Density results for automorphic forms on Hilbert modular groups II
aa r X i v : . [ m a t h . N T ] M a y TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 00, Number 0, Pages 000–000S 0002-9947(XX)0000-0
DENSITY RESULTS FOR AUTOMORPHIC FORMS ON HILBERTMODULAR GROUPS II
ROELOF W. BRUGGEMAN AND ROBERTO J. MIATELLOA bstract . We obtain an asymptotic formula for a weighted sum over cuspidaleigenvalues in a specific region, for SL over a totally real number field F , withdiscrete subgroup of Hecke type Γ ( I ) for a non-zero ideal I in the ring of integersof F . The weights are products of Fourier coe ffi cients. This implies in particularthe existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadraticcase, the regions include floating boxes, floating balls, sectors, slanted strips(see § F . The main tool in the derivation is a sum formula ofKuznetsov type ([3], Theorem 2.1). C ontents Introduction 11. Preliminaries and discussion of main results 52. Sum formula 123. Upper bound 224. Asymptotic formula, first stage 255. Asymptotic formula, second stage 326. Special families 36References 42I ntroduction
Let F be a totally real number field of dimension d , and let O F be its ring of in-tegers. If I is a non-zero ideal in O F , let Γ = Γ ( I ) denote the congruence subgroupof Hecke type of the Hilbert modular group. We allow a character of Γ ( I ) of theform (cid:16) ac bd (cid:17) χ ( d ), with χ a character modulo I .The goal of the present paper is to obtain distribution results for cuspidal auto-morphic representations of G (cid:27) SL(2 , R ) d with eigenvalue parameters in a subset Ω t of the multi-eigenvalue space, as t → ∞ , under some general conditions on thefamily Ω t . Mathematics Subject Classification. c (cid:13) Let V ̟ be a cuspidal automorphic representation, with elements transformingunder the above character of Γ ( I ), and with a compatible central character. TheFourier coe ffi cients of automorphic forms in V ̟ can be normalized so that theyare independent of the chosen automorphic form in V ̟ . This results in coe ffi cients c r ( ̟ ) describing the Fourier expansion at the cusp ∞ . The Fourier term order r runs through the inverse di ff erent O ′ of F .We denote by λ ̟ = ( λ ̟, j ) j ∈ R d the vector of eigenvalues of the Casimir oper-ators at the infinite (real) places of F . For compact sets Ω ⊂ R d , we consider thecounting functions(1) N r ( Ω ) : = N r ξ,χ ( Ω ) : = X ̟, λ ̟ ∈ Ω | c r ( ̟ ) | . The representations ̟ run through an orthogonal system of irreducible subspacesof L , cusp ξ ( Γ ( I ) \ G , χ ), for a fixed choice of the character χ of Γ ( I ) and of thecentral character (determined by ξ ∈ { , } d ).The main result in this paper asserts that if the family t Ω t satisfies certainmild conditions, then(2) N r ( Ω t ) = √| D F | (2 π ) d Pl( Ω t ) + o ( V ( Ω t )) ( t → ∞ )for all non-zero r ∈ O ′ . By D F we denote the discriminant of F over Q , and byPl the Plancherel measure of G . The error term contains a reference measure V which, under some general assumptions, is comparable to Pl.Roughly speaking, we show that the asymptotic formula (2) holds for the family t Ω t under the conditions that Ω t grows in at least one coordinate direction, andthat the boundary ∂ Ω t is small in comparison with Ω t itself. On the other hand, itis often convenient to use, instead of λ ∈ R d , the corresponding spectral parameter ν ̟ ∈ ([0 , ∞ ) ∪ i (0 , ∞ )) d . We use a tilde to indicate that the relevant measures like˜N r , e Pl and ˜ V are taken in the variable ν , and we write˜N r ( ˜ Ω t ) = X ̟,ν ̟ ∈ ˜ Ω t (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) , with ν ̟, j ∈ [0 , ∞ ) ∪ i (0 , ∞ ) such that λ ̟, j = − ν ̟, j .In Theorems 4.6 and 5.3 we prove asymptotic statements in terms of the quan-tities N r ( Ω t ) and ˜N r ( ˜ Ω t ) respectively, and this enables us to show occurrence anddensity of representations for a wide class of families of sets t Ω t . For illustra-tion, we now list some of the distribution results that are obtained in the quadraticcase.(i) Small rectangles . Let d =
2. Let [ α, β ] ⊂ [1 / , ∞ ) and consider for t large Ω t = [ α, β ] × [ t , t + √ t ] . Formula (2) implies(3) N r ( Ω t ) ∼ √ D F π Z βα tanh π q λ − d λ t / ( t → ∞ ) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 3
In particular, there are infinitely many ̟ with λ ̟, ∈ [ α, β ]. These ̟ havea second component of unitary principal series type. A similar result holdswith ̟ of discrete series type.On the other hand if [ α, β ] ⊂ h λ , (cid:17) and Ω t = [ α, β ] × [ t , √ t ], thenN r ( Ω t ) = o (cid:0) t / (cid:1) ( t → ∞ )(4) This gives an upper bound for the weighted density of ̟ of complemen-tary series type.As another example, if one takes Ω t = [ t , t + √ t ] × [ ct , ct + √ t ] , with t large and c ≥
1, then(5) N r ( Ω t ) ∼ C c t ( t → ∞ ) . (ii) Slanting strips . Let d =
2, and put, in terms of the spectral parameter˜ Ω t = n ( ν , ν ) ∈ ( i [1 , ∞ )) : t ≤ | ν | ≤ t , a | ν | + b ≤ | ν | ≤ a | ν | + c o , with a > c > b fixed and t large. Then˜N r ( ˜ Ω t ) ∼ C a , b , c t ( t → ∞ ) . This shows that we see infinitely many points ν ̟ in a slanted direction. Wenote that this slanting strip becomes a sector in λ -space.(iii) Sectors . Let d =
2, and fix 0 < p < q , α > . For t large put Ω t = (cid:26) ( λ , λ ) ∈ [0 , ∞ ) : t ≤ λ ≤ t + t α , p λ ≤ λ ≤ q λ (cid:27) . Then we have N r ( Ω t ) ∼ C p , q t + α ( t → ∞ ) . (iv) Spaces of holomorphic cusp forms.
Take ˜ Ω b equal to the singleton b = ( b , . . . , b j ) ∈ Z d with b j ≥ b j ≡ ξ j mod 2 for 1 ≤ j ≤ d . These are the weights for which there may beholomorphic cusp forms on the product of d copies of the upper half-planefor the character χ of Γ ( I ). Corollary 4.2 shows that for some positiveconstant C we have(6) ˜N r ( ˜ Ω b ) ∼ C d Y j = b j −
12 as d Y j = b j − → ∞ . If we take r totally positive, then the quantity ˜N r ( ˜ Ω b ) has an expressionin terms of Fourier coe ffi cients of holomorphic cusp forms. In particular,this implies that if there are only finitely many weights b with b j ≥ j for which the corresponding space of cusp forms is non-zero. (We donot obtain information concerning weights b for which b j = j .)We will give a much more complete list of applications of the main asymptoticformula in § ROELOF W. BRUGGEMAN AND ROBERTO J. MIATELLO
By using the Selberg trace formula ([21], [8]), some unweighted distributionresults related to those in this paper have been obtained in [9] and [10], while re-sults connected with Weyl’s laws in di ff erent contexts have been proved by severalauthors e.g. [6], [5], [7], [19], [18], [14], [20], [17] and [16].The main tool for the results in the present paper is the Kuznetsov type sumformula in Theorem 3.21 of [3], which we recall in § ffi cients. The results obtained here may be viewed asa generalization of the results in [2]. To explain the di ff erence, we note that thesum formula gives a linear relation between four terms. The two main terms inthe present context are a weighted sum of a test function ϕ over the ν ̟ and anintegral of ϕ against the Plancherel measure. The test function has a product form ϕ ( ν ) = Q j ϕ j ( ν j ) where j runs over the infinite places of the totally real numberfield F . For the terms that are principal in this paper this product form is non-essential. However to show that the other terms are small we need, for one ofthose terms (the sum of Kloosterman sums), estimates of a Bessel transformationof each of the factors ϕ j . This forces the product structure upon us, in contrastwith the case of the Selberg trace formula. There the integral transformation is theFourier transform, which respects rotations.In [2] we chose each factor ϕ j as a Gaussian kernel. For the places in a non-empty set Q of real places, this kernel was an approximation of the constant func-tion one, for the other places it was chosen as an approximation of a delta distribu-tion. This led us to asymptotic results for regions Ω X of the form Y j < Q [ a j , b j ] × Y j ∈ Q (cid:8) ( λ j ) j ∈ Q : X j ∈ Q | λ j | ≤ X (cid:9) . The purpose of this paper is to work with sets having a much more general formin the coordinates in Q . To do this, the test functions have to be chosen in a muchmore complicated way. Our choice is indicated in Lemma 2.2. The idea is to take,for each place in Q a Gaussian kernel of moderate sharpness. We approximate thecharacteristic function of sets in Q j ∈ Q ( R ∪ i R ) by a convolution with this Gaussiankernel. At the real places outside Q we take a general test function to be specified ata later stage. Application of the sum formula gives the relation in Proposition 2.4.The use of a Tauberian argument like in [2] is no longer possible. To be able tohandle the error terms, we first give in § §
5, involves choosing the factors of the test function atthe real places outside Q as approximations of characteristic functions of intervalsin the coordinate λ j . The central result is Theorem 5.3. It is specialized in § ENSITY RESULTS ON HILBERT MODULAR GROUPS II 5
The sum formula involves products c r ( ̟ ) c r ′ ( ̟ ) for two non-zero Fourier termsorders r and r ′ . Its application in the present paper works well if r ′ r is totallypositive. We intend to apply the asymptotic results, under this assumption, in sub-sequent work where we will take eigenvalues of Hecke operators into account.1. P reliminaries and discussion of main results This section serves to recall results and fix notations, and after that to state themain results of this paper.1.1.
Automorphic representations for Hilbert modular groups.
Let F be a to-tally real number field with degree d over Q . The Lie group G = SL ( R ) d is theproduct Q j SL ( k j ) over all infinite places j of F . We fix a non-zero ideal I in thering of integers O of F . The group G contains the discrete subgroup(7) Γ ( I ) = (cid:26) ac bd ! ∈ SL ( O ) : c ∈ I (cid:27) with finite covolume.Let χ be a character of ( O / I ) ∗ . It determines a character of Γ ( I ) by χ (cid:16) ac bd (cid:17) = χ ( d ). Let L ( Γ ( I ) \ G , χ ) be the Hilbert space of classes of functions transformingaccording to f ( γg ) = χ ( γ ) f ( g ) for γ ∈ Γ ( I ). The group G acts unitarily in thisHilbert space by right translation. This space is split up according to central char-acters, indicated by ξ ∈ { , } d . By L ξ ( Γ ( I ) \ G , χ ) we mean the subspace on whichthe center acts by ζ ζ ! , . . . , ζ d ζ d !! Y j ζ ξ j j , where ζ j ∈ { , − } . This subspace can be non-zero only if the following compati-bility condition holds:(8) χ ( − = Y j ( − ξ j . We assume this throughout this paper.There is an orthogonal decomposition(9) L ξ ( Γ ( I ) \ G , χ ) = L , cont ξ ( Γ ( I ) \ G , χ ) ⊕ L , discr ξ ( Γ ( I ) \ G , χ ) . The G -invariant subspace L , cont ξ ( Γ ( I ) \ G , χ ) can be described by integrals of Eisen-stein series and the orthogonal complement L , discr ξ ( Γ ( I ) \ G , χ ) is a direct sum ofclosed irreducible G -invariant subspaces. If χ =
1, the constant functions forman invariant subspace. All other irreducible invariant subspaces have infinite di-mension. They are cuspidal and span the space L , cusp ξ ( Γ ( I ) \ G , χ ), the orthogonalcomplement of the constant functions in L , discr ξ ( Γ ( I ) \ G , χ ).We fix a maximal orthogonal system { V ̟ } ̟ of irreducible subspaces in the Hil-bert space L , cusp ξ ( Γ ( I ) \ G , χ ). This system is unique if all ̟ are inequivalent. Ingeneral, there might be multiplicities, due to oldforms. ROELOF W. BRUGGEMAN AND ROBERTO J. MIATELLO
Each irreducible automorphic representation ̟ of G = Q j SL ( R ) is the tensorproduct N j ̟ j of irreducible representations of SL ( R ). Here and in the sequel, j is supposed to run over the d archimedean places of F .The factor ̟ j can (almost) be characterized by the eigenvalue λ ̟, j of the Casi-mir operator of SL ( R ), and the central character, which is indicated by ξ j .If ξ j =
0, then the eigenvalue λ ̟, j can either be of the form b − b with b ≥ λ ̟, j have a lower bound λ ∈ (cid:16) , i . By the Selberg conjecture, itis expected that one can take λ = . The best result at present is = − (cid:16) (cid:17) ≤ λ ≤ ; see [13]. If ξ j =
1, then the λ ̟, j can either lie in h , ∞ (cid:17) , or be of the form b − b with b ≥ b odd. We call λ ̟ = (cid:16) λ ̟, j (cid:17) ∈ R d the eigenvalue vector of therepresentation ̟ .Spectral theory shows that the set { λ ̟ } is discrete in R d . To see this we usethat the Casimir operator of G has a discrete spectrum with finite multiplicitiesin L , cusp ξ ( Γ \ G ) q , where L , cusp ξ ( Γ \ G ) q is the subspace of L , cusp ξ ( Γ \ G ) with K -type(or weight) q ∈ Z d , q ≡ ξ mod 2. Hence the number of representations ̟ (withmultiplicities) such that V ̟ ∩ L , cusp ξ ( Γ \ G ) q , { } and for which the eigenvalue λ ̟, + λ ̟, + · · · + λ ̟, d of the Casimir operator is under a given bound, is finite.For a given component ̟ j in the discrete series, the weights q j occurring in V ̟, j satisfy | q j | ≥ b j ≥
1, with λ ̟, j = b j (cid:0) − b j (cid:1) . So for a given bounded set Ω ⊂ R d wecan choose the K -type q such that all ̟ with λ ̟ ∈ Ω are present in L , cusp ξ ( Γ \ G ) q .Thus Ω contains only finitely many λ ̟ , counted with multiplicities.The correspondence between values of λ = λ ̟, j and equivalence classes ofunitary representations of SL ( R ) of infinite dimension is one-to-one if λ > ξ = ξ j =
0, and if λ > if ξ =
1. In the other cases, λ = b − b with b ∈ Z ≥ , b ≡ ξ mod 2. In these cases, there are two equivalence classes, one with lowestweight b (holomorphic type), and one with highest weight − b (antiholomorphictype). If b ≥
2, representations of these classes occur discretely in L (cid:0) SL ( R ) (cid:1) , andare called discrete series representations . The representations in the case b = L (cid:0) SL ( R ) (cid:1) .All these classes of representations, discrete series or not, may occur as an irre-ducible component of L , cusp ξ ( Γ ( I ) \ G , χ ).As discussed in § V ̟ determines the Fourier expansion of any automorphic form in V ̟ . We refer to[3] for the normalization. This results in coe ffi cients c r ( ̟ ) describing the Fourierexpansion at the cusp ∞ . The Fourier term order r runs through the inverse di ff erent O ′ .In the choice of the c r ( ̟ ) there is a freedom of a complex factor with absolutevalue one for a given ̟ . Since we shall work with weights | c r ( ̟ ) | this freedomhas no influence on the results we aim at.When dealing with the sum formula, it is technically easier to parametrize theeigenvalues λ ̟, j ∈ R by λ ̟, j = − ν ̟, j , with, for instance, ν ̟, j ∈ (0 , ∞ ) ∪ i [0 , ∞ ). ENSITY RESULTS ON HILBERT MODULAR GROUPS II 7
We put ν ̟ = (cid:0) ν ̟, j (cid:1) , and call ν ̟ and ξ ̟ = (cid:0) ξ ̟, j (cid:1) ∈ { , } d the spectral parameters of ̟ .We have ν ̟ ∈ Y ξ = Q j Y ξ j , with Y : = n b − : b ≥ o ∪ i [0 , ∞ ) ∪ (0 , ν ] , (10) Y : = n b − : b ≥ o ∪ i [0 , ∞ ) , where ν = q − λ .1.2. Discussion of main results.
Counting function.
For compact sets Ω ⊂ R d , we use the counting functions(11) N r ( Ω ) : = N r ξ,χ ( Ω ) : = X ̟, λ ̟ ∈ Ω | c r ( ̟ ) | , with r ∈ O ′ r { } . The representations ̟ run through the orthogonal system ofirreducible subspaces of L , cusp ξ ( Γ ( I ) \ G , χ ) chosen in § c r ( ̟ ) = ̟ and some r . However, varying r we can detect all ̟ .More generally, if f is a function on R d , we can consider the sum(12) N r ( f ) : = N r ξ,χ ( f ) : = X ̟ f ( λ ̟ ) | c r ( ̟ ) | , which converges for suitable f , for instance, compactly supported f . So N r ( Ω ) = N r ( χ Ω ), where χ Ω is the characteristic function of Ω .In the ν -coordinate the counting function is(13) ˜N r ( ˜ Ω ) = X ̟, ν ̟ ∈ ˜ Ω | c r ( ̟ ) | , for sets ˜ Ω ⊂ Y ξ .1.2.2. Plancherel measure.
We will compare N r ( Ω ) with Pl( Ω ) = Pl ξ ( Ω ). Themeasure Pl on R d is the product Pl = N j Pl ξ j , where Pl ξ j is the measure on R given by Pl ( f ) = Z ∞ / f ( λ ) tanh π q λ − d λ (14) + X b ≥ , b ≡ ( b − f (cid:16) b (cid:16) − b (cid:17)(cid:17) , Pl ( f ) = Z ∞ / f ( λ ) coth π q λ − d λ + X b ≥ , b ≡ ( b − f (cid:16) b (cid:16) − b (cid:17)(cid:17) Note that Pl ξ j gives zero measure to the set of exceptional eigenvalues in h λ , (cid:17) . ROELOF W. BRUGGEMAN AND ROBERTO J. MIATELLO
The notations Pl refers to the Plancherel measure of SL ( R ), see, e.g., [15],Chap. VIII, §
4, p. 174.In the ν -coordinate the Plancherel measure on Y ξ is given by e Pl ξ = N j e Pl ξ j ,where(15) e Pl ξ j ( f ) = Z ∞ f ( it ) e pl j ( t ) dt + X β ∈ ξ j + + N f ( β ) e pl j ( β ) , with(16) e pl j ( t ) ξ j = t ∈ i R | t | tanh π | t | ξ j = t ∈ i R | t | coth π | t | t ≡ ξ j − mod 1 t ∈ R r { } | t | t . ξ j − mod 1 t ∈ R r { } f is even, then e Pl ( f ) = i Z Re ν = f ( ν ) ν tan πν d ν + X β ∈ + Z | β | f ( β ) , (17) e Pl ( f ) = − i Z Re ν = f ( ν ) ν cot πν d ν + X β ∈ Zr { } | β | f ( β ) . The reference measure V is more easily given in the ν -coordinate. We leave thereformulation in terms of the λ -coordinate to the reader. The measure has a productform ˜ V = N j ˜ V ,ξ j on the space ((0 , ∞ ) ∪ i [0 , ∞ )) d with Z h d ˜ V , = Z ∞ t h ( it ) dt + Z h ( it ) dt + Z ν h ( x ) dx + X β> , β ≡ (1) β h ( β ) , (18) Z h d ˜ V , = Z ∞ t h ( it ) dt + Z h ( it ) dt + X β> , β ≡ β h ( β ) . So ˜ V is positive everywhere on Y ξ , and e Pl( ˜ Ω ) ≪ ˜ V ( ˜ Ω ) for all Ω . We have also˜ V ( ˜ Ω ) ≪ ε e Pl( ˜ Ω ) if the coordinates of ˜ Ω t stay away from (0 , ν ] ∪ i [0 , ε ) for some ε > Asymptotic formula.
We will prove that for families t Ω t of sets in R d satisfying conditions discussed below:(19) N r ( Ω t ) = √| D F | (2 π ) d Pl( Ω t ) + o (cid:0) V ( Ω t ) (cid:1) ( t → ∞ )for any r ∈ O ′ r { } .Here D F is the discriminant over Q of the totally real number field F of degree d .The character χ of ( O / I ) ∗ and ξ ∈ { , } d are as explained in § ENSITY RESULTS ON HILBERT MODULAR GROUPS II 9
The sets Ω t should get large, in particular, V ( Ω t ) should tend to infinity as t →∞ . Moreover, the boundary ∂ Ω t should be relatively small in comparison with Ω t itself. The precise conditions are discussed in § Ω t ) ≪ V ( Ω t ) the term o (cid:0) V ( Ω t ) (cid:1) need not be small in comparison withPl( Ω t ). If V ( Ω t ) ≪ Pl( Ω t ) holds as well, then the asymptotic formula simplifies to(20) N r ( Ω t ) ∼ √| D F | (2 π ) d Pl( Ω t ) ( t → ∞ ) . In this section we shall be content to discuss a number of families for which theasymptotic formula (19) holds, showing the existence of automorphic forms witheigenvalue (or spectral) parameters lying in such regions Ω t , as t gets large.1.2.4. Small rectangle in the real quadratic case.
Before stating more general re-sults, we consider the case that d =
2, and apply some of the more general state-ments first for this situation.Let [ α, β ] ⊂ [1 / , ∞ ) and consider for t ≥ :(21) Ω t = [ α, β ] × [ t , t + √ t ] . Theorem 1.3 implies(22) N r ( Ω t ) ∼ √ D F π Z βα tanh π q λ − d λ t / ( t → ∞ ) . In particular, there are infinitely many ̟ with λ ̟, ∈ [ α, β ]. These ̟ have asecond component of unitary principal series type. A similar result holds with ̟ of discrete series type: Ω b = [ α, β ] × n b − o with b > , b (cid:27) ξ mod 2 , (23) N r ( Ω b ) ∼ √ D F π Z βα tanh π q λ − d λ · b ( b → ∞ ) . On the other hand if [ α, β ] ⊂ h λ , (cid:17) thenfor Ω t = [ α, β ] × [ t , √ t ] with t ≥ , (24) N r ( Ω t ) = o (cid:0) t / (cid:1) ( t → ∞ ) ;and for Ω b = [ α, β ] × n b − o with b > , b (cid:27) ξ mod 2 , (25) N r ( Ω b ) = o (cid:0) b (cid:1) ( b → ∞ ) . This does not exclude the presence of ̟ of complementary series type, but givesan upper bound for their weighted density.These results also hold with the role of ̟ and ̟ interchanged.One may also consider families of rectangles for which both factors vary:(26) Ω t = [ t , t + √ t ] × [ ct , ct + √ t ] , with t ≥ and c ≥
1. Then(27) N r ( Ω t ) ∼ √ D F π c / t ( t → ∞ ) . Floating boxes.
We consider for general F a small hypercube of fixed sizefloating to infinity in the region ( i [1 , ∞ )) d of the ν -plane. Proposition 1.1.
Let ˜ Ω t = Q j i [ a j ( t ) , a j ( t ) + σ ] with σ > , a j ( t ) ≥ for all jand t, and where lim t →∞ a j ( t ) = ∞ for at least one j. Then ˜N r ( ˜ Ω t ) ∼ √| D F | (2 π ) d e Pl( ˜ Ω t ) ( t → ∞ ) . Note that e Pl( ˜ Ω t ) → ∞ . We have e Pl( ˜ Ω t ) ∼ σ d Q j a j ( t ) if a j ( t ) → ∞ for all j .Proposition 6.1 implies that the size σ may even slowly decrease with t , provided σ ( t ) ≥ γ (cid:18)X j log a j ( t ) (cid:19) − α for any α ∈ (0 ,
12 ) , γ > . We conclude that there are spectral parameters ν ̟ in such hypercube if they aresu ffi ciently far away from the origin. If we reformulate in terms of λ we get boxes Ω t in λ -space with increasing size.1.2.6. Remark.
On the other hand, for hypercubes in λ -space with fixed size ourmethod does not give an asymptotic formula. In fact, the λ ̟ may leave space for asmall hypercube moving around in λ -space avoiding all of the λ ̟ . This can occurif the (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) are not often very small.1.2.7. Discrete series.
For ̟ for which all factors are discrete series representa-tions, we do not need boxes, but can work with single points. Proposition 1.2.
Let p ∈ Q j (cid:16) ξ j + + N (cid:17) and let Ω = Ω p = { p } . Then, if at leastone coordinate of p tends to infinity, we have ˜N r ( { p } ) ∼ √| D F | π d Y j p j . Combinations.
A combination is possible. At some places we take a fixedinterval [ A j , B j ] on the λ -line. We have to require that the endpoints are not equalto a discrete series eigenvalue:(28) A j , B j < n b (cid:16) − b (cid:17) : b > , b ≡ ξ j mod 2 o . Theorem 1.3.
Let E ⊔ Q + ⊔ Q − be a partition of the infinite places of F. Supposethat B E = Q j ∈ E [ A j , B j ] satisfies condition (28) . LetC t = Y j ∈ Q + i h a j ( t ) , a j ( t ) + σ i ENSITY RESULTS ON HILBERT MODULAR GROUPS II 11 with a j ( t ) ≥ for any j ∈ Q + and all t, with δ > fixed. Let p ∈ Q j ∈ Q − (cid:16) ξ j + + N (cid:17) .Then, provided that a j ( t ) → ∞ for at least one j ∈ Q + or p j → ∞ one j ∈ Q − , wehave ˜N r (cid:16) ˜ B E × C t × { p } (cid:17) = √| D F | (2 π ) d e Pl (cid:16) ˜ B E × C t × { p } (cid:17) (29) + o (cid:16) ˜ V ( ˜ B E × C t × { p } ) (cid:17) . (30)In Theorem 1.3 some factors of B E may be in the region h λ , (cid:17) , giving e Pl( ˜ B E × C t × { p } ) =
0. So here we need ˜ V in the asymptotic formula.Again, the constant size σ in Theorem 1.3 can be replaced by(31) σ ( t ) = γ (cid:18) X j ∈ Q + log a j ( t ) + X j ∈ Q − log(2 p j ) (cid:19) − α (0 < α < , γ > . Floating spheres.
The following is analogous to an unweighted distributionresult, based on the Selberg trace formula, given by Huntley and Tepper, [10]:
Proposition 1.4.
Let B ( m , r ) be a sphere in R d with center m and radius r. Let m j ≥ r + for all j, and suppose that m j tends to infinity for all j (not necessarilywith the same speed). Then ˜N r ( iB ( m , r )) ∼ p | D F | v d (cid:16) r π (cid:17) d Y j m j , where v d denotes the volume of the unit sphere in R d . This asymptotic formula holds even if r = r ( m ) is allowed to go down to zeronot faster than (cid:16) log Q j ∈ Q + m j (cid:17) − α , with α < .1.2.10. Slanted strips in the quadratic case.
Proposition 1.5.
Let d = , and put ˜ Ω t = n ( ν , ν ) ∈ ( i [1 , ∞ )) : t ≤ | ν | ≤ t , a | ν | + b ≤ | ν | ≤ a | ν | + c o , with a > , c > b fixed and t large. Then ˜N r ( ˜ Ω t ) ∼ π p | D F | a ( c − b ) t ( t → ∞ ) . Proposition 1.5 shows that we see infinitely many points ν ̟ in a slanted direc-tion. This slanted strip becomes a sector in λ -space. It opens up with a speed ofthe order | λ | / .1.2.11. Density.
Theorem 1.3 directly implies the density results in Propositions3.7 and 3.8 in [2], without the restriction χ = ξ =
0. The idea is that wetake | E | = d − B E to restrict ( λ ̟, j ) j ∈ E to a tiny set withpositive Plancherel measure. The remaining coordinate of λ ̟ is allowed to rangethrough a set in R with increasing Plancherel measure. This shows that there areinfinitely many ̟ with ( λ ̟, j ) j ∈ E ∈ B E . If one of the factors [ A j , B j ] of B E is contained in the interval [ λ , ) of ex-ceptional eigenvalues, the Plancherel measure of B E × C t is zero. The asymptoticformula cannot give the absence of ̟ with λ ̟, j ∈ [ A j , B j ], but only that the densityis small in comparison with ˜ V ( B E × C t ). We note that we project here along a coor-dinate axis in the natural product structure of R d as the product of the archimedeancompletions of F .1.2.12. Weighted Weyl laws.
In this subsection we list some consequences of theasymptotic formula in the λ -parameter. We have that the asymptotic formula holdsfor Ω t = [ − t , t ] d . This implies Proposition 1.6. X ̟, | λ ̟, j |≤ t for all j (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) ∼ √| D F | π d t d ( t → ∞ ) . This result confirms that there are infinitely many cuspidal ̟ for each choice of χ and ξ satisfying (8). With the normalization of the c r ( ̟ ) that we have chosen,the density does not depend on the order of the Fourier coe ffi cients that we use. Proposition 1.7.
Let Q + ⊔ Q − a partition of the real places of F. Then X ̟, P j | λ ̟, j | ≤ t λ ̟, j ≤ , j ∈ Q − λ ̟, j > , j ∈ Q + (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) ∼ √| D F | d ! (2 π ) d t d This variant is Corollary 3.4 in [2]. There we considered only the trivial charac-ter of Γ ( I ) and the trivial central character. Proposition 6.3 implies that all resultsin [2] extend to the more general context in this paper.1.2.13. Sectors in the quadratic case.
Proposition 1.8.
Let d = , and fix < p < q, α > . For t ≥ (1 + p − ) put Ω t = (cid:26) ( λ , λ ) ∈ [0 , ∞ ) : t ≤ λ ≤ t + t α , p λ ≤ λ ≤ q λ (cid:27) . Then N r ( ˜ Ω t ) ∼ q − p t + α ( t → ∞ ) .
2. S um formula
The basis of the result in this paper is the Kuznetsov type sum formula in The-orem 3.21 of [3], which we recall in § § ENSITY RESULTS ON HILBERT MODULAR GROUPS II 13
Statement of the sum formula.
The sum formula as stated in [3] depends ontwo non-zero Fourier term orders r , r ′ ∈ O ′ r { } . For the end results of this paperit su ffi ces to take r ′ = r . In a later paper we intend to work with di ff erent Fourierterm orders and take Hecke operators into account. Then we’ll need to consider r , r ′ as well.The sum formula states an equality with four terms, all depending on a giventest function. We first state the sum formula, and will next recall the description ofthe ingredients. Theorem 2.1.
Spectral sum formula.
For any test function ϕ ∈ T ξ ( τ, a ) the sumsand integrals ˜N r ( ϕ ) , Eis r ( ϕ ) , e Pl( ϕ ) and K r χ (cid:16) B s ξ ϕ (cid:17) converge absolutely, and (32) ˜N r ( ϕ ) + Eis r ( ϕ ) = √| D F | (2 π ) d e Pl( ϕ ) + K r χ (cid:16) B s ξ ϕ (cid:17) . We work with a fixed character χ of Γ and a compatible central character givenby ξ ∈ { , } d ; so condition (8) is satisfied. Fixed is r ∈ O ′ r { } . Fixed are also theparameters τ ∈ (cid:16) , (cid:17) and a >
2, which determine the space of test functions.2.1.1.
Test functions.
The space T ξ ( τ, a ) of test functions consists of functions withthe following product structure:(33) ϕ ( ν , . . . , ν d ) = Y j ϕ j ( ν j ) , where each ϕ j is a function on a set(34) { z ∈ C : | Re z | ≤ τ } ∪ n b − : b ≡ ξ j mod 2 , b ≥ o satisfying the following conditions:(T1) ϕ j is holomorphic on | Re z | ≤ τ ;(T2) ϕ j ( z ) ≪ (1 + | z | ) − a on the domain of ϕ j ,(T3) ϕ j ( − z ) = ϕ j ( z ) on the strip | Re z | ≤ τ .2.1.2. Spectral side.
In the left hand side of (32) are two terms connected to thespectral decomposition of L ξ ( Γ ( I ) \ G , χ ). The first term ˜N r ( ϕ ) is the sum definedin (13). Its convergence already implies the existence of infinitely many cuspidalautomorphic representations in L , cusp ξ ( Γ ( I ) \ G , χ ).The orthogonal complement in L ξ ( Γ ( I ) \ G , χ ) of the cuspidal subspace givesrise to the term Eis r ( ϕ ). Since r is non-zero, the constant functions, in the case χ = ξ =
0, do not contribute to the sum formula. We have:Eis r ( ϕ )(35) = X κ ∈P χ c κ X µ ∈ Λ κ,χ Z ∞−∞ (cid:12)(cid:12)(cid:12) D r ξ ( κ, χ ; it , i µ ) (cid:12)(cid:12)(cid:12) ϕ ( it + i µ ) dt . P χ is a set of representatives of cuspidal orbits suitable for the character χ . Foreach κ , there is a lattice Λ κ,χ in the hyperspace P j x j = R d . In it + i µ , the realnumber t is identified to ( t , t , . . . , t ) ∈ R d . The positive constants c κ come from the spectral formula for the continuous spectrum. The D r ξ ( κ, χ ; it , i µ ) are normalizedFourier coe ffi cients of Eisenstein series. See (2.31) of [3]. There is a positive realnumber q such that(36) D r ξ ( κ, χ ; it , i µ ) ≪ F , I , r (cid:0) log(2 + | t | ) (cid:1) q + (cid:16) log max j ( | µ j | + (cid:17) q if µ , , (cid:0) log(2 + | t | ) (cid:1) q if µ = § q = F , some ideals I and some cusps κ , there might be µ ∈ Λ κ,χ with µ , j | µ j | <
1. This is not intended in [2].The logarithms arise in (71) and (72) of [2]. Checking the reasoning there, we seethat if the bounds for Z ( s , λ, τ ) are negative, they can be replaced by 0. Thus, wecan replace (cid:16) log max j (cid:0) | µ j | + (cid:1)(cid:17) in (36) by (cid:16) log (cid:0) max j ( | µ j | + (cid:1)(cid:17) .There is another reformulation. Put x i = t + µ i . Then t = d P j x j and µ i = x i − d P j x j . This implies that we have | t | ≪ P j | x j | and k µ k ≪ P j | x j | . Thus wearrive at(37) D r ξ ( κ, χ ; it , i µ ) ≪ F , I , r (cid:18) log (cid:0) + X j | t + µ j | (cid:1)(cid:19) . Delta term.
The right hand side in (32) arises from geometrical considera-tions. For the purpose of this paper the first term is the principal one.2.1.4.
Bessel transformation.
The last term in (32) contains a
Bessel transform ofthe test function. It depends on r ∈ O ′ r { } by s = (sign r j ) j ∈ { , − } d . Here weview elements of F as elements of R d = Q j F j . The Bessel transformation has aproduct form:B s ξ ϕ ( t ) = Y j B s j ξ j ϕ j ( t j ) , (38) B η ϕ ( t ) : = − i Z Re ν = ϕ ( ν ) J ν ( | t | ) − J − ν ( | t | )cos πν ν d ν + X b ≥ , b ≡ ( − b / ( b − ϕ (cid:16) b − (cid:17) J b − ( | t | ) , B η ϕ ( t ) : = − η sign( t ) Z Re ν = ϕ ( ν ) J ν ( | t | ) + J − ν ( | t | )sin πν ν d ν − i η sign( t ) X b ≥ , b ≡ ( − ( b − / ( b − ϕ (cid:16) b − (cid:17) J b − ( | t | ) . These Bessel transforms converge absolutely for any test function, and provideus with functions f = B s ξ ϕ on ( R ∗ ) d that satisfy(39) f ( t ) ≪ Y j min (cid:16) | t j | τ , (cid:17) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 15
Sum of Kloosterman sums.
The Kloosterman sums for the present situationare(40) S χ ( r ′ , r ; c ) = X ∗ a mod ( c ) χ ( a ) e π i Tr F / Q (( ra + r ′ ˜ a ) / c ) , with r , r ′ ∈ O ′ and c ∈ I r { } . This is well defined if χ is a character of ( O / ( c )) ∗ ,in particular for the character χ of ( O / I ) ∗ if c ∈ I . For each a ∈ O that is invertiblemodulo ( c ), an element ˜ a is picked such that a ˜ a ≡ c ).For functions f satisfying (39) the sum(41) K r χ ( f ) : = X c ∈ I r { } | N ( c ) | − S χ ( r , r ; c ) f (cid:16) π | r | c (cid:17) converges absolutely. This holds in particular for f = B s ξ ϕ . By π | r | c we mean theelement of ( R ∗ ) d given by the d embeddings F → R of r and c .A trivial estimate of the Kloosterman sums is (cid:12)(cid:12)(cid:12) S χ ( r ′ , r ; c ) (cid:12)(cid:12)(cid:12) ≤ | N ( c ) | . As discussedin § c ) has a representation( c ) = Q p p v p ( c ) in prime ideals, then(42) (cid:12)(cid:12)(cid:12) S χ ( r ′ , r ; c ) (cid:12)(cid:12)(cid:12) ≪ δ (cid:12)(cid:12)(cid:12) N ( rr ′ ) (cid:12)(cid:12)(cid:12) / Y p | / I ( N p ) v p ( c ) / + δ Y p | I ( N p ) v p ( c ) + δ for each δ >
0. This is not the best possible estimate, however it is reasonablysimple and will do for our purpose.2.2.
Application of the sum formula.
In view of the term ˜N r ( ϕ ) in the sum for-mula, we want to choose the test function ϕ such that it approximates the character-istic function of a compact set ˜ Ω in the space Y ξ , in which the spectral parameters ν ̟ take their values. In this paper we first choose a test function approximatingthe delta distribution at q ∈ Q j ∈ Q Y ξ j for a non-empty subset Q of real places. Atthe other archimedean places we leave ϕ j free for the moment in the space of localtest functions. The local factor ϕ j for j ∈ Q such that q j = b − , b (cid:27) ξ j mod 2, b ≥ ϕ j is the delta distribution at ϕ j . For q j ∈ [0 , ν ) ∪ i [0 , ∞ ) the choice is more delicate. It does not su ffi ce that ϕ j approxi-mates the delta distribution at q j . The terms Eis r ( ϕ ) and K r (B s ξ ϕ ) should have goodestimates. Under the additional assumption that q j < (0 , ν ] ∪ i [0 , q j ∈ (0 , ν ] ∪ i [0 ,
1) wehave not found a test function that works well.
Lemma 2.2.
Let { , . . . , d } = E ⊔ Q + ⊔ Q − with Q : = Q + ∪ Q − , ∅ . For q ∈ (cid:16) R r (cid:16) − , (cid:17)(cid:17) Q − × ( i R r i ( − , Q + put ϕ ( q , ν ) = Q j ϕ ( q j , ν j ) where ϕ j ( q , · ) is anarbitrary local test function satisfying the conditions in § ∈ E, and where for j ∈ Q: (43) ϕ j ( q , ν ) j ∈ Q + q U π (cid:16) e U ( q − ν ) + e U ( q + ν ) (cid:17) if | Re ν | ≤ τ elsewherej ∈ Q − if ν = q or − q elsewhereThen there are constants t > , ρ ∈ (1 − τ, such that for U ≥ and A > r (cid:0) ϕ ( q , · ) (cid:1) = √| D F | (2 π ) d e Pl ( ϕ ( q , · ))(44) + O F , I , r , t , t , A (cid:18) N E ( ϕ E ) e t U | Q + | Y j ∈ Q + | q j | ρ Y j ∈ Q − | q j | − A (cid:19) , where ϕ E = N j ∈ E ϕ j andN E ( ϕ E ) = Y j ∈ E N j ( ϕ j ) , (45) N j ( ϕ j ) = sup ν, ≤ Re ν ≤ τ | ϕ ( ν ) | (1 + | ν | ) a + X b ≡ ξ j (2) , b ≥ b a (cid:12)(cid:12)(cid:12)(cid:12) ϕ j (cid:16) b − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . Note that ϕ j chosen in (43) is a local test function for any choice of the parame-ters a > τ ∈ (cid:16) , (cid:17) .By Theorem 2.1 it su ffi ces to estimate Eis r ( ϕ ( q , · )) and K r χ (B s ξ ϕ ( q , · )) by the errorterm in (44). This we carry out in the remainder of this subsection.2.2.1. Bessel transforms.
For the factors j ∈ E we cannot do better than applyLemma 3.12 in [3]. This gives(46) B s j , s ′ j ξ j ϕ j ( t ) ≪ E min (cid:16) | t | τ , (cid:17) . The subscript E in ≪ E , O E and o E indicates not only dependence on the choice ofthe set E , but also on the choice of the test function ϕ E : = N j ∈ E ϕ j . Here and inthe sequel, this dependence goes via the factor N E ( ϕ E ) in (45).For j ∈ Q − we first consider y ≤ √ n , with n ∈ N . Then | J n ( y ) | ≤ ( y/ n e y n ! ≪ y τ n n − τ e n ( n + n + e − n ≪ y τ (cid:16) ne ( n + (cid:17) n n − − τ ≪ b y τ n b for each b >
0. For y ≥ √ n : | J n ( y ) | ≤
1. Hence, for j ∈ Q − and ± q j ∈ ξ j − + N :(47) B s j ξ j ϕ j ( q ; t ) ≪ A min (cid:16) | q | − A | t | τ , | q | (cid:17) for each A > . If | q j | < ξ j − + N , then B s j , s j ξ j ϕ j ( t ) = j ∈ Q + takes more work. The function ϕ j ( q , ν ) is non-zero only for | Re ν | ≤ τ . We need an estimate like (47), in which the dependence on q is explicit. ENSITY RESULTS ON HILBERT MODULAR GROUPS II 17
We may use (3.64) in [3]:(48) B s j ξ j ϕ j ( q ; t ) = − i (cid:16) i s j sign t (cid:17) ξ j Z Re ν = τ ϕ j ( q , ν ) ν J ν ( | t | )cos π (cid:16) ν − ξ j (cid:17) d ν. We proceed as in § α = Re ν = τ and γ ∈ (cid:16) τ, (cid:17) . As in [2] p. 700, this leads to the estimatefor all | t | > J ν ( | t | ) ≪ | t | τ e π | Im ν | (1 + | Im ν | ) − γ − τ . We obtain for q ∈ i R :B s j ξ j ϕ j ( q ; t ) ≪ U / Z ∞−∞ X ± e U Re( ix + τ ± q ) | t | τ | τ + ix | (50) · (1 + | x | ) − τ − γ dx ≪ | t | τ U / Z ∞−∞ e U τ − U ( x −| q | ) (1 + | x | ) − γ − τ dx ≪ | t | τ e U τ Z ∞ e − x (cid:18) + | q | + x √ U (cid:19) − γ − τ dx ≪ | t | τ e U τ Z ∞ e − x (cid:18) (1 + | q | ) − γ − τ + (cid:0) xU − / (cid:1) − γ − τ (cid:19) dx ≪ | t | τ e U τ (cid:18) (1 + | q | ) − γ − τ + U − + γ + τ (cid:19) ≪ | t | τ e U τ (1 + | q | ) ρ , where ρ = − γ − τ ; so ρ ∈ (cid:16) − τ, − τ (cid:17) ⊂ (cid:16) , (cid:17) .The factor e U τ needs attention since it becomes large for large values of U . Wecarry along this factor, and will compensate for it later.These estimates are good for small values of | t | . For large | t | , we use(51) J ν ( | t | ) ≪ e π | Im ν | for Re ν = σ = s j ξ j ϕ j ( q , t ) ≪ U / Z Re ν = (cid:12)(cid:12)(cid:12)(cid:12) e U ( ν + q ) + e U ( ν − q ) (cid:12)(cid:12)(cid:12)(cid:12) | ν | | d ν |≪ Z ∞−∞ e − x (cid:12)(cid:12)(cid:12) xU − / + | q | (cid:12)(cid:12)(cid:12) dx ≪ + | q | ≪ | q | . Thus we have(52) B s ξ ϕ ( q ; t ) ≪ A Y j (cid:16) a j | t j | τ , b j (cid:17) , with(53) a j b j N j ( ϕ j ) N j ( ϕ j ) j ∈ E | q | − A | q | j ∈ Q − e τ U | q | ρ | q | j ∈ Q + Kloosterman term.
We estimate the sum of Kloosterman sums by the sumof the absolute values of the terms:(54) K r , r χ (cid:16) B s , s ξ ϕ ( q ; · ) (cid:17) ≪ X c ∈ I r { } (cid:12)(cid:12)(cid:12) S χ ( r , r ; c ) (cid:12)(cid:12)(cid:12) | N ( c ) | Y j min (cid:18) a j (cid:18) π | r j r ′ j | / | c j | (cid:19) τ , b j (cid:19) , with a j and b j as in (53).For the Kloosterman sums, we use the Weil bound, as stated in (42). This bounddepends only on the ideal ( c ), so we decompose the sum as ≪ r ,δ X ′ ( c ) ⊂ I Y p | / I N p v p ( c )( δ − / Y p | I N p v p ( c ) δ · X ζ ∈O ∗ Y j min (cid:16) p j | ζ j | − τ , q j (cid:17) , with p j = a j (4 π | r j | ) τ , q j = b j , ( c ) = Y p prime p v p ( c ) . The prime denotes that the zero ideal is excluded. We can take δ > α = τ , β = y j = c − j . Thus, we estimatethe sum over ζ ∈ O ∗ by: ≪ (cid:18) + (cid:12)(cid:12)(cid:12) log | N ( c ) | + τ log QP (cid:12)(cid:12)(cid:12)(cid:19) d − min (cid:16) P | N ( c ) | − τ , Q (cid:17) , P = Y j p j = (4 π ) τ d | N ( r ) | τ e τ U | Q + | N E ( ϕ E ) Y j ∈ Q + | q j | ρ Y j ∈ Q − | q j | − A , Q = Y j q j = N E ( ϕ E ) Y j ∈ Q | q j | , PQ = (4 π ) τ d e τ U | Q + | | N ( r ) | τ Y j ∈ Q + | q j | ρ − Y j ∈ Q − | q j | − A − . We have already a small quantity δ . We employ it also for the logarithms. Weuse that for c ∈ O r { } the | N ( c ) | stay away from zero: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log | N ( c ) | + τ log QP (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) log | N ( c ) | (cid:12)(cid:12)(cid:12) + τ (cid:12)(cid:12)(cid:12) log QP (cid:12)(cid:12)(cid:12) ≪ δ | N ( c ) | δ/ ( d − + max (cid:18)(cid:0) QP (cid:1) δ/ ( d − , (cid:0) PQ (cid:1) δ/ ( d − (cid:19) , ENSITY RESULTS ON HILBERT MODULAR GROUPS II 19 (cid:12)(cid:12)(cid:12)(cid:12) log | N ( c ) | + τ log QP (cid:12)(cid:12)(cid:12)(cid:12) d − ≪ | N ( c ) | δ + max (cid:18)(cid:16) QP (cid:17) δ , (cid:16) PQ (cid:17) δ (cid:19) , (cid:18) + (cid:12)(cid:12)(cid:12) log | N ( c ) | + τ log QP (cid:12)(cid:12)(cid:12)(cid:19) min (cid:16) P | N ( c ) | − τ , Q (cid:17) ≪ r e τ U | Q + | (1 + δ ) N E ( ϕ E ) | N ( c ) | − τ (1 − δ ) · Y j ∈ Q + | q j | ρ + (1 − ρ ) δ Y j ∈ Q − | q j | − A + ( A − δ . We have assumed that the first factor in the minimum is the essential one for ourpurpose.Let us define: t = τ (1 + δ ), τ = τ (1 + δ ), ρ = ρ + (1 − ρ ) δ , and A = A + (1 − A ) δ . We take A = −
3, and δ > ffi ciently small such that < τ < ,1 − τ < ρ < A >
2. We find the following estimate for the sum of Kloostermansums in (54): ≪ r ,δ e t U | Q + | N E ( ϕ E ) X ′ ( c ) ⊂ I Y j ∈ Q + | q j | ρ · Y j ∈ Q − | q j | − A Y p | / I N p v p ( c )( δ − / − τ ) Y p | I N p v p ( c )( δ − τ ) ≪ e t U | Q + | N E ( ϕ E ) Y j ∈ Q + | q j | ρ Y j ∈ Q − | q j | − A · Y p | / I − N p δ − / − τ Y p | I − N p δ − τ . Under the additional assumption on δ that 2 τ + − δ >
1, the product converges,and we have obtained:K r , r χ (cid:16) B s , s ξ ϕ ( q ; · ) (cid:17) ≪ F , I , r ,δ e t U | Q + | N E ( ϕ E ) Y j ∈ Q + | q j | ρ Y j ∈ Q − | q j | − A , with the size of the error term in (44).This is the main place where the dependence on the ideal I ⊂ O determining Γ = Γ ( I ) ⊂ SL ( O ) enters the estimates. We leave this dependence implicit.2.2.3. Eisenstein term.
We still have to estimate Eis r ( ϕ ( q , · )). The definition in(35) shows that Eis r ( ϕ ( q , · )) = Q − = j ∈ E , we have ϕ j ( ν ) ≪ E (cid:16) + | ν | (cid:17) − a . In view of (37) it su ffi ces to estimate X κ ∈P χ X µ ∈ Λ κ,χ Z ∞−∞ | N ( r ) | δ N E ( ϕ E ) l ( q , t + µ ) dt , with l ( q , x ) = Y j ∈ E (cid:16) + x j (cid:17) δ − a / · Y j ∈ Q + U / (cid:16) + x j (cid:17) δ (cid:16) e − U ( x j −| q j | ) + e − U ( x j + | q j | ) (cid:17) . In § µ ∈ Λ κ, by an integral over thehyperplane P dj = x j =
0. Since in [2] the quantity corresponding to U went downto zero, there was no problem there. Here U may be large, and we have to take acloser look at the relation between the sum and the integral.The integral gives a contribution, under the assumption a − δ > Z R d l ( q , x ) dx ≪ Y j ∈ E Z ∞−∞ (1 + x ) δ − a / dx (55) · Y j ∈ Q + U / Z ∞−∞ (1 + x ) δ e − U ( x −| q | ) dx ≪ Y j ∈ E Y j ∈ Q + Z ∞−∞ (cid:16) + | q j | + U − x (cid:17) δ e − x dx ≪ Y j ∈ Q + (cid:16) + | q j | (cid:17) δ (cid:16) + U (cid:17) ≪ Y j ∈ Q + | q j | δ . The di ff erence between the value at µ ∈ Λ κ,χ and the integral over µ + V , where V is a compact neighborhood of 0, produces an error estimated by the gradient of l ( q , · ). ∂∂ x j l ( q , x ) ≪ a ,δ l ( q , x ) · | x j | + x j if j ∈ E , | x j | + x j + U (cid:12)(cid:12)(cid:12) x j − | q j | (cid:12)(cid:12)(cid:12)! if j ∈ Q + . The di ff erence between the sum and the integral is estimated by Z R d (cid:18)X m ∈ E O (1) + X m ∈ Q + (1 + U | x m − | q | m | ) (cid:19) l ( q , x ) dx . The terms with m ∈ E can be estimated as in (55). For a term with m ∈ Q + : ≪ Z ∞−∞ (1 + U | x m − | q | m | ) U / (1 + x m ) e − U ( x −| q m | ) dx · Y j ∈ E O (1) Y j ∈ Q + r { m } | q j | δ ≪ Y j ∈ Q + r { m } | q j | δ · Z ∞−∞ (cid:16) + U / | x | (cid:17) (cid:16) + | q m | + U − x (cid:17) δ e − x dx ≪ U / Y j ∈ Q + | q j | δ . Thus, we obtain:(56) Eis r ( ϕ ( q , · )) ≪ F , I , r ,δ N E ( ϕ E ) U / Y j ∈ Q + | q j | δ . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 21
Since δ >
Delta term.
Lemma 2.3.
Let e pl j as in (16) . For E, Q + , Q − and ϕ ( q , · ) as in Lemma 2.2: e Pl ξ ( ϕ ( q , · )) = Y j ∈ E e Pl ξ j ( ϕ j ) Y j ∈ Q e pl j ( q j )(57) + U − / N E ( ϕ E ) | Q + | Q j ∈ Q | q j | min j ∈ Q + | q j | if Q + , ∅ , if Q + = ∅ . Proof.
Since e Pl ξ j ( ϕ j ( q , · )) = | q | if 2 q ≡ ξ j − , , we have to consider the discrepancy between e Pl ξ j ( ϕ j ( q , · )) and 2 e pl j ( q j ) for j ∈ Q + .The function t e pl j ( it ) is even and smooth on R . If ξ j =
0, then e pl j (0) =
0, andif ξ j =
1, then e pl j (0) = π . We have pl j ( it ) ∼ | t | as | t | → ∞ , and ddt pl j ( it ) = O (1) for t ∈ R . e Pl ξ j (cid:16) ϕ j ( q j , · ) (cid:17) − e pl j ( q j )(58) = r U π Z ∞ (cid:16) e − ( x −| q j | ) + e − ( x + | q | ) (cid:17) e pl j ( ix ) dx − e pl j ( q j ) = r U π Z ∞−∞ e − U ( x −| q j | ) e pl j ( x ) dx − e pl j ( q ) = √ π Z ∞−∞ e − x (cid:18)e pl j (cid:18) q j + ix √ U (cid:19) − e pl j (cid:16) q j (cid:17)(cid:19) dx . We write e pl j (cid:18) q j + ix √ U (cid:19) − e pl j (cid:16) q j (cid:17) = U − / x dd ϑ e pl j ( i ϑ ) for | x | ≤ b , with b ≥
1, and ϑ between | q j | and | q j | + x √ U . e Pl ξ j (cid:16) ϕ j ( q j , · ) (cid:17) − e pl j ( q j ) ≪ Z b − b e − x O (1) U − / | x | dx + Z | x |≥ b e − x O (cid:16) | q j | + | x | U − / (cid:17) dx ≪ U − / + | q j | e − b b + U − / e − b . Here we have used that for b ∈ R and l ≥ Z ∞ b | x | l e − x dx ≪ l b ≤ , b l − e − b if b ≥ We assume that U ≥ e , and choose b = b ( q , U ) = q log | q j | + log U , whichsatisfies b ≥
1. This gives e Pl ξ j (cid:16) ϕ j ( q j , · ) (cid:17) − e pl j ( − iq j ) ≪ U − / . Furthermore, we have e pl j ( q j ) ≪ | q j | ( j ∈ Q + ) , (60) e Pl ξ j (cid:16) ϕ j ( q j , · ) (cid:17) ≪ U − / + | q j | ≪ | q j | ( j ∈ Q + ) , e pl ξ j (cid:16) ϕ j ( q j , · ) (cid:17) ≪ | q j | ( j ∈ Q − ) , e Pl ξ j ( ϕ j ) ≪ N j ( ϕ j ) ( j ∈ E ) . These local estimates imply that Y j ∈ Q + e Pl ξ j ( ϕ j ( q j , · )) − Y j ∈ Q + e pl j ( q j ) ≪ X m ∈ Q + U − / Y j ∈ Q + , j , m | q j |≪ U − / | Q + | Q j ∈ Q + | q j | min j ∈ Q + | q j | . Hence we have shown the estimate in (57). (cid:3)
We now fix A >
2. From here on we view the quantities t > ρ ∈ (1 − τ, τ and a in the sum formula. We apply Lemmas 2.2and 2.3 to obtain: Proposition 2.4.
For E, Q + , Q − , ϕ ( q , · ) as in Lemma 2.2, with U ≥ e : ˜N r ( ϕ ( q , · )) = | Q | + √| D F | (2 π ) d e Pl E ( ϕ E ) Y j ∈ Q e pl j ( q j )(61) + O F , I , r (cid:18) N E ( ϕ E ) e t U | Q + | Y j ∈ Q + | q j | ρ Y j ∈ Q − | q j | − A (cid:19) + O (cid:18) N E ( ϕ E ) U − / | Q + | Q j ∈ Q | q j | min j ∈ Q + | q j | (cid:19) . This is the basis for the results in the next sections.The main term in (61) can be estimated by O F (cid:16) N E ( ϕ E ) Q j ∈ Q | q j | (cid:17) . This implies(62) ˜N r ( ϕ ( q , · )) ≪ F , I , r , U N E ( ϕ E ) Y j ∈ Q | q j | .
3. U pper bound ] The next step is to derive by integration of (62) an upper bound for ˜N r ( f ) forfunctions of the form f = ϕ E ⊗ A : = ϕ E ⊗ χ A , where χ A is the characteristic function ENSITY RESULTS ON HILBERT MODULAR GROUPS II 23 of a set A . To integrate, we fix a non-negative measure d Q on ((0 , ∞ ) ∪ i [0 , ∞ )) Q of the form d Q q = N j ∈ Q d j q j ,(63) Z h ( q ) d j q = Z ∞ h ( it ) dt + Z ν h ( x ) dx + X β> , β ≡ ξ j − (1) h ( β ) . We shall use d R q = N j ∈ R d j q j for any set R of real places.We define for b ∈ R and for bounded measurable sets B ⊂ ( R ∪ i R ) R with R ⊂ Q :˜ V b ( B ) = Z B Y j ∈ R p ( q j ) b d R q , (64) p ( q j ) = q j ∈ (0 , ν ] ∪ i [0 , , | q j | otherwise . (65)The set R of real places is not visible in the notation ˜ V b , and should be clear fromthe set B . Note that with b = ffi ces to estimate ˜N r ( ϕ ⊗ A ) for bounded sets A of the form A = A + × A × A − , A + ⊂ ( i [1 , ∞ )) R + , (66) A = ((0 , ν ) ∪ i [0 , R , A − ⊂ Y j ∈ R − (cid:16) ξ j + + N (cid:17) , for any partition Q = R + ⊔ R ⊔ R − . This choice reflects that q in (62) has no factorsin (0 , ν ) ∪ i [0 , r ( ϕ E ⊗ A ) by N E ( ϕ E ) ˜ V ( A ). Given perfect knowledgeof the spectral set { ν ̟ } one can choose A + as the union of tiny boxes around many ν ̟, A + = ( ν ̟, j ) j ∈ A + in such a way that N r ( ϕ E ⊗ A ) is large while ˜ V ( A ) stays arbi-trarily small. This shows that we need a further assumption on the factor A + .Let dist be the distance along i [0 , ∞ ) ∪ (0 , ν ) given by(67) dist ( ν, q ) = | q − ν | if q , ν ∈ i [0 , ∞ ) or q , ν ∈ (0 , ν ) , | q | + | ν | otherwise . For ε > ν ∈ ( i [0 , ∞ ) ∪ (0 , ν )) B , where B is a set of real places:(68) A ( ν, ε ) = n q ∈ ( i [0 , ∞ ) ∪ (0 , ν )) B : dist ( q j , ν j ) ≤ ε for any j ∈ B o . Again, the set B should be clear from the context. Definition 3.1.
Let w, ε >
0, and let B be a set of real places. A ( w, ε ) -blunt subset H ⊂ ( i [0 , ∞ ) ∪ (0 , ν ]) B is a d B q -measurable set such that(69) Z A ( ν R ,β ) ∩ H d B q ≥ w vol B A ( ν, β ) for all ν ∈ H and all β ∈ (0 , ε ] . By vol B we mean the volume for d B .Note that ( w, ε )-bluntness implies ( w, ε )-bluntness for any ε ∈ (0 , ε ). Boxeswith size at least ε in all coordinate directions are (1 , ε )-blunt. Proposition 3.2.
Let A = A + × A × A − be as in (66) . Suppose that A + is ( w, ε ) -bluntfor some w > and ε ∈ (0 , e − ] . Then for any ϕ E : (70) N r ( ϕ E ⊗ A ) ≪ F , I , r w − N E ( ϕ E ) ˜ V ( A ) . Proof.
We apply Proposition 2.4 with E replaced by ˆ E = E ∪ R , Q ± replaced by R ± , and the test function ˆ ϕ ( q , · ) chosen as follows j ∈ E R + ∪ R − R ˆ ϕ j = ϕ p ϕ j ( q , · ) with U ≥ e e ν if | Re ν | ≤ τ, ϕ p ( ν ) = ( p − ν ) − a / if | Re ν | ≤ τ, ( p + ν ) − a / otherwise , (71)with some fixed p > τ and q ∈ ( i [1 , ∞ )) R + × Q j ∈ R − (cid:16) ξ j + + N (cid:17) . We put(72) ϕ p , E ( ν ) = O j ∈ E ϕ p ( ν j ) . From (62) we obtain ˜N r ( ˆ ϕ ( q , · )) ≪ F , I , r N E ( ϕ p , E ) Y j ∈ R + ∪ R − | q j | , where we have used that (45) implies N j ( ϕ j ) = O (1) for j ∈ R . Integration over q gives(73) Z A + × A − ˜N r ( ˆ ϕ ( q , · )) d R + ∪ R − q ≪ F , I , r ˜ V ( A + × A − ) . We have omitted N E ( ϕ p , E ) since it is O (1) for the fixed choice of p .Now we note that for a given ν ∈ Y ξ we have ˆ ϕ ( q , ν ) ≥
0. With the obviousmeaning ( ν j ) j ∈ B of ν B for sets B of real places, we have Z A + × A − ˆ ϕ ( q , ν ) d R + ∪ R − = Z A + ×{ ν R − } ˆ ϕ ( q , ν ̟ ) d R + ∪ R − q ≥ Z A ( ν R + ,ε ) ∩ A + ϕ p , E ( ν E ) Y j ∈ R e ν j Y j ∈ R − Y j ∈ R + ϕ j ( q j , ν j ) d R + q ≥ ϕ p , E ( ν E ) e −| R | q U π e − U ε ! | R + | Z A ( ν R + ,ε ) ∩ A + d R + q ≥ wϕ p , E ( ν E ) e −| R | (cid:18) ε U / e − U ε π / (cid:19) | R + | . With the choice U = ε − :(74) Z A + × A − ˆ ϕ ( q , ν ) d R + ∪ R − ≥ w e −| R | π −| R + | / ϕ p , E ( ν E ) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 25
Since ˆ ϕ ( q , · ) ≥ Y ξ , we can reverse the order of summation and integrationin Z A + × A − ˜N r ( ˆ ϕ ( q , · )) d R + ∪ R − q = X ̟ (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) Z A + × A − ˆ ϕ ( q , ν ̟ ) d R + ∪ R − q . Hence ˜N r ( ϕ E ⊗ A ) = X ̟, ν ̟, Q ∈ A (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) ϕ p , E ( ν ̟, E ) ≤ e | R | w π | R + | / X ̟, ν ̟ ∈ A (cid:12)(cid:12)(cid:12) c r ( ̟ ) (cid:12)(cid:12)(cid:12) Z A + × A − ˆ ϕ ( q , ν ) d R + ∪ R − q ≪ d w Z A + × A − ˜N r ( ˆ ϕ ( q , · )) d R + ∪ R − q With (73):(75) ˜N r ( ϕ E ⊗ A ) ≪ F , I , r w ˜ V ( A ) , where we have used that ˜ V ( A ) = O (1). (cid:3)
4. A symptotic formula , first stage Now we start a more precise approximation of N r ( ϕ E × C ) where ϕ E is still anarbitrary test function, and where C is a product C + × C − , with bounded closed sets C + ⊂ ( i [1 , ∞ )) Q + and C − ⊂ [ , ∞ ) Q − . Only the intersection C − ∩ Q j ∈ Q − (cid:16) ξ j + + N (cid:17) matters for the present purpose, not C − itself. For C + we define, with c > C + (0) = C + , (76) C + ( c ) = n ν ∈ ( i [0 , ∞ ) ∪ (0 , ν ]) Q − : A ( ν, c ) ∩ C + , ∅ o , C + ( − c ) = (cid:8) ν ∈ C + : A ( ν, c ) ⊂ C + (cid:9) , C + [ c ] = C + ( c ) r C + ( − c ) . Proposition 4.1.
Let r ∈ O ′ r { } . Let a > and τ ∈ [ , ] as in Theorem 2.1, anddecompose the set of real places of F as E ⊔ Q + ⊔ Q − with Q = Q + ∪ Q − , ∅ . Thereare t > , D > , ρ ∈ (1 − τ, and A > such that for any U > De and any ε ∈ [( DU ) / , e − ] , for all products ϕ E = N j ∈ E ϕ j of local test functions and for allbounded d Q -measurable sets C = C + × C − , C + ⊂ ( i [1 , ∞ )) Q + and C − ⊂ [ , ∞ ) Q − : (77) ˜N r ( ϕ E × C ) = √| D F | (2 π ) d e Pl( ϕ E ⊗ C ) + N E ( ϕ E ) O F , I , r ( E ( C , U , ε )) , where if Q + = ∅ (78) E ( C , U , ε ) = ˜ V − A ( C − ) , and if Q + , ∅ E ( C , U , ε ) = e t U | Q + | ˜ V ρ ( C + ) ˜ V − A ( C − ) + e − U ε ˜ V ( C )(79) + ˜ V ( C + [2 ε ] × C − ) + U − / ˜ V ( C ) . At this point we can derive the statement in example (iv) in the introduction. Wedenote by S b ( Γ , χ ) the space of holomorphic cusp forms on the product H d of d copies of the upper half plane for the group Γ with character χ and weight b ∈ N d satisfying b j ≥ b j ≡ ξ j mod 2 for all j . Corollary 4.2.
The space S b ( Γ ( I ) , χ ) is non-zero for all but finitely many weightsin the set (cid:8) b ∈ N d : b j ≥ , b j ≡ ξ j mod 2 for all j (cid:9) .Proof. We apply Proposition 4.1 with E = Q + = ∅ , and C = C − equal to thesingleton C b = (cid:8)(cid:0) b j − (cid:1) j (cid:9) . Then e Pl( C b ) = Q dj = b j − and ˜ V − A ( C b ) = Q dj = (cid:0) b j − (cid:1) − A .We obtain (6) with the constant C equal to 2 √| D F | (2 π ) − d .If we take r totally positive, the ̟ entering ˜N r ( C b ) form an orthogonal system ofcuspidal representations for which each factor ̟ j is a discrete series representationwith lowest weight b j . (See (2.29) in [3].) Thus these ̟ correspond to an orthogo-nal basis of S b ( Γ ( I ) , χ ). So ˜N r ( C b ) can be non-zero only if S b ( Γ ( I ) , χ ) , { } . (cid:3) Proof of Proposition 4.1.
The proof of the proposition is rather long and willrequire some intermediate steps that we shall give in a series of lemmas.4.1.1.
Integration.
We integrate (61) over C . Taking into account (15), we obtain: Z c ˜N r ( ϕ ( q , · )) d Q q − √| D F | (2 π ) d e Pl E ( ϕ E ) e Pl Q ( C )(80) ≪ F , I , r N E ( ϕ E ) e t U | Q + | ˜ V ρ ( C + ) ˜ V − A ( C − ) + N E ( ϕ E ) U − / V ( C ) . In this term we have left out the denominator min j ∈ Q + | q j | , since we have already asmall factor U − / .To prove Proposition 4.1 we will estimate the di ff erence(81) Z c ˜N r ( ϕ ( q , · )) d Q q − ˜N r ( ϕ e ⊗ C ) . Local comparison.
Let X j = X + j = ξ j + + N if j ∈ Q − , and X j = (0 , ν ] ∪ i [0 , ∞ ), X + j = i [1 , ∞ ) if j ∈ Q + . We consider for ν ∈ X j : I j α ( ν ) = Z q ∈ X j , dist ( q ,ν ) ≤ α ϕ j ( q , ν ) d j q , (82) J j α ( ν ) = Z q ∈ X j , dist ( q ,ν ) ≥ α ϕ j ( q , ν ) d j q . Lemma 4.3. If α ≥ U − / , then for j ∈ Q + , ν ∈ X j :if ν ∈ i [1 + α, ∞ ) : I j α ( ν ) = + O (cid:16) e − U α (cid:17) , J j α ( ν ) = O (cid:16) e − U α (cid:17) , if ν ∈ i [0 , + α ) : I j α ( ν ) = O (1) , J j α ( ν ) = O (cid:16) e − U α (cid:17) . For j ∈ Q + , ν ∈ X j , U − / ≤ ε ≤ e − , and ν ∈ i [0 , − ε ) ∪ (0 , ν ] :I j ε ( ν ) = , J j ε ( ν ) = O (cid:16) e − U ε (cid:17) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 27
For j ∈ Q − , ν ∈ X j : I j α ( ν ) = , J j α ( ν ) = . Proof.
The results for j ∈ Q − are immediate. We consider the case j ∈ Q + . Thebest situation is ν ∈ i [1 + α, ∞ ). Then, with (59): I j α ( ν ) = r U π X ± Z | ν | + α | ν |− α e − U ( x ∓| ν | ) dx (83) = √ π (cid:18)Z α √ U − α √ U e − x + Z (2 | ν | + α ) √ U (2 | ν |− α ) √ U e − x dx (cid:19) = + O (cid:18) e − U α α √ U (cid:19) + O (cid:18) e − U (2 | ν |− α ) (2 | ν | − α ) √ U (cid:19) = + O (cid:18) e − U α α √ U (cid:19) , J j α ( ν ) = (cid:18)Z | ν |− α + Z ∞| ν | + α (cid:19) q U π X ± e − U ( x ∓| ν | ) dx ≪ e − U α α √ U + e − U (2 | ν |− α ) (2 | ν | − α ) √ U ≪ e − U α α √ U . We ignore the denominator α √ U ≥ ν ∈ i [0 , + α ], part of the integral for I j α ( ν ) is omitted. This leads to theestimate O (1). The quantity J j α ( ν ) is at most as large as in the previous case.We consider finally a small value α = ε ∈ [ U − / , e − ]. For ν ∈ i [0 , − ε ] ∪ (0 , ν ],the integral I j ε ( ν ) vanishes. If ν ∈ i [0 , − ε ) we have already obtained J j ε ( ν ) = O (cid:16) e − U ε (cid:17) . For ν ∈ (0 , ν ] ⊂ (0 , ]: J j ε ( ν ) = r U π Z ∞ e U ( ν − t ) cos 2 t ν dt ≪ e U ν Z ∞√ U e − x dx ≪ e − U (1 − ν ) √ U ≤ e − U (1 − ν ) ≤ e − U ε . (cid:3) Global comparison.
Lemma 4.4.
Let ν ∈ Y ξ , ν j ∈ X j for j ∈ Q. Let α ≥ U − / and ε ∈ [ U − / , e − ] .Then Z q ∈ A ( ν Q + ,α ) ×{ ν Q − } ϕ ( q , ν ) d Q q ≪ ϕ E ( ν E ) ;(84) Z q ∈ ( C + r A ( ν Q + ,α )) ×{ ν Q − } ϕ ( q , ν ) d Q q ≪ ϕ E ( ν E ) e − U α . (85) If ν j ∈ i [0 , − ε ) ∪ (0 , ν ] for some j ∈ Q + then (86) Z q ∈ A ( ν Q + ,ε ) ×{ ν Q − } ϕ ( q , ν ) d Q q = . If A ( ν Q + , α ) ⊂ C + , then (87) Z q ∈ A ( ν Q + ,α ) ×{ ν Q − } ϕ ( q , ν ) d Q q = ϕ E ( ν E ) (cid:16) + O ( e − U α ) (cid:17) . Proof.
We have Z q ∈ A ( ν Q + ,α ) ×{ ν Q − ϕ ( q , ν ) d Q q = ϕ E ( ν E ) Y j ∈ Q + I j α ( ν j ) . This implies directly (84). If A ν Q + , α ) ⊂ C + , then ν j ∈ i [1 + α, ∞ ) for any j ∈ Q + .Hence (87) follows. Equality (86) follows also from Lemma 4.3.For (85) we use Z q ∈ ( C + r A ( ν Q + ,α )) ×{ ν Q − } ϕ ( q , ν ) d Q q ≪ ϕ E ( ν E ) X m ∈ Q + J m α ( ν m ) Y j ∈ Q + r { m } (cid:16) I j α ( ν j ) + J j α ( ν j ) (cid:17) ≪ ϕ E ( ν E ) X m ∈ Q − O ( e − U α ) O (1) | Q + |− . (cid:3) Error term.
We will use these comparison results to estimate the followingdi ff erence: ˜N r ( ϕ E ⊗ C ) − Z C ˜N r (cid:18) ϕ ( q , · ) (cid:19) d Q q (88) = X ̟, ν ̟ Q ∈ C | c r ( ̟ ) | ϕ E (cid:0) ν ̟, E (cid:1) − Z C ϕ ( q , ν ̟ ) d Q q ! − X ̟, ν ̟, Q < C | c r ( ̟ ) | Z C ϕ ( q , ν ̟ ) d Q q , with X j as in § ff erence in (88) as T i + T b + T o , given by the respective conditions ν ̟, Q + ∈ C + ( − ε ), ν ̟, Q + ∈ C + [ ε ], and ν ̟, Q + < C + ( ε ).4.1.5. Inner error term. C + ( − ε ) is contained in the subset X i = [ ν ∈ C + ( − ε ) A ( ν, ε )of C + , which is (1 , ε )-blunt. With (87) and Proposition 3.2: T i ≪ X ̟, ν ̟, Q + ∈ C + ( − ε ) (cid:12)(cid:12)(cid:12) | c r ( ̟ ) | (cid:12)(cid:12)(cid:12) e − U ε | ϕ E ( ν ̟, E ) | (89) ≪ F , I , r N E ( ϕ E ) ˜ V ( X i × C − ) e − U ε ≤ N E ( ϕ E ) e − U ε ˜ V ( C ) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 29
Boundary error term.
With (84) T b ≪ X ̟, ν ̟, Q + ∈ C + [ ε ] | c r ( ̟ ) | | ϕ E ( ν ̟, E ) | . We put C + [ ε ] in the (1 , ε )-blunt set S ν ∈ C + [ ε ] A ( ν, ε ) contained in C + [2 ε ]. This leadsto(90) T b ≪ F , I , r N E ( ϕ E ) ˜ V ( C + [2 ε ] × C − ) . Outer error term.
Now we use (85), and use that the (1 , ε )-blunt set [ ν ∈ C + ( ε ( n + r C + ( ε n ) A ( ν, ε )is contained in C + ( ε ( n + r C + ( ε ( n − T o ≪ F , I , r N E ( ϕ E )(91) · ∞ X n = e − U ε n (cid:16) ˜ V ( C + ( ε ( n + − ˜ V ( C + ( ε ( n − (cid:17) . Growth on shells.
Estimate (91) has the disadvantage that the bound is givenby an infinite sum. Let us consider D n = C + ( ε ( n + − C + ( ε n ). The size of thesum ∞ X n = e − U ε n ˜ V ( D n − ∪ D n ∪ D n + )depends mainly on the size of ˜ V ( D n ) for small values of n . Lemma 4.5.
There is a constant R = R ( | Q + | ) > , not depending on ε ∈ (0 , e − ] ,such that ˜ V ( D n ) ≤ R n C + [ ε ] for any n ≥ .Proof. The sets D n are subsets of ( i [0 , ∞ ) ∪ (0 , ν ]) Q + , which we identify with[ − ν , ∞ ) A + : q j ∈ i [0 , ∞ ) is replaced by q j ∈ [0 , ∞ ), and q j ∈ (0 , ν ] by − q j .Now in each factor, the distance dist in (67) is for each coordinate given by theabsolute value of the di ff erence. The measure d Q + corresponds to the Lebesguemeasure on R Q + .If q ∈ D n , then there is ν ∈ C + such that dist ( q j , ν j ) ≤ ε ( n +
1) for all j anddist ( q l , ν l ) > ε n for some l . For the latter l we define ˜ q l ∈ [ − ν , ∞ ) such thatdist ( ˜ q l , ν l ) ≤ ε n and dist ( ˜ q l , q l ) = ε . For the other coordinates we put ˜ q j = q j .This implies that each point of D n + can be moved into D n by a translation T v in R Q + over a vector v with coordinates in { , ε, − ε } . Hence(92) D n + ⊂ [ v T v D n . There are 3 | Q + | translates. For each of these translates˜ V ( T v D n ) = Z D n p ( x + v ) d Q x . Now we have p ( x + v ) ≤ | Q + | X m = | Q + | m ! ε m p ( x ) . Hence we have ˜ V ( T v D n ) ≤ R ˜ V ( D n ), with R = P | Q + | m = (cid:16) | Q + | m (cid:17) e − m , for any ε ∈ (0 , e − ]. This implies ˜ V ( D n + ) ≤ R ˜ V ( D n ) with(93) R : = R ( | Q + | ) = (cid:16) + e − ) (cid:17) | Q + | . Hence ˜ V ( D n ) ≤ R n ˜ V ( D ) ≤ R n ˜ V ( C + [ ε ]). (cid:3) The factor 3 | Q + | is much too large in most cases, since the translates T v D n overlapa lot, and cover more than D n + .To use this lemma in an estimate of the sumin (91) we assume that U ε ≥ D with D : = log R . Then n e − U ε n R n is adecreasing function, and ∞ X n = e − U ε n ( R n + + R n + R n − ) ≤ ( R + R + e − U ε + ( R + + R − ) Z ∞ x = e − U ε x R x dx ≪ | Q + | e − U ε + e − U ε + log C q − log C U ε ≪ e − U ε . Therefore, under the assumption U ε ≥ D , where D = log( R ), the outer error term(91) can be estimated by O | Q + | (cid:16) e − U ε ˜ V ( C + [ ε ] × C − ) (cid:17) , and hence be absorbed into the term O (cid:16) ˜ V ( C + [2 ε ] × C − ) (cid:17) . This concludes estima-tion of the error term, hence the proof of Proposition 4.1 is now complete.4.2. Choice of the parameters U and ε . We now arrive at the delicate pointwhere the parameters U , ε will be linked to the volume quantities, depending onthe set C . Let us rewrite the error term E in (79): E = E ( C , U , ε ) = (cid:16) e t U | Q + | m ρ ( C ) + e − U ε + U − / + β ε ( C ) (cid:17) ˜ V ( C ) , m ρ ( C ) = ˜ V ρ ( C + ) ˜ V − A ( C − )˜ V ( C ) , β ε ( C + ) = ˜ V ( C + [2 ε ])˜ V ( C + ) . (94)We will require that m ρ ( C ) and β ε ( C ) get small, to be able to control the error termin Proposition 4.1. Furthermore we will need to choose U , ε suitably. It turns outthat a convenient election will be to let U (resp. ε ) tend slowly to ∞ (resp. ) insuch a way that U ε still tends to ∞ . Keeping e t U | Q + | m ρ ( C ) and U − / in mind, wechoose(95) U = U ( C ) = t | Q + | (cid:18) | log m ρ ( C ) | −
12 log | log m ρ ( C ) | (cid:19) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 31
The condition U ≥ e D with D = log R ( | Q + | ), as in Proposition 4.1, is satisfied if m ρ ( C ) is su ffi ciently small. With this choice e t U | Q + | m ρ ( C ) + U − / ≪ | log m ρ ( C ) | − / . The contribution e − U ε should also be small. We take ε only slightly larger than U − / : ε = ε ( C ) = r log | log m ρ ( C ) | U (96) = vuut t | Q + | log | log m ρ ( C ) | | log m ρ ( C ) | (cid:18) −
12 log | log m ρ ( C ) || log m ρ ( C ) | (cid:19) . The quantity ε tends to zero as m ρ ( C ) tends to zero, and ε U = log | log m ρ ( C ) | tendsto ∞ as m ρ ( C ) tends to 0. Thus, ε satisfies the conditions in Proposition 4.1 forsu ffi ciently small values of m ρ ( C ).Let us consider the term U − / ˜ V ( C ) in the error term. With the choice of U and ε just indicated, this term is slightly larger than˜ V ( C ) (cid:0) log m ρ ( C ) (cid:1) = ˜ V ( C ) (cid:0) log ˜ V ( C ) − log( ˜ V ρ ( C + ) ˜ V − A ( C − ) (cid:1) . So the size of the error term will in general di ff er from the size of the main term by alogarithmic factor. Therefor we switch now from giving O -estimates to asymptoticestimates with an o -term.In this way we obtain as the endpoint of the first stage of the derivation of theasymptotic formula: Theorem 4.6.
Let r ∈ O ′ r { } . Divide up the set of real places of F as E ⊔ Q + ⊔ Q − with Q = Q + ∪ Q − , ∅ . Let C be the collection of bounded d Q -measurable setsC = C + × C − such that C + ⊂ ( i [1 , ∞ )) Q + , C − ⊂ h , ∞ (cid:17) Q − . Let ρ ∈ (0 , be as inProposition 4.1, and let ε ( C ) be as chosen in (96) .For each product ϕ E = N j ∈ E ϕ j of local test functions, and for each familyt C t in C such that as t → ∞ (97) ˜ V ρ ( C + t ) ˜ V − A ( C − t ) = o (cid:16) ˜ V ( C t ) (cid:17) , if Q + , ∅ , then ˜ V ( C + t [2 ε ( C + t )]) ˜ V ( C − t ) = o (cid:16) ˜ V ( C t ) (cid:17) , the following asymptotic result holds as t → ∞ : (98) ˜N r ( ϕ E ⊗ C t ) = √| D F | (2 π ) d Pl( ϕ E ⊗ C t ) (cid:0) + o F , I , r (1) (cid:1) . For families t C t as in the theorem, the quantities ˜ V ( C t ) and e Pl( C t ) have thesame size. So we have replace o (cid:0) ˜ V ( C t ) (cid:1) by e Pl( C t ) in (19). When formulating the asymptotic formula in λ -space, the quantity ˜ V b ( C t ) corre-sponds to V b ( C t ), given by the measure V b = L j V b , j , V b , j ( f ) = Z ∞ / f ( λ )( λ − / ( b − / d λ + Z / λ f ( λ ) d λ q(cid:12)(cid:12)(cid:12) λ − (cid:12)(cid:12)(cid:12) (99) + X β ∈ ξ j + + N | β | b f (cid:18) β − β (cid:19) . Unions.
It is also useful to state an asymptotic formula for families of disjointunions t c t = F n C ( n ) t where C ( n ) t = C ( n ) + t × C ( n ) − t , with n in a countable indexset. Then we have to replace (94) by m ρ ( C t ) = V ( C t ) X n ˜ V ρ ( C ( n ) + t ) ˜ V − A ( C ( n ) − t ) , (100) β ε ( C t ) = V ( C t ) X n ˜ V ( C ( n ) + t [2 ε ]) ˜ V ( C ( n ) − t ) . Proceeding with these choices, we obtain:
Proposition 4.7.
Let r ∈ O ′ r { } . Let E ⊔ Q be a partition of the set of real placesof F, with Q , ∅ . Let t C t be a family of bounded d Q -measurable sets such thatfor each t (101) C t = G n C ( n ) t , with each C ( n ) t in the collection C in Theorem 4.6. The decomposition Q = Q + n ⊔ Q − n may depend on n. Under the conditions (102) X n ˜ V ρ ( C ( n ) + t ) ˜ V − A ( C ( n ) − t ) = o (Pl( C t )) , if Q + , ∅ , then X n ˜ V ( C ( n ) + t [2 ε ( C t )]) ˜ V ( C ( n ) − t ) = o (Pl( C t )) , the asymptotic formula (98) holds for each choice of ϕ E as product of local testfunctions.
5. A symptotic formula , second stage We have still the freedom to choose the test function ϕ E . In the second stagewe use this freedom to fill in the region i [0 , ∪ (0 , ν ] for the coordinates of ν ̟ in E . More generally, by specializing ϕ E we can make the asymptotic formula looksharply at the coordinate of ν ̟ in E .We shall choose the test functions ϕ j with j ∈ E as an approximation of the char-acteristic functions of “intervals” in i [0 , ∞ ) ∪ (0 , ∞ ). Going over to a descriptionin terms of the eigenvalue vectors λ ̟ , we obtain Theorem 5.3, which givesN r ( B × ˆ C + t × ˆ C − t ) = √| D F | (2 π ) d Pl( B × ˆ C + t × ˆ C − t ) (1 + o (1)) , ENSITY RESULTS ON HILBERT MODULAR GROUPS II 33 where ˆ C + t and ˆ C − t are the sets corresponding to C + t and C − t under the transformation ν λ . The set B is a fixed box in R E . This result is strong, but not adequate forsome obvious families. Proposition 4.7 gives a generalization allowing us to applythe asymptotic formula to families that are countable disjoint unions of families ofthe form t B × ˆ C + t × ˆ C − t .5.1. Compactly supported functions at the places in E . For a family t C t satisfying the conditions in Theorem 4.6 or in Proposition 4.7 we rewrite (98) asfollows:(103) lim t →∞ √| D F | (2 π ) d ˜N r ( ϕ E ⊗ C t )Pl( C t ) = e Pl E ( ϕ E ) . Proposition 5.1.
Let r ∈ O ′ r { } and the decomposition E ⊔ Q + ⊔ Q − be as before.For f E = N j ∈ E f j with f j : R → R , define ˜ f E = N j ∈ E ˜ f j by ˜ f j ( ν ) = f j ( − ν ) . Iff j ∈ C c ( R ) for any j ∈ E, then (104) lim t →∞ √| D F | (2 π ) d ˜N r ( ˜ f E ⊗ C t )Pl( C t ) = Pl E ( f E ) . The proof is given in the remainder of this subsection. We can follow the ap-proach in [2] closely.5.1.1.
Functionals.
First we formulate two lemmas to be used in the proof.For f : R E → C put(105) A rt ( f ) = √| D F | (2 π ) d ˜N r ( ˜ f ⊗ C t ) e Pl( C t ) , where ˜ f is defined by ˜ f ( ν ) = f (cid:18)(cid:0) − ν j (cid:1) j ∈ Q (cid:19) . This defines a measure on R E . Wewant to compare it to the measure f e Pl( ˜ f ) = Pl( f ). Lemma 5.2.
Let r ∈ O ′ r { } , let T f T be a family of real-valued functions on R E , and let f and h also be real-valued on R E , such that i) f , h and every f T is integrable for all A rt and for Pl . ii) lim t →∞ A rt ( f T ) = Pl( f T ) for all T . iii) lim t →∞ A rt ( h ) = Pl( h ) . iv) There is a function T a ( T ) such that a ( T ) = o (1) as T → ∞ , and | f T ( x ) − f ( x ) | ≤ a ( T ) h ( x ) for all x ∈ R E . Then lim t →∞ A rt ( f ) = Pl( f ) .Proof. The measures A rt is non-negative, hence(106) A rt ( f T ) − a ( T ) A rt ( h ) ≤ A rt ( f ) ≤ A rt ( f T ) + a ( T ) A rt ( h ) . Taking the limit as t → ∞ of both terms on the side shows that for all T :0 ≤ lim sup t →∞ A rt ( f ) − lim inf t →∞ A rt ( f ) ≤ a ( T ) A rt ( h ) . Since a ( T ) = o (1), the limit lim t →∞ A rt ( f ) exists. With the non-negativity of Pl we derive from iv) that for all T Pl( f T ) − a ( T )Pl( h ) ≤ Pl( f ) ≤ Pl( f T ) + a ( T )Pl( h ) . Hence for all T : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Pl( f ) − lim t →∞ A rt ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ a ( T )Pl( h ) . This gives the statement of the lemma. (cid:3)
Approximation.
Now we start the proof of Proposition 5.1. For given f j ∈ C c ( R ) we take(107) ϕ j ( ν ) = r T π Z ∞−∞ e − T ( λ − + ν ) f j ( λ ) d λ, with T a large positive parameter. Again we use a Gaussian kernel functions, butnow in the λ -space. This defines ϕ j ( ν ) as an even holomorphic function of ν ∈ C ,with exponential decay on the strip | Re ν | ≤ τ , and for ν ∈ R . So ϕ j is a testfunction in the sense of § ν ∈ R ∪ i R , it is given by(108) ϕ j ( ν ) = r T π Z ∞−∞ e − T λ f j (cid:16) λ + − ν (cid:17) d λ. We view ϕ j as an approximation of ˜ f j . Similarly, ϕ E is an approximation of ˜ f E .5.1.3. Local estimates.
Since f j is real-valued, we have | ϕ j ( ν ) | ≤ k f j k ∞ for any ν ∈ R ∪ i R . Take N large, such that Supp f j ⊂ [ − N , N ] for any j ∈ E . If ν ∈ R ∩ i R with | ν | ≥ √ N +
1, and − N ≤ λ ≤ N , then λ − + ν ≤ N − − | ν | ≤ − | ν | − if ν ∈ i R , ≥ − N − + | ν | ≥ | ν | + if ν ∈ R . Hence e − T ( λ − + ν ) ≤ e − T ν for such values of ν and λ . Together, these facts givefor ν ∈ R ∪ i R :(109) | ϕ j ( ν ) | ≤ k f j k ∞ if | ν | < √ N + , N q T π e − T ν k f j k ∞ if | ν | ≥ √ N + . We recall the positive test function ϕ p , E = N j ∈ E ϕ p in (72), which gives for p > τ : ϕ p ( ν j ) = ( p − ν j ) − a / if | Re ν | ≤ τ, ( p + ν j ) − a / otherwise . We take T ≥ T =
4. For ν ∈ R ∪ i R , | ν | ≥ √ N +
1, we have
T e − T | ν | ≤ T e −| ν | ≪ | ν | − a , and hence(110) | ϕ j ( ν ) | ≪ T − / N k f j k ∞ | ν | − a ≪ N T − / ϕ p ( ν ) . For ν ∈ R ∪ i R , | ν | ≤ √ N + ϕ j ( ν ) − ˜ f j ( ν ) = √ π Z ∞−∞ e − y (cid:18) f j (cid:18) y √ T + − ν (cid:19) − f j (cid:16) − ν (cid:17)(cid:19) d y (111) ENSITY RESULTS ON HILBERT MODULAR GROUPS II 35 ≪ k f ′ j k ∞ Z T / e − y y √ T d y + k f j k ∞ Z ∞ T / e − y d y ≪ T − / k f ′ j k ∞ + k f j k ∞ e − T T / ≪ T − / (cid:16) k f j k ∞ + k f ′ j k ∞ (cid:17) . Under the assumption | ν | ≤ √ N + ν ∈ R ∪ i R , we have ϕ p ( ν ) ≫ N − a / .Hence(112) ϕ j ( ν ) − ˜ f j ( ν ) ≪ f j T − / N a / ϕ p ( ν ) . Global approximation.
We apply the Lemma 5.2 with f T ( λ ) = ϕ E ( ν ) , h ( λ ) = ϕ p , E ( ν ) , f ( λ ) = f E ( λ ) = O j ∈ E ˜ f j ( ν j ) , with λ j = − ν j , ν j = ± q − λ f . Condition i) is satisfied by continuity. We haveii) and iii) from the assumption that t C t is a family for which Theorem 98 holds.To check Condition iv) we note that if ν ∈ ( R ∪ i R ) E , such that | ν j | ≤ √ N + j , then:(113) (cid:12)(cid:12)(cid:12) ϕ E ( ν ) − ˜ f E ( ν ) (cid:12)(cid:12)(cid:12) ≪ f T − / ϕ p , E ( ν ) . If there is at least one j ∈ E with | ν j | ≥ √ N +
1, then by (112)(114) (cid:12)(cid:12)(cid:12) ϕ E ( ν ) − ˜ f E ( ν ) (cid:12)(cid:12)(cid:12) = | ϕ E ( ν ) | ≪ f T − / ϕ p , E ( ν ) . Application of the lemmas in § Remark.
We refrain from extending the asymptotic formula to compactlysupported functions on R E that have no product structure.5.2. Boxes.
Proposition 5.1 works with compactly supported continuous functionswith product structure. The last step in stage two is the extension to boxes in R E .We now formulate the asymptotic formula in terms of the coordinate λ , and use thenotation(115) ˆ C = n ( − ν j ) j ∈ Q : ν ∈ C o , for C ⊂ ( i R ∪ R ) Q . Theorem 5.3.
Let r ∈ O ′ r { } . Let E ⊔ Q + ⊔ Q − a decomposition of the real placesof F with Q = Q + ∪ Q − , ∅ . Let t C t be a family of bounded d Q -measurable setsin the collection C in Theorem 4.6 or as considered in Proposition 4.7. In particularwe suppose that the conditions in (97) or in (102) hold. Let B E = Q j ∈ E [ A j , B j ] besuch that (116) A j , B j < n b (1 − b ) : b > , b ≡ ξ j mod 2 o . Then, as t → ∞ : (117) N r ( B E × ˆ C t ) = √| D F | (2 π ) d Pl( B E × ˆ C t ) + o (cid:0) Pl Q ( ˆ C t ) (cid:1) . Proof.
Let χ E be the characteristic function of B E . It has the form χ E = N j ∈ E χ j .The local characteristic functions χ j are integrable for Pl ξ j and for all A rt as definedin (105).The conditions in the theorem on the endpoints A j and B j make it possible tofind for a given T ≥ u T , j , U T , j ∈ C c ( R ) such that 0 ≤ u T , j ≤ χ j ≤ U T , j on R , and such that(118) Pl ξ j ( U T , j − u T , j )) ≤ T . Put u T = N j ∈ E u T , j , U T = N j ∈ E U T , j , and h = N J ∈ E h j . The asymptoticformula holds for h and for all u T (Proposition 5.1). We have0 ≤ U T ( λ ) − u T ( λ ) ≤ X m ∈ E ( U T , m ( λ m ) − u T ( λ m )) Y j ∈ E r { m } h j ( λ j ) , ≤ Pl E ( U T − u T ) ≤ X m ∈ E T − Y j ∈ E r { m } Pl ξ j ( h j ) = O ( T − ) . (119)Since the A rt are non-negative measures, we have: A rt ( u T ) ≤ A rt ( B E ) ≤ A rt ( U T ) , Pl E ( u T ) ≤ lim inf t →∞ A rt ( B E ) ≤ lim sup t →∞ A rt ( B E ) ≤ Pl E ( U T ) , lim sup t →∞ A rt ( B E ) − lim inf t →∞ A rt ( B E ) = O ( T − ) . So lim t →∞ A rt ( B E ) exists, and satisfiesPl E ( u T ) ≤ lim t →∞ A rt ( B E ) ≤ Pl E ( U T ) . Again applying (119) we conclude that this limit is equal to Pl E ( B E ). HenceN r ( B E × ˆ C t ) = √| D F | (2 π ) d Pl( B E × ˆ C t ) + o (cid:0) Pl Q ( ˆ C t ) (cid:1) , which is (117). (cid:3)
6. S pecial families
Theorem 5.3 describes a large class of families of sets for which the asymp-totic formula (117) holds. It has the limitation that the factor C t is a subset of( i [1 , ∞ ) ∪ [1 / , ∞ )) Q , while the region i [0 , ∪ (0 , ν ] is treated only in the coordi-nates in E . To avoid technical complications we have chosen not to try to derive anasymptotic formula for a larger class of families of sets, but to apply Theorem 5.3in a number of special cases. This will su ffi ce to give many applications.6.1. Boxes.
Directly from Theorem 5.3 we get families of boxes of the type ˜ B E × C t with(120) C + t = Y j ∈ Q + i [ a j ( t ) , b j ) t )] , C − t = Y j ∈ Q − [ a j ( t ) , b j ( t )] , where for all t (121) 1 ≤ a j ( t ) ≤ b j ( t ) if j ∈ Q + , ≤ a j ( t ) ≤ b j ( t ) if j ∈ Q − . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 37
A computation of the quantities in (94) shows that m ρ ( C t ) ≪ Y j ∈ Q + b j ( t ) ρ − Y j ∈ Q − V ( C − t ) , (122) β ε ( C + t ) ≪ ε X m ∈ Q + b m ( t )( a m ( t ) + b m ( t ))( b m ( t ) − a m ( t ) + ε ) . In the uninteresting case when C − t does not intersect Q j ∈ Q − (cid:16) ξ + + N (cid:17) , we have˜N r ( B E × C t ) = e Pl( B E × C t ) =
0. So we assume that Q − = ∅ or ˜ V ( C − t ) >
0. Then˜ V ( C − t ) ≥ Q − , ∅ .The conclusion is that m ρ ( C t ) ↓ j ∈ Q we have b j ( t ) → ∞ .With ε = ε ( C t ) as in (96) it su ffi ces to require in (122) that for any m ∈ Q + :( b m ( t ) − a m ( t ) + ε ) = o (1) . This can be achieved by requiring that b j ( t ) − a j ( t ) ≥ γ | log m ρ ( C t ) | for any j ∈ Q + and all t large, for any α ∈ (0 , ) and any γ >
0. Thus we have:
Proposition 6.1.
The asymptotic formula holds for a family of boxes t ˜ Ω = ˜ B E × C t with B E any box in R E satisfying (116) and C t as in (120) and (121) underthe conditions a) b j ( t ) → ∞ for some j ∈ Q. b) b j ( t ) − a j ( t ) ≥ σ ( t ) for any j ∈ Q + and all t, with σ ( t ) = γ (cid:18) (1 − ρ ) X j ∈ Q + log b j ( t ) + log ˜ V ( C − t ) (cid:19) − α ( γ > , < α <
12 ) . c) [ a j ( t ) , b j ( t )] ∩ (cid:16) ξ j + + N (cid:17) , ∅ for all t and for any j ∈ Q − . This proposition implies Theorem 1.3, and its consequences Propositions 1.1and 1.2.For boxes in the λ -parameter we have the following result: Proposition 6.2.
The asymptotic formula holds for families of boxes t Ω t , where Ω t = Y j [ A j ( t ) , B j ( t )] satisfies the following conditions: a) If for a fixed j A j and B j are constant then A j = A j ( t ) and B j = B j ( t ) satisfy condition (116) . b) If A j ( t ) is not constant, then A j ( t ) ≤ for all t, or A j ( t ) ≥ for all t.Similarly for B j ( t ) . c) There is a constant σ > such that if B j ( t ) ≥ thenB j ( t ) − max( A j ( t ) , ) ≥ σ (cid:18) q | B j ( t ) + q max( A j ( t ) , ) (cid:19) . d) If A j ( t ) ≤ , then the interval [ A j ( t ) , B j ( t )] intersects for all t the set of b (1 − b ) , b > , b ≡ ξ j mod 2 non-trivially for all t. (This intersection maydepend on t.) e) lim t →∞ A j ( t ) = −∞ or lim t →∞ B j ( t ) = ∞ for at least one j. This is not the most general statement for boxes. We have decided not to com-plicate the proposition by considering non-constant endpoints that have values in(0 , ). Proof.
Let E be the set of places for which A j and B j are constant. We considerpartitions Q + ⊔ Q ⊔ Q − of the remaining infinite places of F . For each of thesepartitions P we form Ω Pt = Ω t ∩ (cid:16) R E ∪ ( i [1 , ∞ )) Q + ∪ ( i [0 , ∪ (0 , ν ]) Q ∪ [ , ∞ ) Q − (cid:17) . Suppose Q + , ∅ . For j ∈ Q + we write A · j ( t ) = max( A j ( t ) , ). In the ν -description, the factor Ω Pt , j is of the form i [ a j ( t ) , b j ( t )] with a j ( t ) = q A · j ( t ) − and b j ( t ) = q B j ( t ) − . Condition c) implies that condition b) in Proposition 6.1is satisfied.If Q − , ∅ for P , then condition d) implies condition c) in Proposition 6.1.For the partition P we take E = E ∪ Q , and try to apply Proposition 6.1 to t Ω Pt . This gives the asymptotic formula for Ω Pt provided either there is j ∈ Q + for which B j ( t ) → ∞ , or there is j ∈ Q − for which A j ( t ) → −∞ . Otherwise, the set Ω Pt is bounded.Condition e) implies that the asymptotic formula holds for at least some parti-tion P . Thus ˜ V ( Ω Pt ) → ∞ for such P . Adding the corresponding finitely manyterms we get the asymptotic formula for the union of the Ω Pt . For the remainingpartitions P , the set Ω Pt stays bounded. Adding the corresponding terms to the as-ymptotic formula does no harm. This gives the asymptotic formula for t Ω t . (cid:3) Now an approximation of Pl (cid:16) [ − X , X ] d (cid:17) gives Proposition 1.6.6.2. Simplices.
The results in [2] are for sets of the form(123) ˆ C t = (cid:8) λ ∈ [0 , ∞ ) Q + × ( −∞ , Q − : X j ∈ Q | λ j | ≤ t (cid:9) . By showing that the asymptotic formula holds for sets of this form, we extendthe results in [2] to general character χ and general compatible central charactergiven by ξ . Proposition 6.3.
Let E ⊔ Q + ⊔ Q − be a partition of the infinite places of F withQ = Q + ∪ Q − , ∅ . Let B E be any box in R E satisfying (116) . The asymptoticformula holds for t B E × ˆ C t , and (124) Pl( ˆ C t ) ∼ | Q | ! t | Q | ( t → ∞ ) . ENSITY RESULTS ON HILBERT MODULAR GROUPS II 39
Proof.
Let us first consider W n ( Y ) = (cid:8) λ ∈ (cid:2) , ∞ (cid:1) n : X j λ j ≤ Y (cid:9) . We have V ( W ( Y )) = ( Y − ) for Y ≥ . From V ( W n ( Y )) = Z Y / V (cid:0) ( W n − ( Y − λ ) (cid:1) d λ , we obtain by induction that:(125) V ( W n ( Y )) = n n ! (cid:0) Y − n (cid:1) n + . We use ( x ) + = x <
0, and ( x ) + = x if x ≥ ρ ∈ (0 ,
1) we find by the inclusion W n ( Y ) ⊂ [5 / , Y ] n that(126) V ρ (cid:0) W n ( Y ) (cid:1) = O n (cid:0) Y n ( ρ + / (cid:1) . Furthermore, for ε small in comparison with Y , the part of W n ( Y )[2 ε ] on which λ j > for all j ∈ Q + is contained in W n ( Y + ) − W n ( Y − ) with Y + = Y + ε nY / + ε n and Y − = Y − ε nY / . The other parts of W n ( Y )[2 ε ] are contained in boxes of theform h − ε, Y + ε i n − × h − ε, + ε i . Hence˜ V (cid:0) ˜ W n ( Y )[2 ε ] (cid:1) ≤ V ( W n ( Y + )) − V ( W n ( Y − )) + nO ( Y n − ε )(127) ≤ n n ! (cid:16) ( Y + ε nY / + ε n − n ) n − ( Y − ε nY / − n ) n (cid:17) + O n ( ε Y n − ) ≪ n ε Y n − . If Y − n is small, we get at least O ( ε ), which is O ( ε Y n − / ) as well.Now we apply Proposition 4.7 to the following subset of ˆ C t :(128) W t = G p W | Q + | (cid:18) t − X j ∈ Q − (cid:0) p j − (cid:1)(cid:19) × { p } , where p runs over Q j ∈ Q − (cid:16) − ξ j + N (cid:17) for which P j ∈ Q − (cid:0) p j − (cid:1) ≤ t . For each given t , this is a finite union. But as a family depending on t it is an infinite union. Weobtain V ( W t ) = X p | Q + | | Q + | ! (cid:18) t − | Q + | + | Q − | − X j ∈ Q − p j (cid:19) | Q + | + Y j ∈ Q − p j , m ρ ( W t ) V ( W t ) ≪ Q + X p (cid:18) t + | Q − | − X j ∈ Q − p j (cid:19) | Q + | ( ρ + / Y j ∈ Q − p − Aj ,β ε ( W t ) V ( W t ) ≪ Q + ε X p (cid:18) t + | Q − | − X j ∈ Q − p j (cid:19) | Q + |− . We compare the sum for V ( W t ) with the integral Z x ∈ [1 , ∞ ) Q − , P j ∈ Q − x j ≤ t + | Q − | t + | Q − | − | Q + | − X j ∈ Q − x j | Q + | · Y j ∈ Q − x j d x = Z y ∈ [5 / , ∞ ) Q − , P j ∈ Q − y j ≤ t t − | Q + | − X j ∈ Q − y j | Q + | · −| Q − | d y = | Q | | Q | ! (cid:16) t − | Q | (cid:17) | Q | ∼ t | Q | | Q | | Q | ! . The transition from sum to integral gives a contribution O ( t | Q |− ).The other sums can be treated similarly. This leads to the estimates m ρ ( W t ) ≪ t | Q + | ( ρ − / and β ε ( W t ) ≪ t − / . This implies that the asymptotic formula holds for t B E × W t .The definition of e Pl shows that Pl( W t ) ∼ | Q | | Q | | Q | ! V ( W t ).The remaining parts in ˆ C t r W t are contained in unions of sets of the form Y = [0 , ] × [ − t , t ] | Q |− for which V ( Y ) ≪ t | Q |− and Pl( Y ) ≪ t | Q |− . So adding theseparts do not influence the asymptotic formula or the asymptotic behavior of Pl( ˆ C t ). (cid:3) Proposition 1.7 is a corollary of this result.6.3.
Sectors.
Under the assumption | Q + | = S p , q ,α, t = (cid:26) ( λ , λ ) ∈ (cid:16) , ∞ (cid:17) : t ≤ λ ≤ t + t α , p λ ≤ λ ≤ q λ (cid:27) , with 0 < p < q , α ≤
1, and t → ∞ . Lemma 6.4.
For c ≤ : (128) V c ( S p , q ,α, t ) ∼ c + (cid:18) q c + − p + c (cid:19) t c + α . Furthermore, for ε > small in comparison to t as t → ∞ :V ρ ( S p , q ,α, t ) ≪ p , q t ρ − V ( S p , q ,α, t ) , (129) V ( ˜ S p , q ,α, t [2 ε ]) ≪ p , q ε t max( ,α + ) V ( S p , q ,α, t ) . (130) Proof. (128) is obtained by direct computation based on (99). It immediately im-plies (129).The description of S p , q ,α, t in ν -space is as follows:˜ S p , q ,α, t = ( i ( t , t ) : a ≤ t ≤ b , q pt + p − ≤ t ≤ q qt + q − ) , ENSITY RESULTS ON HILBERT MODULAR GROUPS II 41 with a = q t − , b = q t + t α − . The left hand side of ˜ S p , q ,α, t [2 ε ] is contained in[ a − ε, a + ε ] × (cid:2) p pt − / + ε, p qt − / − ε (cid:3) . Its contribution to ˜ V ( ˜ S p , q ,α, t [2 ε ]) is O q − p ( ε a · t ) = O ( t / ). The contribution of theright hand side of ˜ S p , q ,α, t [2 ε ] has the same order.On the lower side remains a piece for which ˜ V can be estimated by Z bt = a t Z q pt + p − + ε (1 + p ) t = q pt + p − − ε (1 + p ) t dt dt . The inner integral can be estimated by O p ε q pt + p − ! . This gives for the totalintegral ≪ p ε (cid:16)(cid:0) pb + p − (cid:1) / − (cid:0) pa + p − (cid:1) / (cid:17) ≪ p ε ( pb − pa ) (cid:0) pb + p − (cid:1) / ≪ p ε t α + . Similarly, the upper part contributes O q ( ε t α + / ). (cid:3) This lemma shows that if α ≥ , the asymptotic formula holds for the family t B E × S p , q ,α, t for any choice of the box B E as before, where E contains all infiniteplaces except the two places we put in Q + . In particular, we obtain Proposition 1.8.6.4. Spheres.
We consider the sphere S Q + ( m , r ) ⊂ ( i [1 , ∞ )) Q − with radius r andcenter m : The set of ν with P j ∈ Q + (cid:16) | ν j | − | m j | (cid:17) ≤ r . We suppose that | m j | − r ≥ j ∈ Q + , and that | m j | → ∞ for at least one j ∈ Q + . A computation byinduction on | Q + | leads to(131) ˜ V (cid:0) S Q + ( m , r ) (cid:1) = v | Q + | r | Q + | Y j ∈ Q + | m j | , where v n is the volume of the unit sphere in R n .Furthermore ˜ V ρ (cid:0) S Q + ( m , r ) (cid:1) ≪ r | Q + | Y j ∈ Q + | m j | ρ , (132) ˜ V (cid:0) S Q + ( m , r )[2 ε ] (cid:1) ≪ ε r n − Y j ∈ Q + | m j | . These estimates follow from the inclusion S Q + ( m , r ) ⊂ Y j ∈ Q + i [ | m j | − r , | m j + r ] , S Q + ( m , r )[2 ε ] ⊂ S Q + ( m , r + ε √ n ) r S Q + ( m , r − ε √ n ) . For the latter inclusion we use that if ν is on the boundary of S Q + ( m , r ), then P j ∈ Q + (cid:16) | ν j | + ε (cid:17) ≤ r + ε P j ∈ Q + | ν j | + ε n ≤ r + ε r √ n + ε n , and simi-larly P j ∈ Q + (cid:16) | ν j | − ε (cid:17) ≥ (cid:16) r − ε √ n (cid:17) . These estimates show that the asymptotic formula holds for families m B E × S Q + ( m , r ), where B E is a box as earlier, and Q − = ∅ . It works for constant radius r ,or even for r going down as a multiple of (cid:16) log Q j ∈ Q + | m j | (cid:17) − α with α < . As aspecial case we obtain Proposition 1.4. There we have required that all m j go toinfinity, in order to have e Pl (cid:0) S Q + ( m , r ) (cid:1) ∼ d ˜ V (cid:0) S Q + ( m , r ) (cid:1) .6.5. Slanted strips.
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