MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS OF COHOMOLOGICAL TYPE ON GL 2

Transcription

1 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 24, Number 4, October 20, Pages S ) Article electronically published on April 6, 20 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS OF COHOMOLOGICAL TYPE ON GL 2 SIMON MARSHALL Introduction One of the central problems in the subject of quantum chaos is to understand the behaviour of high energy Laplace eigenfunctions on a Riemannian manifold M There is an important conjecture of Rudnick and Sarnak [32] which predicts one aspect of this behaviour in the case when M is compact and negatively curved, namely that the microlocal lifts of eigenfunctions tend weakly to Liouville measure on the unit tangent bundle This is known as the quantum unique ergodicity conjecture and has as a corollary that the L 2 mass of eigenfunctions becomes weakly equidistributed on M We refer the reader to [20, 2, 32, 37, 38, 42, 43] for many illuminating discussions and interesting results related to this conjecture In this paper we shall deal with a variant of Rudnick and Sarnak s conjecture which replaces Laplace eigenfunctions with certain modular forms This may be described most easily in the case of the modular surface X = SL2, Z)\H 2,where the objects we shall consider are holomorphic modular forms of large weight, or equivalently sections of high tensor powers of the line bundle of holomorphic differentials on X If f is a holomorphic modular cusp form of weight k, the analogue of the L 2 mass of f is the Petersson measure μ f = y k fz) 2 dv, where dv denotes the hyperbolic volume The measure μ f is invariant under SL2, Z), and we may suppose that f has been normalised so that it descends to a probability measure on X The analogue of the quantum unique ergodicity conjecture for holomorphic forms is then to show that the measures μ f tend weakly to the hyperbolic volume as the weight of f tends to infinity This is very much in the spirit of the original conjectures, with the Cauchy-Riemann equations replacing the Laplace operator and the weight k playing the role of the eigenvalue, and was considered in [23, 34] There are two main differences between this conjecture and the classical form of QUE The first is that no microlocal lift is known for holomorphic forms, so we are restricted to considering an equidistribution on X rather than its unit tangent bundle, and ergodic methods may not presently be applied to this problem The second is that the literal analogue of the conjecture fails because the space of cusp forms is large and contains elements such as Δ k where Δ is Ramanujan s cusp form) whose mass is not equidistributing From a number-theoretic point of view it is Received by the editors July, 200 and, in revised form, December 0, 200, and February 22, Mathematics Subject Classification Primary F4, F; Secondary F75 c 20 American Mathematical Society Reverts to public domain 28 years from publication 05

2 052 SIMON MARSHALL natural to deal with this multiplicity issue by requiring f to be a Hecke eigenform, which gives a refinement of the conjecture known as arithmetic QUE This is a natural condition to impose, as Watson s triple product formula [4] illustrates that the generalised Riemann hypothesis would imply QUE for holomorphic Hecke eigenforms with the optimal rate of equidistribution The first unconditional results on this conjecture were obtained by Sarnak [34], who showed that it was true for dihedral forms, and by Luo and Sarnak [23], who showed that it was true for almost all eigenforms of weight at most k In [5, 6, 39], Holowinsky and Soundararajan established QUE for all holomorphic Hecke eigenforms on the modular surface X, or more generally any noncompact congruence hyperbolic surface Their proof is a combination of two different approaches, one based on bounding the L-value appearing in Watson s triple product formula and the other on bounding shifted convolution sums, and which complement each other in a remarkable way to produce the full result In this paper we extend Holowinsky and Soundararajan s methods to prove QUE for holomorphic Hecke eigenforms on GL 2 over a totally real number field, or more generally for automorphic forms of cohomological type on GL 2 over an arbitrary number field and which satisfy the Ramanujan bounds For simplicity, we assume our fields to have narrow class number one throughout the paper, but this is not essential and can be avoided by working adelically as in the work of Nelson [29] Structure of the paper We introduce the manifolds and automorphic forms with which we shall work in section 2, before stating our results in section 3 We describe the structure of the proof in section 4 As our proof is a direct generalisation of the methods used by Holowinsky and Soundararajan over Q, wedothis by first giving an overview of their proof before explaining the modifications which must be made to extend it to a number field Sections 5 to 7 contain the generalisation of Holowinsky s method of shifted convolution sums, and section 8 contains the extension of Soundararajan s approach of triple product identities and weak subconvexity In section 9 we combine these two approaches to establish our main result, and in section 0 we prove the generalisation of Rudnick s theorem on the equidistribution of zero divisors of holomorphic forms Section is an appendix which contains various computations which are needed in the course of the proofs 2 Definitions and notation 2 Arithmetic manifolds We begin by introducing the manifolds on which we shall work Let F be a number field of narrow class number one with discriminant D and regulator R LetF have degree n and r infinite places, of which r are real and r 2 are complex Let F = F Q R, and let F + be the subset of totally positive elements If O is the ring of integers of F,letO + = O F + We shall denote the multiplicative group of F + by F +, and the group of totally positive units by O + Define μ + to be the group of totally positive roots of unity in F, which is the ordinary unit group if F is totally complex and trivial otherwise Let G i = GL + 2, R) fori r and GL2, C) otherwise,andg = G G r = GL + 2, F) Z i will denote the centre of G i,andg i = G i /Z i N will denote the usual unipotent subgroup of G and G, anda and M the maximal split and compact diagonal subgroups with lower entry equal to K = K K r will be the maximal compact Let Γ = GL + 2, O) be the integral matrices with totally positive

3 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 053 determinant, and define Γ =Γ B and Γ U =Γ U, whereb and U are the upper triangular Borel and unipotent subgroups of GL 2 F ) Let H F = G/K be identified with the product H 2 ) r H 3 ) r 2 of two and three dimensional hyperbolic spaces, and introduce on it the following coordinates: z = z,,z r ), { xt + iy z t = t, x t,y t R for t r, x t + jy t, x t C, y t R for t>r, x = x,,x r ) F, y = y,,y r ) R r + We let dv = yt 2 t r dx t dy t 3 yt t>r 2i dx tdx t dy t be the product of standard hyperbolic measures on H F We define X =Γ\G and Y =Γ\H F, so that automorphic forms on GL 2 /F of full level are equivalent to Hecke eigenforms on X Throughout the paper, we will use a multi-index notation for coordinates on H F and the weights of automorphic forms; for instance, if y is the coordinate on H F introduced above and k is an r-tuple of integers, the expression y k will denote y k i i Ifδ i is defined to be for i r and 2 otherwise, for any r-tuple x we denote x δ i i by Nx, and the maximum of x i by x If a =a i )andb =b i )aretwo elements of F, the inequality a>bwill mean that a i >b i when ν i is a real place, and that a i > b i when ν i is complex In particular, the set {x O: x>0} is just O + 22 Eisenstein series In addition to the usual complete Eisenstein series, we will work with two kinds of incomplete Eisenstein series which we term pure incomplete Eisenstein series and unipotent Eisenstein series To define them, we must introduce the multiplicative characters of the group F +/O + following Hecke Let {ɛ j : j r } be a generating set for O +, and denote the image of ɛ j at the ith Archimedean place of F by ɛ i j Define the matrix A to be A = /n log ɛ log ɛ r /n log ɛ r log ɛ r r, and denote its inverse by A = 2 e e 2 e r e r e r 2 e r r,

4 054 SIMON MARSHALL where the first row of A contains r s and r 2 2 s We may now define the characters λ m y) form Z r by the following formula: r r λ m : y =y,,y r ) F + y p 2πim qe q p p= q= r ) = exp βm, p)log y p, p= r ) where βm, p) =2πi m q e q p As λ m is invariant under the action of O + on F +, it may be extended to a Hecke character on F via the isomorphism F /O F + /O + When defining Eisenstein series, we shall frequently think of λ m as a character of the multiplicative group R +) r /O + Having defined λ m, we may let Es, m, z) denote the usual Eisenstein series associated to the character Ny s λ m y) ofthecuspofx The pure incomplete Eisenstein series are formed by automorphising a function on Γ \H F which is invariant under U and transforms according to λ m under the norm one elements of the diagonal They are specified by an index m Z r and a function ψ C0 R + ) and are defined as Eψ, m z) = ψnyγz))λ m yγz)) γ Γ \Γ The unipotent Eisenstein series are formed by automorphising a function on Γ U \H F which is only invariant under U They are specified by a function g C0 R r +) and defined as Eg z) = gyγz)) γ Γ U \Γ We note that it is less standard to form Eisenstein series by symmetrising a function over Γ U \Γ in this way, and while these series do not play a major part in the proof, their appearance is related to the key fact that the correct way in which to generalise Holowinsky s methods is by unfolding over the unipotent, as will be discussed in section Representation theory of SL2, C) For m N, letρ m denote the irreducible m + )-dimensional representation of SU2) SL2, C) withhermitian inner product,, and let denote the associated conjugate linear isomorphism between ρ m and ρ m We choose an orthonormal basis {v t } t = m, m 2,, m) for ρ m and dual basis {vt } for ρ m, consisting of eigenvectors of M satisfying e iθ 0 0 e iθ ) v t = e itθ v t, q= e iθ 0 0 e iθ ) v t = e itθ v t If r C and k Z, leti k,r) be the representation of SL2, C) unitarily induced from the character ) z x χ : 0 z z/ z ) k z 2ir

5 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 055 These are unitarisable for k, r) intheset U = {k, r) r R} {k, r) k =0,r i, )}, and two such representations I k,r), I k,r ) are equivalent iff k, r) =±k,r ) Furthermore, these are all the irreducible unitary representations of SL2, C) other than the trivial representation We choose a set U U representing every equivalence class in U to be U = {k, r) r 0, )} {k, r) r =0,k 0} {k, r) k =0,r i0, )} Given π SL2, C) nontrivial, we shall say that π has weight k and spectral parameter r if it is isomorphic to I k,r) with k, r) U AsweshallworkonGL 2 with trivial central character, we may specify the Archimedean components of our automorphic representations by specifying their restrictions to SL 2 At complex places we shall use the parameters just introduced, and at real places we shall use the customary weight and spectral parameter 24 Automorphic forms We shall consider QUE for cuspidal automorphic forms π on GL 2 /F of full level, trivial central character and cohomological type This means that their local factors at real places are holomorphic discrete series of even weight, and the factors at complex places have even weight and spectral parameter 0 In the notation of section 2, these correspond to automorphic forms on X of the prescribed Archimedean type and which are eigenfunctions of the Hecke operators Our assumptions on the class number of F and the level of π imply that π is not dihedral We denote the weight of π by an r-tuple k =k i ), and its normalised Hecke eigenvalues by λ π p) Define ρ k to be the representation ρ k = i r χ ki i>r ρ ki of K, noting that, in the presence of complex places, K will be nonabelian and ρ k will have dimension greater than one for most choices of the weight As ρ k occurs as a K-type in the Archimedean component of π, there is an embedding R π in Hom K ρ k,l 2 cuspx)) corresponding to π We may associate to R π a section F k of the principal bundle X K ρ k on Y, where we recall that for a representation τ of K, X K τ is the quotient of X τ by the right K-action x, v)k =xk, τk) v) so that sections of X K τ may be thought of as sections of X τ satisfying τk)vxk) =vx) F k may be defined by the relation R π v)x) =F k x),v)forv ρ k and x X, which may be unwound to give F k x) = k i +) /2 R π v t )x)vt, i>r t F k x) 2 = k i +) R π v t )x) 2, i>r t where {v t } is a basis of M-eigenvectors for ρ k Note that F k x) 2 descends to a function on Y Alternatively, we may define E k to be the restriction to Γ of the

6 056 SIMON MARSHALL representation ) Sym k i 2 Sym ki/2 Sym ki/2 i r i>r of G, and let V k be the associated local system on Y, which we equip with a certain canonical positive definite norm Then F k may be thought of as a harmonic -form which represents a cohomology class in H Y,V k ) this is why π is referred to as being of cohomological type) However, we will not use this point of view in this paper and shall only refer the reader to the book of Borel and Wallach [] where correspondences of this kind are described in detail We wish to establish the equidistribution of the probability measures F k 2 dv on Y, in generalisation of holomorphic QUE over Q Because the K-integrals of R π v t ) 2 are independent of t, we may let v k ρ k be the vector of highest weight and think of the measure F k 2 dv as the pushforward of R π v k ) 2 dx from X In the case where F is totally real, the reader may instead let f be a holomorphic Hecke eigenform with associated representation π, and let F k be the mass function F k = y k/2 f In particular, the results stated in the next section may all be read with this simpler definition in mind To simplify the transition from Fourier expansions to shifted convolution sums in the next chapter, we will write the Fourier expansions of all our automorphic forms using sums over the ring of integers O rather than the inverse different O as follows: φz) = ξ O a ξ y)etrξκx)), where κ will denote a fixed totally positive generator of O throughout As the F k are vector-valued, it turns out that they may be expanded in Fourier series more simply by enlarging their domain H F, in a manner which we now describe We identify H F with the subgroup NA of G in the standard Iwasawa factorisation, and let H F be the subgroup NAM We then have an inclusion of H F in H F,and we extend our hyperbolic coordinate system to H F by allowing y i to take complex values for i>r TheK-covariance of F k means that it is determined by its values on H F, and these determine the embedding R π by the formula R π v)g) =ρk)v, F k z)), where g = zk is the Iwasawa factorisation of g On H F, we may expand F k in a Fourier series as F k z) = a f ξ)k k ξκy)etrξκx)), ξ>0 where K k y) = r t= K ty t )andthek t y t ) are defined by 2) K t y t )=y t ) kt/2 exp 2πy t ) for t r, k t ) /2 K t y t )= y t k kt 3) t/2+ K kt/2 j4π y t )e k t 2j)iθ t /2 vk j t 2j, t > r, j=0 and θ t is the argument of y t The formula for the Whittaker functions K t at complex places is taken from Jacquet-Langlands [9] The coefficients a f ξ) are proportional to the Hecke eigenvalues λ π ξ), 4) a f ξ) =λ π ξ)nξ /2 a f ),

7 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 057 and the first Fourier coefficient is determined by the L 2 normalisation of F k to be 5) a f ) 2 = 27r2 π r +3r 2 4π)ki 2π)ki D L, sym 2 π) Γk i ) Γk i r i>r i /2+) 2 See section 2 for this calculation) 3 Statement of results Our main result is Theorem, which establishes QUE for the sections F k under the assumption that the associated cohomological representations π satisfy the Ramanujan bound; this is known when F is totally real or imaginary quadratic, as discussed below We must also assume that the weight k =k i )ofπ tends to infinity in a weaky uniform way, namely that there exists ν>0such that k i k ν for all i Theorem If φ is a Hecke-Maass cusp form, we have φf k,f k φ,ɛ,ν log k ) /30+ɛ If φ is a pure incomplete Eisenstein series, we have φf k,f k = VolY ) φ, + O φ,ɛ,νlog k ) 2/5+ɛ ) Let us give two simple illustrations of the content of this theorem in the cases where Ramanujan is known Suppose that F is totally real and Γ = GL + 2, O) is the subgroup of GL2, O) of elements with totally positive determinant Fix ν>0, and let {f n } be a sequence of holomorphic Hecke modular forms for Γ whose weights k n =k i,n )satisfyk i,n k ν for all i Theorem implies Corollary 2 The normalised Petersson probability measures μ n = y k n f n z) 2 dv tend weakly to the uniform measure on Γ\H 2 ) n as k An equidistribution theorem of this kind for Hilbert modular forms was first conjectured by Liu in [22] As a consequence of Corollary 2, we prove that if k is a fixed positive weight and {f T } a sequence of holomorphic Hecke forms of weight Tk =Tk,,Tk n ), then the zero divisors Z T of f T become equidistributed on Γ\H 2 ) n as T, either as Lelong, )-currents or as measures defined by integration over Z T with respect to the volume form of the induced Riemannian metric This result is stated more fully in section 3, and generalises a theorem of Rudnick [3] on the equidistribution of zeros of Hecke modular forms on SL2, Z) In the imaginary quadratic case, let Y be the Bianchi manifold Γ\H 3,whereO is the ring of integers in an imaginary quadratic field F and Γ = SL2, O) Theorem proves the equidistribution of a measure μ π on Y which may be associated to any cuspidal automorphic form π of cohomological type on Γ\SL2, C) If φ π is a vector of minimal K-type, we may define μ π to be the pushforward of φ 2 dg to Y Alternatively, we may define E d to be the representation Sym d Sym d of SL2, C) as in section 24, and let V d be the associated local system on Y which we equip with a certain canonical positive definite norm If π has weight k =2d +2, φ corresponds to a -form ω A Y,V d ) which is harmonic with respect to the norm on V d and is an eigenform for the Hecke operators If we define the norm on T Y V d to be the tensor product of the Riemannian norm on T Y and the norm on V d mentioned earlier, and dv is the hyperbolic volume, it may be shown that μ π

8 058 SIMON MARSHALL is equal to ω 2 dv Theorem then implies Corollary 3 If {π n } is a sequence of cohomological automorphic forms on Γ\SL2, C) whose weights are tending to, then the associated measures μ πn tend weakly to the hyperbolic volume on Y Theorem is proven by combining the following two results, which summarise the extensions of Holowinsky and Soundararajan s respective approaches to proving the equidistribution of F k Their statements are almost identical to those of the original theorems over Q, which are recalled in section 4, with the only significant difference being that we must impose our assumption on the uniform growth of the weight in Theorem 4 Theorem 4 Fix an automorphic form φ, and suppose that there exists a ν>0 such that k i > k ν for all i Define M k π) = log k ) 2 L, sym λ ) πp) π) Np Np k If φ is a Hecke-Maass cusp form, then 6) φf k,f k φ,ɛ,ν log k ) ɛ M k π) /2 for any ɛ > 0 If φ is a pure incomplete Eisenstein series, then 7) φf k,f k = VolY ) φ, + O φ,ɛ,νlog k ) ɛ M k π) /2 + R k π))) for any ɛ>0, where R k π) = NkL, sym2 π) m + Theorem 5 If φ is a Hecke-Maass cusp form, we have log k ) /2+ɛ 8) φf k,f k φ,ɛ L, sym 2 π) If E 2 + it, m, ) is a unitary Eisenstein series, we have L/2+it, sym 2 π λ m ) t + m +) A dt 9) E 2 + it, m, )F 2n log k ) +ɛ k,f k ɛ + t + m ) L, sym 2 π) We shall prove Theorem 4 in sections 5 to 7 and Theorem 5 in section 8, before combining them to give our main result in section 9 The presence of these two components and the way in which they interact makes the overall proof somewhat elaborate, and so we begin by reviewing its basic outline in the case of SL2, Z) and giving an overview of our modifications in section 4 Our assumption that π satisfies the Ramanujan bound is needed in the proofs of both Theorems 4 and 5, in the first case to establish the weak form of Ramanujan required by Soundararajan s weak subconvexity theorem, and in the second case as an ingredient in bounding shifted convolution sums It is known when F is totally real or imaginary quadratic, and so we have an unconditional theorem in these cases In the totally real case this is derived from Deligne s theorem by Blasius in [3], while in the imaginary quadratic case this relies on deep work of Harris, Soudry, Taylor, Berger, Harcos

9 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 059 et al [2, 2] and requires the construction of a theta lift from GL 2 /F to GSp 4 /Q, where complex geometry is available The generalisation of their results to other fields with complex places is not yet established, and consequently we have no unconditional result outside totally real and imaginary quadratic fields On the other hand, Ramanujan will hold for forms lifted from totally real subfields and so our theorem becomes unconditional if the family of cohomological forms of fixed level has the structure suggested by the results of [7], ie if base change and CM constructions account for all but finitely many forms The assumption we have made on the uniform growth of the weight is a purely technical one Nelson [29] shows that it may be removed in the case when F is totally real, and by combining the triple product identities of section 8 with the Lindelöf hypothesis we see that it should be unnecessary in general The reason we have adopted it is so that when we come to the point in the generalisation of Holowinsky s theorem at which we apply the sieve, it will ensure that we are sieving over a rounded subset of the ring of integers rather than a narrow box Nelson s improvement relies on an explicit formula for the integral appearing as a weight in the shifted convolution sum 32) of section 53, and it would be interesting to know if this can be extended to the case of one or more complex places Theorem establishes QUE for any sequence of sections F k over a totally real or imaginary quadratic field whose weights tend to infinity with the required uniformity However, we should ask whether such a sequence exists for these fields When F is totally real, Riemann-Roch ensures that the dimension of the space S k of holomorphic cusp forms of weight k is Nk, with an exact formula established by Shimizu in [36] Over a general field, base change from Q is expected to provide k forms of parallel weight k on a sufficiently deep congruence subgroup of Γ, where the term parallel means that the weights at all places are equal as in the totally real case In particular, for F imaginary quadratic it has been proven by Finis, Grunewald and Tirao [7] that base change produces forms of full level and so our result is not vacuous for the Bianchi manifolds The proof may be easily modified to allow a nontrivial level in any case, so by restricting to forms base changed from Q or another totally real subfield) which are known to satisfy Ramanujan, we may view it as having content over any solvable field F 3 Equidistribution of zero currents One consequence of QUE for holomorphic modular forms over Q is that the zeros of a sequence of forms become equidistributed with respect to hyperbolic measure as k, as was proven by Shiffman and Zelditch [35] for compact hyperbolic surfaces and extended to SL2, Z)\H 2 by Rudnick [3] Using their methods, we have derived the analogous statement about the equidistribution of the zero divisors of holomorphic modular forms from our proof of holomorphic QUE We may prove this equidistribution either in the sense of measures of integration over the smooth parts of the) zero divisors, or in the more refined sense of Lelong,)-currents, which we now describe As we shall only consider totally real number fields from this point until the end of section 5, in these sections we shall use H n to denote the product of n copies of the upper half-plane, so that the holomorphic forms f we consider live on Y =Γ\H n In higher dimensions we may replace the sum of delta measures at the zeros of f by the current of integration over its zero divisor Z f, which is a distribution on differential forms of bidegree n,n ) If Z f = i ord V i f)v i

10 060 SIMON MARSHALL is the expression of Z f as the sum of irreducible subvarieties, then 0) Z f,φ)= ord Vi f) φ i V i for all smooth, compactly supported forms φ on Γ\H n To define these notions in the presence of torsion in Γ, we use the standard procedure of choosing Γ Γto be of finite index and torsion free, and defining forms, subvarieties etc on Γ\H n to be those on Γ \H n which are invariant under Γ Integrals such as 0) are defined to be the lifted integral on Γ \H n divided by Γ :Γ We shall use w to denote weak- convergence of currents With these notions in mind, we may state our result Theorem 6 Fix a weight k =k i ), k i > 0, andlet{f T } be a sequence of holomorphic Hecke modular forms of weight Tk =Tk i ) Define ω = i log yk 2π = 4π ki y 2 i dx i dy i If Z T are the zero divisors of f T,then T Z T ω, ie lim T T Z T,φ ) = ω φ Y for all continuous, compactly supported n,n )-forms φ Inparticular, if k =2,,2), then T Z w T ω 0,theKähler form of Y with the product hyperbolic metric This theorem is based on ideas from complex potential theory as developed for problems in quantum chaos in [30, 3, 35] It may be loosely interpreted as saying that not only do the smooth parts of the) submanifolds Z T become equidistributed as measures of integration with respect to the induced Riemannian volume, but that the directions in which their tangent subspaces lie are also becoming equidistributed We prove Theorem 6 in section 0 4 Outline of the proof 4 The proof over Q We begin by giving an outline of Holowinsky and Soundararajan s proof over Q, as our proof over a number field runs on the same lines as theirs Suppose f is a holomorphic Hecke eigenform of weight k on Y = SL2, Z)\H, with associated mass function F k = y k/2 f We wish to show that the normalised probability measure μ f = F k 2 y 2 dxdy tends weakly to the hyperbolic measure 3 π y 2 dxdy as k tends to infinity, ie that for all h C0 X), μ f h) = h F k 2 y 2 dxdy 3 h, Y π In [5, 6, 40], Holowinsky and Soundararajan have established this by decomposing h in two different bases for smooth functions on X, the first a complete set of eigenfunctions for the Laplacian and the second the incomplete Poincaré series P m, defined by P m ψ z) = emxγz))ψyγz)) γ Γ \Γ w

11 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 06 for m Z and ψ C 0 R + ) The chosen basis of Laplace eigenfunctions consists of the constant function, Hecke-Maass cusp forms φ and unitary Eisenstein series E 2 + it, ), and the corresponding integrals which must be estimated are φf k,f k and E 2 + it, )F k,f k These integrals may be expressed in terms of central L- values, using Watson s formula in the first case and the classical Rankin-Selberg formula in the second, and so one may hope that the theory of L-functions would provide nontrivial upper bounds for them The convex bound just fails to be of use here; however, by strengthening the convex bound by a factor of log C) +ɛ, where C is the analytic conductor, Soundararajan obtains the following result: Theorem 7 If φ is a Hecke-Maass cusp form, we have log k) /2+ɛ φf k,f k φ,ɛ L, sym 2 f) If E 2 + it, ) is a unitary Eisenstein series, we have E 2 + it, )F 2 log k) +ɛ k,f k ɛ + t ) L, sym 2 f) The equidistribution of μ f would follow from Theorem 7 if one knew that L, sym 2 f) log k) /2+δ for some δ > 0 This is certainly expected, as it follows from the generalised Riemann hypothesis that L, sym 2 f) is bounded below by a power of log log k The best unconditional bound in this direction is due to Hoffstein and Lockhart [3], and Goldfeld, Hoffstein and Lieman [0], who prove that L, sym 2 f) log k) ; this is a deep result analogous to proving that there is no Siegel zero The bound L, sym 2 f) log k) /2+δ is known unconditionally for all but K ɛ eigenforms of weight K by a zero density argument; however, one cannot yet rule out the existence of forms with small values of L, sym 2 f)for which Soundararajan s approach is insufficient Holowinsky s approach is to test μ f against incomplete Poincaré and Eisenstein series This is equivalent to testing μ f against Hecke-Maass cusp forms and incomplete Eisenstein series and evaluating the inner products φf k,f k by regularising them with a second incomplete Eisenstein series and then unfolding In doing this one is led to estimating the shifted convolution sums λ f n)λ f n + l) n k for fixed l as k,whereλ f are the automorphically normalised Hecke eigenvalues of f Quite strikingly, Holowinsky is able to obtain useful bounds for these by taking absolute values of the terms and forgoing any additive cancellation The idea behind this is that the eigenvalues λ f p) not only satisfy the Ramanujan bound λ f p) 2, but are distributed in the interval [ 2, 2] according to Sato-Tate measure and so on average λ f p) will be significantly smaller than 2 we do not need to consider dihedral forms as we are working at full level) Moreover, as a typical λ f n) is a product of many λ f p) s this leads to a gain on average over the bound λ f n) τn) This phenomenon may also be seen in the work of Elliott, Moreno and Shahidi [6] where they prove the bound τn) x 3/2 log x) /8, n x

12 062 SIMON MARSHALL where τ here denotes Ramanujan s τ-function Holowinsky uses this idea, combined with a large sieve to show that n and n + l seldom both have small prime factors, to prove the following: Theorem 8 If λ f are the normalised Hecke eigenvalues of f as above, define M k f) = log k) 2 L, sym λ ) f p) f) p p k If φ is a Hecke-Maass cusp form, we have φf k,f k φ,ɛ log k) ɛ M k f) /2 If Eψ ) is an incomplete Eisenstein series, we have Eψ )F k,f k 3 π Eψ ), ψ,ɛ log k) ɛ M k f) /2 + R k f)), where L/2+it, sym 2 f) R k f) = k /2 L, sym 2 f) + t ) 0 dt One can see the appeal to Sato-Tate in the quantity M k f) appearing in Theorem 8; if we only apply the bound λ f p) 2 to this, one finds that M k f) log k) 2 L, sym 2 f), which is of no use However, under certain natural assumptions about the distribution of λ f p) it may be shown that M k f) is small; more precisely, in [4], Holowinsky shows that if neither L, sym 2 f)norl, sym 4 f)are small, then we have M k f) log k) δ for some δ>0 As with Soundararajan s theorem, these assumptions may also be shown to hold for almost all eigenforms using zero density estimates Surprisingly, while both of these approaches may fail it can be shown that together they cover all cases completely Intuitively speaking, if L, sym 2 f) < log k) /2+δ is small, then we should have λ f p 2 ) formostprimesp k a Siegel zero type phenomenon) However, M k f) isprovenin[6]tosatisfythe upper bound ) M k f) log k) ɛ exp λ f p) ) 2, p p k and if λ f p 2 ), then λ f p) 2, so that λ f p) 0formostp k and the right-hand side of ) should be small The precise bound Holowinsky and Soundararajan prove based on this argument is M k f) log k) /6 log log k) 9/2 L, sym 2 f) /2 This inequality may be used to show that if φ is a cusp form and L, sym 2 f) < log k) /3 δ for some δ>0, then M k f), and hence φf k,f k, is small However, if L, sym 2 f) > log k) /3 δ > log k) /2+δ, then Theorem 7 shows that φf k,f k is small This shows how Theorems 7 and 8 complement each other in the cusp form case, and a similar relationship holds between them in the incomplete Eisenstein case

13 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS Extension to a number field We now describe the steps which we have taken to generalise the method of section 4 to a number field Soundararajan s approach is the easier of the two to extend, as one has the triple product formula of Ichino [7] available to generalise Watson s formula, and Soundararajan s weak subconvexity theorem is sufficiently general to also be applicable to the central L-value which appears there The only technical difficulty is in making Ichino s formula sufficiently quantitative, which requires estimating certain Archimedean integrals The necessary computation at complex places was carried out in [24] using a result of Michel and Venkatesh appearing in [25], while at real places it may be obtained by comparison with Watson s formula Applying weak subconvexity is then straightforward, with the only consideration being that Soundararajan s theorem is stated for L-functions over Q rather than a number field However, it is easy to show that our L-functions still satisfy the required hypotheses when viewed as Euler products over Q These steps are carried out in section 8 The modifications which we have made to Holowinsky s approach are more substantial, and we shall now describe his method in detail before illustrating how we have adapted it in the simple case of a real quadratic field Holowinsky s approach for SL2, Z) is similar to calculating the integral of F k 2 against a Poincaré series in terms of shifted convolution sums For a Hecke-Maass form or incomplete Eisenstein series φ, he defines a regularised unfolding of φf k,f k in terms of a fixed positive g C0 R + ) and a slowly growing parameter T by 2) I φ T )= gty)φz) F k z) 2 dμ Γ \H This behaves like the integral of φ F k 2 over T copies of a fundamental domain for SL2, Z), and this may be quantified by taking the Mellin transform G of g and expressing 2) in terms of Eisenstein series as I φ T )= G s)t s Es, z)φz) F k z) 2 dμ 2πi σ) Y Shifting the contour to σ =/2 thengives I φ T ) = ct φf k,f k + OT /2 ), where c = 3 Eg z), π Holowinsky then calculates I φ T ) in a second way using the Fourier expansions of φ and f, fz) = n a f n)enz), φz) = l a l y)elx) Only those l with l T +ɛ make a significant contribution, and for those l Holowinsky defines S l T ) = gty)a l y)elx) F k z) 2 dμ 3) Γ \H a l T ) n ) a f n)a f n + l) gty)y k 2 e 2π2n+l)y dy 0

14 064 SIMON MARSHALL so that I φ T )= S l T )+OT /2 ) l T +ɛ When l 0, the regularising factor gty) effectively truncates the sum in 3) to n Tk, and we end up with an upper bound for S l T )of S l T ) a lt ) kl, sym 2 λ f n)λ f n + l) f) n Tk The expected main term 3 π φ, appears in S 0T ), and so to prove that 3 π φ, and φf k,f k are close one needs to bound the off-diagonal terms S l T ) and hence n x λ f n)λ f n + l) Having given up additive cancellation in this sum, Holowinsky instead proceeds by using the ideas discussed in section 4 to show that λ f n)λ f n + l) is small on average We have extended this method to work over an arbitrary number field F,withthe key innovation being the way the unfolding is carried out in the presence of units For simplicity, we will briefly describe the method in the case of a real quadratic field F = Q d), and f a holomorphic Hecke modular form of parallel weight k, k) with associated automorphic representation π Letφ be a Hecke-Maass cusp form, and write the Fourier expansions of f and φ as fz) = a f η)etrηκz)), η>0 φz) = ξ 0 a ξ y)etrξκx)) The totally positive units O + of O act on the terms of these expansions, and when unfolding we must do so in a way which breaks this symmetry so that the resulting shifted convolution sums are over well-rounded sets in O The correct approach is to unfold to Γ U \H 2 R 2 + R 2 /O) and localise in a set of the form B T R 2 /O), where B is a ball in R 2 + and we multiply it by T in each coordinate to get B T This lets us largely ignore the units, and when we form the analogues of S l T ) it will allow us to truncate the resulting shifted convolution sum over O at each place separately We therefore define I φ T )astheintegral 4) I φ T )= gty)φz) F k z) 2 dv, Γ U \H 2 where now we let h C0 R + ) be a positive function and g C0 R 2 +)beits square We extract a main term ct 2 φf k,f k from this as before, by forming the symmetrised function gy) = guy) u O + and expanding it in the multiplicative characters of R 2 +/O + to express I φ T )in terms of integrals against Eisenstein series When we calculate I φ T )intermsof the Fourier expansion of φ it may again be shown that only those ξ with ξ T +ɛ contribute, and for these we define S ξ T )= gty)a ξ y)exp2πitrξκx)) F k z) 2 dv Γ U \H 2

15 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 065 The analogue of the upper bound 3) for S ξ T )is S ξ T ) a ξ T ) a f η)a f η + ξ) gty)y k 2 exp2πtr2η + ξ)κy))dy η>0 R 2 + The key feature of the integral appearing here is that it factorises over the places of Q d), and the value of each factor depends only on the image of 2η + ξ at that place This lets us truncate the sum to the ball of radius k in O and leaves us with bounding 0<η<k λ πη)λ π η + ξ) This round set is well suited to the application of the large sieve for lattices in R n, and we may carry out Holowinsky s sieving approach as before by translating congruences modulo primes p of O to sieve conditions in O/pO without significant interference from the units We carry this method out in detail in sections 5 to 7 The proof splits into two parts, the first of which is reducing bounds on φf k,f k to ones on shifted convolution sums, and the second of which is bounding these sums using the large sieve The bulk of the work lies in the first step, and we have divided it into the case of totally real fields, carried out in section 5, and the modifications which are needed in the presence of complex places which are described in section 6 The application of the large sieve is carried out in section 7 5 Sieving for mass equidistribution: The totally real case In this section we prove Proposition 9 below, which reduces the problem of bounding φf k,f k to one of bounding shifted convolution sums We shall assume for simplicity in this section that F is totally real, so that the key modifications in the unfolding argument can be seen more clearly, and leave the treatment of complex places for section 6 We will work with holomorphic forms rather than vector-valued ones, and so let f be an L 2 normalised holomorphic Hecke eigenform of weight k with associated automorphic representation π We continue to assume that there exists ν>0such that k i k ν for all i Proposition 9 Let T and ɛ>0 Fix h C0 R + ) positive and let g C0 F + ) be its n-fold product, and define C g = Eg z), /VolY ) Fix an automorphic form φ with Fourier expansion φz) = ξ O a ξ y)etrξκx)) If φ is a Hecke-Maass cusp form, then 5) φf k,f k = Cg T n 0< ξ <T +ɛ S ξ T )+OT n/2 ) If φ is a pure incomplete Eisenstein series, then ) 6) φf k,f k = φ, + C g T n +Rk π) S ξ T )+O VolY ) T n/2 0< ξ <T +ɛ with 7) R k π) = NkL, sym2 π) m + L/2+it, sym 2 π λ m ) t + m +) A dt

16 066 SIMON MARSHALL Furthermore, we have the bound 8) S ξ T ) a ξt ) NkL, sym 2 λ π η)λ π η + ξ) π) η>0 n ) T ki ) h 4πη i + ξ i /2) ) + ONk k ν+ɛ T n+ɛ ) The bound we shall apply to the shifted convolution sums appearing in Proposition 9 is given below; it will be proven in section 7 following Holowinsky, although it should be noted that this result may also be derived from the works of Nair [27] and Nair-Tenenbaum [28] Proposition 0 Let λ and λ 2 be multiplicative functions on O + satisfying λ i η) τ m η) for some m For any x =x i ) sufficiently large with respect to ɛ and satisfying x i x ν,andanyfixedξ Osatisfying 0 < ξ x ν, we have 0<η<x λ η)λ 2 η + ξ) τξ)nx log x ) 2 ɛ where z = x /s and s = ɛ log log x Np z i= + λ ) p) + λ 2 p), Np To deduce Theorem 4 in the totally real case from Propositions 9 and 0, first apply Proposition 0 with λ = λ 2 = λ π and x = Tk Because we shall choose T to be bounded by any positive power of k, weseethatz = o k ) sothatwehave 9) n λ π η)λ π η +ξ) η>0 i= ) T ki ) h τξ)t n Nk 4πη i + ξ i /2) log k ) 2 ɛ Np k + 2 λ ) πp) Np InthecaseofφaMaass form, we substitute this into 8) and bound a ξ T ) by ρξ) T n/2+ɛ using Lemma from section 5 below ρξ) denotes the Hecke eigenvalue of φ) As T is bounded above by any positive power of k, 8) then becomes S ξ T ) ρξ) τξ)t n/2+ɛ L, sym 2 π)log k ) 2 ɛ Np k Np k + 2 λ ) πp) + O k ν+ɛ ) Np Substituting this into 5) and applying the Ramanujan bound on average 20), we obtain φf k,f k T n/2 T log k ) ɛ log k ) 2 L, sym λ ) πp) + OT n/2 ) π) Np which gives 6) on choosing T n = M k π) The derivation of 7) in the pure incomplete Eisenstein series case is similar The organisation of this section is as follows In section 5 we prove some results we shall need on the Fourier coefficients of φ and f In section 52 we introduce the regularised unfolding integral which lies at the heart of our proof, before using it to relate φf k,f k to shifted convolution sums in section 53

17 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS Fourier coefficient calculations In this section we present some bounds and normalisations we shall need for the Fourier coefficients of φ and f Ifφ is an automorphic form on Γ\H n, we may expand it in a Fourier series as φz) =a 0 y)+ ξ 0 a ξ y)etrξκx)) with a 0 y) =0ifφ is a cusp form If φ is a fixed Maass cusp form with spectral parameter t =t i ), then we have the expansion φz) = Ny n ρξ) K itp 2π ξ p κ p y p )etrξκx)), ξ 0 p= where the Hecke eigenvalues ρξ) satisfy the Ramanujan bound on average, ie 20) ρξ) T n ξ T If φ is a pure incomplete Eisenstein series Eψ, m z), we may determine its Fourier coefficients in terms of the coefficients of the complete Eisenstein series Es, m, z) The Fourier expansion of these series was calculated by Efrat [5] to be Es, m, z) =Ny s λ m y)+φs, m)ny s λ m y)+ 2n π ns Ny /2 D n Nξκ) s /2 λ m ξκ) ξ 0 p= where βm, p) isasin)and φs) = πn/2 D K s+βm,p) /2 2π ξ p κ p y p ) σ 2s, 2m ξκ) Γs + βm, p)) ζ2s, λ 2m ) etrξκx)), n p= Γs + βm, p) /2) ζ2s,λ 2m ) = Γs + βm, p)) ζ2s, λ 2m ) n θs) = π ns D s Γs + βm, p))ζ2s, λ 2m ), p= σ 2s, 2m ξκ) = λ 2m c) Nc 2s c) ξ/c O θs /2), θs) We have Eψ, m z) = Ψ s)es, m, z)ds, 2πi 2) where Ψ is the Mellin transform of ψ, and we may use this to calculate the Fourier coefficients of Eψ, m z) In the case of the constant term, we have a 0 y) = Ψ s)ny s λ m y)+φs, m)ny s λ m y))ds 2πi 2) = ψny)λ m y)+ony ) Moving the line of integration to σ =/2 weobtain a 0 y) = VolY ) Eψ, m z), + ONy/2 ),

18 068 SIMON MARSHALL where the main term is only nonzero for m = 0 Doing the same for nonzero ξ gives a ξ y) = 2n π n/2 Ny /2 2πi D π nit Ψ /2 it)nξκ) it λ m ξκ) n p= We may apply the bound + r 2) K ir y) Γ/2+ir) y valid for any integer A 0andɛ>0, to this to obtain a ξ y) τξ)ny /2 ξy A K it+βm,p) 2π ξ p κ p y p ) σ 2it, 2m ξκ) Γ/2+it + βm, p)) ζ2s, λ 2m ) dt ) A + + r ) ɛ, y n i= + ξ i y i Similar bounds are valid for φ a Maass cusp form The bounds for both varieties of the form are summarised in the following lemma: Lemma Let φ be an automorphic form on Y with Fourier series expansion φz) =a 0 y)+ ξ 0 a ξ y)etrξκx)) ) ɛ If φ is a Maass cusp form, then a 0 y) =0and for ξ 0we have n a ξ y) ρξ) Ny /2 ξy A + ) ɛ ξ i y i for any integer A 0 and any ɛ>0 If φ is a pure incomplete Eisenstein series, then a 0 y) = VolY ) φ, + ONy/2 ) and for ξ 0we have n a ξ y) τξ)ny /2 ξy A + ) ɛ ξ i y i for any integer A 0 and any ɛ>0 As we shall work with the Fourier expansion of f rather than that of F k,thel 2 normalisation of a f ) differs slightly from the one given in section 24 We write f as fz) = a f η)etrηκz)), η>0 where the coefficients a f η) continue to satisfy 4), and the bound λ π ξ) τξ) is known by the work of Blasius [3] and Deligne The correct normalisation of a f ) so that F k,f k =is 22) a f ) 2 = i= i= π n /2 DL, sym 2 π) κk n i= 4π) k i Γk i )

19 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS The regularised unfolding integral I φ T ) In this section, we introduce the integral I φ T ) Proposition 9 will follow from comparing two different asymptotic formulae for this integral, the first of which has the main term T n φf k,f k,and the second of which is expressed in terms of shifted convolution sums Choose a positive function h C0 R + ), and let g C0 F + )beitsn-fold product Let g = guy) u O + be the symmetrisation of g under the action of O +, and let 23) Gs, m) = gy)ny s λ m y)dy F + be the Mellin transform of g thought of as a function on F + /O + If F + denotes the multiplicative subgroup of norm elements of F, we may use the formula of Efrat [5] for the volume of F +/O + to invert this, obtaining 24) gy) = 2 n G s, m)ny s λ m y)ds πir m σ) Let T, and define I φ T ) to be the integral ) 25) I φ T )= gty)ny 2 φz) F k z) 2 dx dy F + F/O By symmetrising under the action of O + on F+, substituting 24) and refolding the Eisenstein series, we see that I φ T ) may also be written in the form 26) I φ T )= 2 n G s, m)t ns Es, m, z)φz) F k z) 2 dvds πir m σ) Y The following lemma relates I φ T ) to the inner product φf k,f k Lemma 2 For φ a fixed Hecke-Maass cusp form or pure incomplete Eisenstein series we have 27) I φ T )=C g φf k,f k T n + OT n/2 ), where C g is as in Proposition 9 Proof Starting with equation 26) and moving the contour of integration to the line Res) =/2, we write I φ T )=C g φf k,f k T n + R φ T ) with C g coming from the pole of the Eisenstein series at s = see section 3 for this calculation) R φ T ) is the remaining integral along Res) =/2, R φ T )= pz)φz) F k z) 2 dv, Y with pz) = 2 n G s, m)t ns Es, m, z)ds πir m /2) The Fourier series expansion of Es, m, z) and the bound 2) for K ir give Es, m, z) Ny /2 + Ny n /2 s + m ) n+2 + s + m )Ny /n ) ɛ,

20 070 SIMON MARSHALL so that pz) NyT n/2 if Ny It follows from this and the rapid decay of φz) F k z) 2 that R φ T ) φ,g T n/2 Substituting the formula 27) for I φ T )into25)gives ) 28) C g φf k,f k T n + OT n/2 )= gty)ny 2 φz) F k z) 2 dx dy, F + F/O and we shall express the RHS of this in terms of shifted convolution sums after truncating our fixed form φ Recall that this had a Fourier expansion 29) φz) = ξ O a ξ y)etrξκx)) with the a ξ y) bounded as in Lemma If φ is a pure incomplete Eisenstein series, then we find that the contribution to I φ T ) from the tail of 29) with ξ T +ɛ for any ɛ>0 is bounded by I T )T n/2+a+ɛ τξ) ξ A+ɛ T 3n/2+ɛn+ A) ξ T +ɛ by the support of g and Lemma which is the source of the ɛ ) Here I T )is equal to I φ T ) with φ chosen to be the constant function It follows from this that the contribution of these terms to I φ T )is T n/2 after choosing A sufficiently large with respect to ɛ, and a similar argument works when φ is a fixed cusp form If we define φ to be the truncated function φ z) = a ξ y)etrξκx)), ξ <T +ɛ we therefore have ) 30) C g φf k,f k T n = gty)ny 2 φ z) F k z) 2 dx dy + OT n/2 ) F + F/O 53 Extracting shifted convolution sums In this section we shall expand the RHS of 30) using the Fourier expansion of φ,writing C g φf k,f k T n = S 0 T )+ S ξ T )+OT n/2 ), 0< ξ <T +ɛ where for any ξ Owe define ) S ξ T )= gty)ny 2 a ξ y)etrξκx)) F k z) 2 dx F + F/O dy Note that 5) of Proposition 9 follows from this definition The aim of this section is to analyse the objects S ξ T ) so that when we divide through by C g T n we have the remaining equations and bounds of Proposition 9 We first note that S 0 T )=0whenφ is a cusp form, and by Lemma we have 3) ) φ, S 0 T )= VolY ) + OT n/2 ) I T )

21 MASS EQUIDISTRIBUTION FOR AUTOMORPHIC FORMS 07 when φ is a pure incomplete Eisenstein series We shall treat the two cases I T ) and S ξ T )forξ 0 seperately, beginning with the latter Squaring out F k z) 2 and integrating in x gives 32) S ξ T )= D ) a f η)a f η + ξ) gty)a ξ y)y k 2 e 2πtr2η+ξ)κy) dy η>0 F + As the exponentials and g are positive, this satisfies S ξ T ) a ξ T ) ) a f η)a f η + ξ) gty)y k 2 e 2πtr2η+ξ)κy) dy η>0 F + Appealing to the Mellin transform H of h and applying the normalisations 4) and 22) of a f η) anda f η + ξ), we may integrate in y to obtain 33) S ξ T ) a ξt ) NkL, sym 2 λ π η)λ π η + ξ) π) η>0 ) ki n ) s ηi η i + ξ i ) T Γs + k i ) H s) ds η i + ξ i /2 2πi 4πκ i η i + ξ i /2) Γk i ) i= σ) Note that η i η i + ξ i ) η i + ξ i /2, so that these factors may be omitted We may simplify this expression using a lemma seen in the work of Luo and Sarnak [23] Using Stirling s formula, they prove the asymptotic 34) Γs + k i ) Γk i ) =k i ) s + O a,b s +) 2 k i )) in any vertical strip 0 <a Res) b If we apply this to 33) we may invert the Mellin transform of h to obtain 35) S ξ T ) a ξt ) λ π η)λ π η + ξ) NkL, sym 2 π) n h i= η O η i +ξ i /2 Tk i,i t η>0 T ki ) 4πη i + ξ i /2) ) + O ki ɛ T η i + ξ i /2 ) )) +ɛ The final step in proving 8) from this is showing that when the product in 35) is expanded out, the total contribution from all the error terms is Nk k ν+ɛ T n+ɛ It is enough to consider one such term which contains main term factors at the first t places and error term factors at the last n t As the factors of h truncate the sum at the first t places, the contribution from this term is bounded by 36) λ π η)λ π η + ξ) ) +ɛ ki ɛ T η i>t i + ξ i /2 If we let τ = η + ξ/2, then τ 2 O+ becausewemayassumeη i and η i + ξ i are positive), and because ξ i T +ɛ we have τ i + T 2 maxη i,η i + ξ i ) for all i Therefore by Deligne s bound, λ π η)λ π η + ξ) Nη ɛ Nη + ξ) ɛ τ ɛ + T ɛ,

22 072 SIMON MARSHALL and the expression 36) may be simplified to k ɛ i T +ɛ τ ɛ i τ i Tk i,i t τ ɛ + T ɛ ) i>t Because τ i Tk i for i t, this may be further reduced to Nk ɛ T n t+ɛ τ ɛ i τ i Tk i, i t If we project the set {τ 2 O+ : τ i Tk i for i t} onto the last n t real places, we obtain a set O R n t, any two of whose elements are at a distance δ =T t i t k i) /n t) from each other and the origin The sum above may therefore be bounded by T t k i x ɛ i dx i i t x i δ i>t T t+ɛ i t k +ɛ i, i>t so that the total contribution of our error term is Nk ɛ T n+ɛ i t k+ɛ i Ast<n, we are omitting a factor of size at least k ν from Nk,sothisis Nk k ν+ɛ T n+ɛ, as required Therefore S ξ T ) NkL, sym 2 π) a ξt ) λ π η)λ π η + ξ) η>0 n i= ) T ki ) h 4πη i + ξ i /2) ), + ONk k ν+ɛ T n+ɛ ) which is the bound 8) We now deal with the case ξ =0 Squaringout F k z) 2 and integrating in x gives I T )= D ) a f η) 2 gty)y η>0 F k 2 e 4πtrηκy) dy + Expressing a f in terms of λ π and symmetrising by the action of O +, this becomes I T ) = D a f ) 2 κ ) k λ π η) 2 gty)ηκy) k e 4πtrηκy) dy η>0 F + = D a f ) 2 κ ) k λ π η) 2 gty) ψηκy)dy F + where η)>0 = D a f ) 2 κ k gy) = guy), η)>0 ) λ π η) 2 gty) ψηκy)dy, F + /O + ψy) = uy) k exp 4πtruy)) u O + u O +