of pairs in the present case is Combining (3.1), (3.2) and (3.3), we complete the proof.
We will apply Proposition 3.1 to the multiplicative functions n → |λ f i (n)|, i = 1, 2, where f 1 , f 2 are cuspidal (holomorphic or Maaß) Hecke eigenforms with Hecke eigenvalues λ f i (n) satisfying the Ramanujan-Petersson conjecture, so that for all n 1.
-If f is a dihedral form (in which case the Ramanujan-Petersson conjecture is automatic), then for any z 5 we have -If f is not a dihedral form, then for any z 5 we have Proof.Let N denote the level of f and χ be its nebentypus.If f is dihedral, we recall that it comes from some Hecke character (not of order 1 or 2 since f is cuspidal) of a quadratic number field K/Q and hence λ f (p) vanishes if p is inert in K.For prime p ∤ N split in K we have where f 2 is the theta series induced from the square of the Hecke character mentioned above.We conclude that for a smooth non-negative function w supported on [0, 2) and equal to 1 on [0, 1].The desired bound (3.5) follows after integration by parts from Perron's formula applied to −d log ζ K (s) and −d log L(s, f 2 ) which are holomorphic and logarithmically bounded in a standard (Hadamard/de la Vallée-Poussin type) neighbourhood of the line ℜs = 1 except for a pole at s = 1 (for the former) with residue +1 and (possibly) a pole at some Siegel zero with residue −1 (which if it exists, then contributes a negative amount).
If f is not dihedral, the bound (3.6) is similar, but requires a few higher symmetric power L-functions of f (cf.[EMS84]).The starting point is the following computation, valid for p ∤ N and whenever |λ f (p)| 2, namely The bounds for the last two sums are then consequences of the holomorphy and the logarithmic bounds satisfied by −d log L(s, sym 2ν f × χ ν ) for ν = 1, 2 when s is within the standard neighbourhood of the line ℜs = 1.For ν = 1, the required properties follow from the cuspidality of L(s, sym 2 f × χ) (since f is non-dihedral), and the absence of a Siegel zero established in [HL94,Ban97].For ν = 2, we know from [Kim03, KS02] that L(s, sym 4 f × χ 2 ) is automorphic but not necessarily cuspidal.It is cuspidal unless f is tetrahedral or octahedral.In the tetrahedral case, we have by [KS02, §3.2] for χ 3 = χ 3,f a non-trivial character of order 3.In particular L(s, sym 4 f × χ 2 ) has no zeros or poles in a standard region along the line ℜs = 1.In the octahedral case we have [KS02, Thm.3.3.7] f ) where η = η f is a quadratic character and π(χ) is the dihedral representation induced from a non-trivial cubic Hecke character χ = χ f of the quadratic field determined by η.It follows that L(s, sym 4 f × χ 2 ) has no zeros or poles in a standard region along the line ℜs = 1.
Finally if f is not of the above type then L(s, sym 4 f × χ 2 ) is cuspidal and has no zeros or poles in a standard region along the line ℜs = 1 except for a possible Siegel zero.If χ f is trivial, the existence of a Siegel zero was ruled out by Ramakrishnan and Wang in [RW03, Theorem B'].As was pointed out to us by D. Ramakrishnan, the proof extends to the case of a general nebentypus with adequate modifications: let Π be the degree 9 isobaric sum Π = 1 ⊞ sym 2 f × χ ⊞ sym 4 f × χ 2 ; its degree 81 Rankin-Selberg L-function (which has non-negative coefficients) factors as say.Following the proof of [RW03, Theorem B'] it is sufficient to prove that L 2 (s) is holomorphic along the interval (1/2, 1) (cf.[RW03, Prop.5.21]).Using the factorisation and its remains to prove the holomorphy of L(s, sym 3 f ; sym 2 × χ 3 ) along (1/2, 1).One then proceeds as in [RW03, Lem.5.25 & §7].The only difference is that, when χ 3 is non-trivial, one has to use the work of Takeda [Tak14] in place of the work of Bump-Ginzburg [BG92] regarding the holomorphy of the incomplete L-function L S (s, sym 3 f ; sym 2 × χ 3 ) (where S is the union of the archimedean places and the finite places where f is ramified).
Remark 3.1.In this paper we will need only the case where f has trivial nebentypus.Remark 3.2.In the non-CM case the Sato-Tate conjecture predicts the constant 8/(3π) in place of 17/18 in (3.6), and it is known when f is holomorphic [CHT08, HSBT10, Tay08, NT21].In the CM case, the best possible constant is 2/3 which is attained if f comes from a character of order 3.

Equidistribution for the cuspidal spectrum
In this section, we establish the equidistribution statements of Theorems 1.1 and 1.3 for pairs of cuspidal newforms (assuming the Ramanujan-Petersson conjecture).
Proof.Clearly (n 1 , n 2 ) ∈ Λ q;b implies (n 1 d, n 2 ) ∈ Λ q;bd , so that s(bd; q) ds(b; q).Similarly, s(b d; q) ds(b; q).Replacing b with b d and bd we obtain the inequalities in the other direction.The last statement follows from exchanging n 1 and n 2 .
With future applications in mind, we consider slightly more general Weyl sums than in (1.4).
Theorem 4.1.Let q be a large prime, b ∈ (Z/qZ) × and s = s(b; q).Let f j , j = 1, 2, be two L 2 -normalized cuspidal Hecke-Maaß newforms of levels N j with Hecke eigenvalues λ j (n) for which the Ramanujan-Petersson conjecture holds.Let x 0 ∈ R, y 0 ∈ R × , r 0 ∈ Q where the denominator of r 0 is coprime to N 1 N 2 q, and let for any ε > 0 with an implied constant depending polynomially on |x 0 |, y 0 + y −1 0 , the denominator of r 0 and the conductors of f 1 and f 2 .In other words, for fixed (x 0 , y 0 , r 0 ) and fixed cusp forms f 1 , f 2 we obtain decay as soon as s → ∞.
Remark 4.1.The reason to include general triples (x 0 , y 0 , r 0 ) (and not just the triple (0, 1, 0) as in (1.4)) is that we will later apply this to pairs of cusp forms (f 1 , f 2 ) where f 2 , lifted to a function on an S-adic quotient of for two fixed primes q 1 , q 2 , is acted on by a fixed group element g ∈ PGL 2 (R)×PGL 2 (Q q 1 )× PGL 2 (Q q 2 ).When translated back to classical language, this action has the effect of introducing the extra parameters (x 0 , y 0 , r 0 ).See Section 5 for details.
We remark that a similar bound can be obtained for more general test functions, for instance Maaß forms of fixed weights k j , or even more generally automorphic forms whose archimedean component is a fixed test function (for instance compactly supported) in the Kirillov model.
Remark 4.2.To see that the condition s → ∞ cannot be dropped completely or replaced with a simpler condition of the kind q b/q → ∞, consider the case where b = (q + 1)/2 (for odd q).Then the Fourier expansion (4.4) below yields essentially The congruence is equivalent to q | 2n 1 − n 2 , and so we obtain a diagonal term which in the case f 1 = f 2 and λ 2 (2) = 0 does not decay in q.In other words, the underlying condition for equidistribution is really of diophantine nature.
Proof.We start with the Fourier expansion (4.2) where ǫ j is the parity, λ j (n) is the n-th Hecke eigenvalue (we have assumed that the f j are newforms) and 1 t j 0 is the spectral parameter of f j for j = 1, 2. For notational simplicity let us write 1 by the Selberg eigenvalue conjecture, although this is not essential for the argument, unlike the

Ramanujan-Petersson conjecture at finite places
We use the simple bound for j ∈ N 0 , A, ε > 0 with polynomial dependence in t.In the following we use the convention that all implied constants may depend polynomially on cond(f j ) and |x 0 |, y 0 + y −1 0 , den(r 0 ) without further mention.
Summing over a (mod q), we conclude This vanishes unless ǫ 1 = ǫ 2 = ǫ, say, in which case we get By trivial estimates we can truncate the n 1 , n 2 -sum at q 1+ε at the cost of a negligible error.Moreover, if r 0 = c 0 /d 0 , we can replace the additive character n 2 → e(n 2 r 0 ) with (a linear combination of) Dirichlet characters χ modulo d 0 , so that it suffices to bound 1 q for some choice of ±.
The character χ may not be primitive; if it is induced by the primitive character χ * , we obtain by Möbius inversion and the Hecke multiplicativity relations 1 q The point of this maneuvre is to show that in (4.4) we may assume without loss of generality that χ is primitive.Since (d 0 , N 2 ) = 1, the function n → λ 2 χ(n) describes the Hecke eigenvalues of the newform f 2 × χ, which allows us to apply Proposition 2.1.To this end we have to replace b by some fixed multiple, but by Lemma 4.1 we have s(b; q) ≍ s(f gb; q), so that the subsequent analysis remains unchanged.We may therefore return to (4.4) under the assumption that χ is primitive.The (n 1 , n 2 )-sum runs through a lattice Λ q,±b with basis ∓b 1 , q 0 of volume q and minima 0 < s(b; q) = s(−b; q) ≪ q 1/2 .Let us fix one sign and drop it from the notation.Let be a reduced basis of Λ q,b with the first vector of minimal length so that Both x 1 , x 2 are nonzero (since q ∤ b) and coprime (since s is minimal).In terms of this basis, the congruence condition reads We now use two different methods to estimate the n 1 , n 2 -sum depending on the size of s.If s is not too big, then we interpret the n 1 , n 2 -sum as a shifted convolution problem and apply first Proposition 2.1 to the inner sum with summation condition (4.5) which we write as an equality n 1 x 2 − n 2 x 1 = h with q | h (see [BFK + 17] for a similar argument).Since s ≪ q 1/2 and n 1 , n 2 ≪ q 1+ε , we have automatically that (h/q, q) = 1.We need to treat the diagonal term ∆ with h = 0 separately.In this case x 1 | n 1 , x 2 | n 2 , and by Rankin-Selberg theory (and the Ramanujan conjecture, although this could be dispensed with) it is easy to see the contribution is x 0 , y 0 , r 0 ) the remaining portion of (4.4).Applying a smooth partition of unity to the n 1 , n 2 , h-sums, we obtain by Proposition 2.1 with . Note that the θ-dependence in Proposition 2.1 comes from the entire spectrum of forms of level x 1 x 2 , not from the two forms f 1 , f 2 .
Alternatively, we apply absolute values to the inner sum in (4.4), and we can then apply Proposition 3.1 with z = q γ for some very small γ > 0 and More precisely, we estimate the contribution of the terms with q | n 1 n 2 trivially by O(q ε−1 ).For the remaining terms we apply a decomposition into annuli (k − 1)q n 2 1 + n 2 2 kq for k = 1, 2, . . ., q ε , and by Proposition 3.1 we obtain the bound Employing Lemma 3.2, we obtain (4.8) W f 1 ,f 2 (b; q; x 0 , y 0 , r 0 ) ≪ 1 (log q) 1/9 + q 3γ+ε s .
Remark 4.3.The bound (4.8) is almost sufficient alone.The shifted convolution argument is only needed when s grows below the q ε -scale, say logarithmically with q.On the other hand, however, the shifted convolution argument is more robust and provides a power saving.
4.2.The continuous case.For the continuous joint equidistribution problem in Theorem 1.3 we need the following analogue of Theorem 4.1.
Theorem 4.2.Let f 1 , f 2 be two L 2 -normalized cuspidal Hecke-Maaß newforms of levels N j with Hecke eigenvalues λ j (n) for which the Ramanujan-Petersson conjecture holds.Let x 0 ∈ R, y 0 ∈ R × , r 0 ∈ Q such that the denominator of r 0 is coprime to N 1 N 2 .Let T > 1, I ⊆ (0, ∞) a fixed compact interval, y ∈ I and W a fixed smooth weight function with compact support in R and write y = a/q + O(1/qQ) for positive coprime integers a, q with q Q := T 0.99 .Then for any ε > 0 with an implied constant depending polynomially on the conductors of f 1 and f 2 and on |x 0 |, y 0 + 1/y 0 and the denominator of r 0 .
Proof.We start again with the Fourier expansion (4.2) and perform the x-integration.As in (4.4) this leaves us with bounding where Ŵ denotes the Fourier transform of W , G is as in (4.3) and χ is some primitive character whose conductor divides the denominator of r 0 .Inserting the rational approximation, the n 1 , n 2 -sum is, up to a negligible error, restricted by For now, let us only assume log Q ≍ log T .Applying Proposition 2.1 with d = 1 and treating the diagonal contribution qn 1 ± an 2 = 0 separately as in (4.6), we obtain the first bound (4.9) On the other hand, we can apply absolute values to R f 1 ,f 2 (y, T ) and use Proposition 3.1.To this end, let S ± (a, q, H ′ , T ′ ) for H ′ ∈ N and T ′ 1 be the number of (n 1 , n 2 ) such that T ′ < n 2 2T ′ and H ′ < qn 1 ± an 2 2H ′ .Then λ∈S ± (a,q,H ′ ,T ′ ) Theorem 3.1 and Lemma 3.2 we conclude for some sufficiently small γ > 0 similarly as in the preceding proof We may now choose, for instance, Q = T 0.99 (with γ as above sufficiently small) and combine (4.9) and (4.10) to obtain the desired bound and the last term is absorbed in the previous two terms.

Application of the joinings theorem of Einsiedler-Lindenstrauss
In this section we use the powerful measure classification theorem of Einsiedler-Lindenstrauss [EL19].
5.1.Passage to an S-arithmetic quotient.For any place v of Q we denote by Q v the corresponding completion and for a finite place v = p by Z p ⊆ Q p the closure of Z.We denote by the ring of adeles (the first component is always the infinite place).
We fix an integer D 1 and a non-empty finite set of places S = {∞} ∪ S f containing ∞ and such that the primes in S are coprime to D; we set Let G = PGL 2 and let K(D) (S) be the open-compact subgroup where for p ∈ S we put in particular one has K(D) p = G(Z p ) for p ∤ D. We also set By the strong approximation property for SL 2 and the fact that det(K(D) , where is a discrete subgroup of G(Q S ).In the sequel we set For S = {∞} and D = 1, the space X(1) ∞ is a covering of the level 1 modular curve with compact fibers.More generally the S-arithmetic quotient X(D) S is a covering of the usual modular curve X 0 (D) of level D. The compact fiber of a point z ∈ H is where g z ∈ G(R) is such that g z .i= z.We denote by π S : X(D) S → X 0 (D) → X 0 (1) = X the composite of this projection and the usual finite covering.Since the fibers of this map are compact, Γ(D) S ⊆ G(Q S ) has finite covolume (i.e. is a lattice).

Lifting the measures.
To simplify notations, we omit the dependency in D and write X S for X(D) S , Γ S for Γ(D) S etc.Given a prime q ∈ S and a ∈ Z we set u a/q := 1 a/q 0 1 .

We define
x S q,a := Γ S u a/q 1 0 0 q , u a/q , • • • , u a/q ∈ X S .
We observe that since q ∈ S the Γ S -coset x S q,a depends only on the congruence class a (mod q): for p ∈ S and k ∈ Z we have u (a+qk)/q = u k .ua/q ∈ Γ S u a/q .

Let
H S q := {x S q,a | a ∈ Z/qZ} ⊆ X S .By construction we have π S (H S q ) = H q .
Let us pass to the product: given b ∈ (Z/qZ) × we set Let us denote by µ S q,b the uniform probability measure on X S × X S supported on H S q,b .There are multiple advantages of lifting the whole situation to the S-arithmetic group quotient X S = Γ S \G(Q S ).
• The space X S is endowed with an action of G(Q S ) by right multiplication (noted x.g = Γ S ug for x = Γ S u ∈ X S and g ∈ G(Q S )) and a left action on its space of functions: for f : • We can test the lifted measures µ S q and µ S q,b against more general functions, e.g.corresponding to automorphic forms not necessarily of weight 0.
• The lifted measures have additional invariance from the places in S.

Properties of the measures µ S
q,b .We now specify the set of places S we will use: given D 1, let q 1 , q 2 be two fixed primes coprime with D and put S = {∞, q 1 , q 2 }.
In this section we verify that any weak-⋆ limit of the µ S q,b is a joining.Let µ S q,b,i , i = 1, 2, denote the image of µ S q,b on X S under the first and second coordinates projections π i : X S × X S → X S .Lemma 5.1.As q → ∞, the measures µ S q,b,i , i = 1, 2, converge to the Haar probability measure µ S G on X S .
Proof.For q large enough, we have q ∈ S and (q, D) = 1, and the two projections of H S q,b are given by the set H S q .Since q is prime, H S q is the q-Hecke orbit of the identity class Γ S Id ∈ X S minus one point, namely x S q,∞ defined like x S q,a but with with u a/q replaced by diag(q, q −1 ); equidistribution H S q follows from the equidistribution of q-Hecke orbit.
Remark 5.1.In particular any weak-⋆ limit of the (µ S q,b ) q (the limit of a converging subsequence) is a probability measure.
We now make use of the auxiliary places q 1 and q 2 .We introduce the two diagonal matrices which we view as embedded diagonally into G(Q S ) and also diagonally embedded into Lemma 5.2.The set H S q,b is invariant under right multiplication by the elements t ∆ 1 , t ∆ 2 .
5.4.Excluding the diagonal measures.In this section we use the bounds for certain well chosen Weyl sums established in Theorem 4.1 to show that the coefficient c of µ G ∆ in the decomposition of is zero, effectively concluding the proof of Theorem 1.1.The proof of Corollary 1.2 is a consequence of this result and [EL19, Cor 1.5].
Consider a subsequence of the µ S q,b converging to some measure of the shape µ ∞ = (1 − c)µ G×G + cµ G ∆ and suppose that c > 0.
Let Π be a cuspidal automorphic representation such that its space Π K (S) of K (S)invariant vectors occurs in X S , and let ϕ 1 , ϕ 2 be two smooth L 2 -normalized vectors in Π.Consider the function on X S × X S given by We have therefore ϕ 2 (g.h)dλ(h) The integral λ * ϕ 2 converges since the measure λ is finite and cusp forms have rapid decay.
Roughly speaking, we will choose ϕ 1 = λ * ϕ 2 .On the one hand, Theorem 4.1 will imply that under certain conditions µ ∞ (ϕ) = 0. On the other hand, we will show under certain conditions λ * ϕ 2 = 0, which is only possible if c = 0. We now make this precise.We make the following choices: let K = Q( √ 229) (notice that 229 is a prime fundamental discriminant); this quadratic field has class number 3. Let χ be a class group character of order 3 ; it lifts to an automorphic form f χ of level 229 with central character the Legendre symbol χ 229 = ( 229. ).Since 229 ≡ 1 (mod 4), there exists primitive Dirichlet character ψ modulo 229 of order 4 and its square is χ 229 .The automorphic form (5.1) f = f χ × ψ is an automorphic form of level 229 2 with trivial central character and Laplace eigenvalue 1/4 (i.e.spectral parameter t = 0).Let q 1 = 37, q 2 = 53.
These are split primes in K lying below principal ideals.Moreover, q 1 and q 2 are 4-th power residues modulo 229, so ψ is trivial on q 1 , q 2 .We conclude that the automorphic representation Π generated by f satisfies the following properties: • it is cuspidal; • it has trivial central character; • it satisfies the Ramanujan conjecture at all places; • it is ramified only at 229 which is disjoint from S; • it has trivial Langlands parameters at all three places in S = {∞, q 1 , q 2 }.Of course these properties could have been obtained in a rather general way, the explicit construction above is only for illustrative purposes.The key point is the last property, for the following reason: We consider the representation Π S = Π ∞ ⊗ Π q 1 ⊗ Π q 2 .Since Π is locally unramified principal series at all places in S, we can consider its induced model I S which is the space of functions F : G(Q S ) → C that transform like where we use that the Langlands parameters at places in S are trivial.(Note that y is only well-defined up to units in the non-archimedean case, but this is irrelevant for the transformation rule.)We have a G(Q S )-equivariant vector space homomorphism ι : Π S → I S .The image of the weight 0 Maaß form f defined in (5.1), lifted to a vector in the adelic representation Π, projected onto Π S in the model I S , corresponds to the function F on G(Q S ) that is 1 on PSO(2) × G(Z q 1 ) × G(Z q 2 ) which is obviously non-negative on all of G(Q S ) by (5.2).In particular, λ * F is not the zero function, so there exists some δ > 0 with 0 < δ < λ * ι −1 F 2 where ι −1 F is an L 2 -normalized spherical vector in Π S (or in Π).
By strong approximation, we can translate this back into classical language.Clearly, ϕ 2 (g q,ba ) = f ba + i q .
The entries x 0 Y, y 0 Y, r 0 , Y are bounded in terms of R and as q → ∞ (through primes), we have s(q; bY ) ≍ Y s(q; b) → ∞, by our assumption.By Theorem 4.1 and Lemma 4.1 we obtain that W f ,f ( bY, q; x 0 Y, y 0 Y, r 0 ) → 0, uniformly for g R and therefore (5.4) tends to zero.This contradicts (5.3) unless c = 0 and this completes the proof of Theorem 1.1.
The proof of Theorem 1.3 is similar, based on Theorem 4.2 instead which we need to apply with yr 0 instead of y.If y = a/q + O(1/qQ), then yr 0 = ar 0 /q + O(r 0 /qQ), so as long as r 0 is fixed, this changes (as in the previous proof) only the implicit constants.
Dedicated to Peter Sarnak with admiration 1. Introduction 1.1.Main results.A classical result of Sarnak [Sar81] says in a special case that the image of a low-lying horocycle {x + i/T | x ∈ [0, 1]} of height 1/T equidistributes on the modular curve X = SL 2 (Z)\H with respect to the usual hyperbolic probability measure dµ X (z