Orthogonal Eisenstein series at harmonic points and modular forms of singular weight

We investigate the behaviour of orthogonal non-holomorphic Eisenstein series at their harmonic points by using theta lifts. In the case of singular weight, we show that the orthogonal non-holomorphic Eisenstein series that can be written as a theta lift have a simple pole at s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = 1$$\end{document} whose residues yield holomorphic orthogonal modular forms that are Eisenstein series on the boundary. Moreover, we will investigate the image of this construction and give sufficient conditions for the surjectivity.


Introduction
Let L be an even lattice in a rational quadratic space V of signature (2, l).There is an index 2 subgroup of the corresponding orthogonal group O + (2, l) ⊆ O(2, l) acting on the orthogonal upper half-plane H l .Similar to the case of elliptic modular forms, it is possible to define orthogonal modular forms, which can be seen to be global sections of a hermitian line bundle.For l = 1 we obtain the classical case of elliptic modular forms and for l = 2 we obtain Hilbert modular forms for real quadratic number fields.
An important problem is the construction of such modular forms, in particular for low weight.It turns out that there is a minimal weight for non-zero holomorphic modular forms, which is given by κ = l 2 − 1 for l > 2, see [Bun01].The weight κ = l 2 − 1 is called the singular weight, and modular forms of singular weight have many vanishing Fourier coefficients.In particular, there are no cusp forms of singular weight (for an analogous theory in the Siegel case see [Fre83], where it is shown that every holomorphic modular form of singular weight is a linear The author was partially supported by the LOEWE research unit USAG.Partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124.combination of theta functions).In contrast to the symplectic case, there are no holomorphic theta series for the orthogonal group (except for some low dimensional examples where we have an exceptional isomorphism from the orthogonal group to the symplectic group).Moreover, the usual constructions of holomorphic modular forms do not work in low weight.For example Eisenstein series of low weight do not converge.On the other hand, using the celebrated multiplicative Borcherds lift of [Bor98,Theorem 13.3], examples of holomorphic modular forms of singular weight can be constructed.For results in this direction see [DHS15], [OS19], [Sch17].Another method is to use the additive Borcherds lift of [Bor98, Theorem 14.3] using holomorphic modular forms of weight 0, i.e. invariant vectors, as input functions.
Our aim is to investigate Eisenstein series E κ,λ (Z) of low weight, in particular of singular weight κ = l 2 − 1 by considering the non-holomorphic Eisenstein series E κ,λ (Z, s), get a meromorphic continuation to all s ∈ C and hope that they yield holomorphic modular forms for special values of s.This will be done using the results of [Kie21].
We will give more details now.As above, let L be an even lattice of signature (2, l) and L ′ the corresponding dual lattice.Throughout we will assume that L has Witt rank 2 and that l > 2. For a primitive isotropic vector z ∈ L and z ′ ∈ L ′ with (z, Then H l is a hermitian symmetric domain and there is an index 2 subgroup O + (L ⊗ R) ⊆ O(L ⊗ R) acting on H l .Let Γ(L) be the kernel of the natural map O + (L) → O(L ′ /L).By the theory of Baily-Borel (see [BB66], [BJ06]) we can add 0-dimensional boundary components (points) and 1-dimensional boundary components (isomorphic to H) to obtain H * l , such that the quotient Γ(L)\H * l is a compact complex analytic space, which can be shown to be projective and hence, by the Chow's theorem, algebraic.For an holomorphic orthogonal modular form F : H l → C of weight κ ∈ Z with respect to Γ(L) one can now define its restriction F | I : H → C to a 1-dimensional boundary component I.One easily sees that F | I is a holomorphic modular form of weight κ for an appropriate arithmetic subgroup of SL 2 (Q).In particular, it makes sense to talk about the space M ∂ Eis κ (Γ(L)) of holomorphic orthogonal modular forms that are linear combinations Eisenstein series on the boundary.If F | I vanishes for every boundary component, we say that F is a cusp form and we write S κ (Γ(L)) for the space of cusp forms.
For an easier exposition, we assume throughout the introduction that L splits two hyperbolic planes over Z. Then the 0-dimensional cusps of Γ(L)\H l are in bijective correspondence to the isotropic elements in L ′ /L (which we denote by Iso(L ′ /L)) up to ±1.For an appropriate automorphic form f : H → C[L ′ /L] of weight k consider the additive Borcherds lift defined by [Bor98] Φ(Z, f ) : where Θ L is a certain Siegel theta function of weight k in τ (see Section 6).In [Kie21] we have seen that the additive Borcherds lift Φ k,β (Z, s) of a vector-valued non-holomorphic Eisenstein series E k,β (τ, s), β ∈ Iso(L ′ /L) of weight k for the Weil representation ρ L defined by [BK03] (in [BK03] the dual Weil representation ρ * L is considered) is a linear combination of orthogonal non-holomorphic Eisenstein series G κ,δ (Z, s) of weight κ = l 2 − 1 + k with respect to Γ(L) corresponding to a 0-dimensional cusp δ ∈ Iso(L ′ /L).To obtain holomorphic Eisenstein series one would try to evaluate this at the harmonic point s = 0.By calculating the Fourier expansion, one can split up Φ k,β (Z) := Φ k,β (Z, 0) into a holomorphic part Φ + k,β (Z) and a non-holomorphic part Φ − k,β (Z).We will show Proposition 1.1 (see Proposition 8.6).The holomorphic part Φ + k,β (Z) is given by where N z and N λ are the levels of z and λ, is a divisor sum over positive and negative divisors and b(λ) are certain Fourier coefficients.
Since only the terms with q(λ) = 0 in the Fourier expansion contribute to the restrictions to boundary components, one easily sees that the holomorphic part restricted to a boundary component is given by an Eisenstein series, see [DS05, Theorem 4.2.3] for their Fourier expansions.By inspecting the non-holomorphic part and using that vector-valued non-holomorphic Eisenstein series converge at s = 0 for k > 2 we obtain Theorem 1.2 (see Theorem 8.9).The non-holomorphic part Φ − k,β (Z) vanishes for k > 2 and hence Φ k,β (Z) = Φ + k,β (Z) is a holomorphic orthogonal modular form that is an Eisenstein series on the boundary.Moreover, every holomorphic orthogonal modular form that is a linear combination of Eisenstein series on the boundary is obtained as a theta lift (up to cusp forms).
For k = 0 and hence singular weight κ = l 2 − 1 this does not work immediately, since the non-holomorphic part does not vanish in general.Using the functional equation relating the values at s and 1 − s we will instead inspect the behaviour at s = 1.We obtain Theorem 1.3 (see Theorem 8.10).The additive Borcherds lift Φ 0,β (Z, s) has a simple pole at s = 1 with residue given by where c 0,β (0, 0, s) is a Fourier coefficients of the vector-valued non-holomorphic Eisenstein series E 0,β (τ, s).In particular, it is a holomorphic orthogonal modular form of singular weight that is an Eisenstein series on the boundary.
As in the higher weight case, it is natural to ask whether all holomorphic orthogonal modular forms of singular weight that are linear combinations of Eisenstein series on the boundary can be obtained in this way.Therefore we consider the adjoint theta lift Φ * mapping holomorphic orthogonal modular forms of singular weight to modular forms of weight 0. For an orthogonal modular form F : H l → C it is given by We will show that if F : H l → C is holomorphic of singular weight, then Φ * (τ, F ) is a harmonic function of weight 0 (see Lemma 9.2).In Section 9 we prove the following main theorem.

As a corollary one obtains
Corollary 1.5 (see Corollary 9.6).Assume that L splits two hyperbolic planes.The theta lift Φ * (τ, F ) vanishes if and only if F vanishes in every 0-dimensional cusp.
Since the theta lifts (1) and (2) are adjoint to each other we obtain Corollary 1.6 (see Corollary 9.8).Assume that L splits two hyperbolic planes.Then every holomorphic orthogonal modular form of singular weight for Γ(L) that is an Eisenstein series on the boundary is the residue at s = 1 of some nonholomorphic Eisenstein series and the additive Borcherds lift is an isomorphism Since maximal lattices of Witt rank 2 always split two hyperbolic planes over Z, we obtain for singular weight κ = l 2 − 1.
If L does not split two hyperbolic planes over Z, then we can still fully determine the image of the theta lift (see Corollary 9.7).
Acknowledgment.I would like to thank my advisor J.H. Bruinier for suggesting this topic as part of my doctoral thesis.Moreover, I would like to thank him for his support and our helpful discussions.

Vector-Valued Non-Holomorphic Eisenstein Series
We will now introduce the Weil representation and vector-valued modular forms.Therefore, let H := {τ = u + iv ∈ C | v > 0} be the usual upper half-plane.For z ∈ C we write e(z) := e 2πiz and we denote by where ( a b c d ) τ = aτ +d cτ +d is the usual action of SL 2 (R).By Mp 2 (Z) we denote the inverse image of SL 2 (Z) under the covering map.It is generated by We have the relation S 2 = (ST ) 3 = Z, where Z = −1 0 0 −1 , i is the standard generator of the center of Mp 2 (Z).Furthermore we will write if c = 0 and by √ i For a vector-valued function f : is smooth and invariant under the action of T , i.e. f | k,L T = f , then we have a Fourier expansion We call f a holomorphic modular form if f is holomorphic with Fourier expansion Obviously, non-trivial modular forms only exist for weights with 2k + b − − b + = 0 mod 2.
Assume now that b + − b − is even and let k ∈ Z. Moreover set κ = b − −b + 2 + k.Let β ∈ Iso(L ′ /L) and similar to [BK03] define the vector-valued non-holomorphic Eisenstein series of weight k by Observe that [BK03] consider the dual Weil representation ρ L .For β ∈ Iso(L ′ /L) of order N β and a character χ : (Z/N β Z) × we define More generally, for v ∈ Iso(C[L ′ /L]) we define We have E k,v * = (−1) κ E k,v and a Fourier expansion of the form where W s is a special Whittaker function.For the precise coefficients see [BK03], [Wil19] (or [BK01], [Sch06], [Sch18] for the holomorphic case), we will not need them here.It can be easily seen that the vector-valued Eisenstein series are Mp 2 (Z)translates of usual scalar-valued Eisenstein series.They are normalized such that their meromorphic continuation to C are holomorphic in s = 0 and have a simple pole for k = 0 at s = 1 whose residue is an invariant vector.For an invariant vector , where E k (τ, s) is the suitably normalized Eisenstein series for SL 2 (Z).For k > 2 the series converges at s = 0 and defines a holomorphic Eisenstein series For k = 2 they have a Fourier expansion of the form [Miy06], [DS05] Applying the lowering operator shows that w ∈ Inv(C[L ′ /L]) is an invariant vector.
For k = 0 their residue at s = 1 yields an invariant vector and if v ∈ Inv(C[L ′ /L]) we obtain res s=1 E 0,v (τ, s) = res s=1 E 0 (τ, s)v, in particular these residues span the space of invariants.

Orthogonal Modular Forms
From now on let L be an even lattice of signature (2, l) and let V = L⊗Q, V (R) = V ⊗ R. Write P (V (C)) for the corresponding projective space.For elements is the hermitian symmetric domain associated to O(V ).It has two connected components which are interchanged by We choose one of them and call it K + .The action of O(V (R)) on V (R) induces an action on K. Let O + (V (R)) be the subgroup which preserves the connected components of K and let be the preimage of K + under the projection.Write Iso 0 (L) for the set of primitive isotropic elements of L. For z ∈ Iso 0 (L) and Then d, d, d 3 , . . ., d l is a basis of K ⊗ R. We define the orthogonal upper half plane as We will write Z = z 1 d+z 2 d+Z D with Z D ∈ D⊗C and analogously for X, Y ∈ W (R).
For Z ∈ H l we define Z L := Z − q(Z)z + z.
be the factor of automorphy, so that we have and the cocycle relation According to [Bru02, Lemma 3.20] we have For a modular form F : K+ → C of weight κ ∈ Z with respect to Γ define for all σ ∈ Γ and we have a bijective correspondence between modular forms and functions with this transformation property.If we define the weight κ slash operator then the modular forms on H l are exactly the functions that are invariant under the slash operator for all σ ∈ Γ.If F z is holomorphic on H l it has a Fourier expansion of the form where e(λ, Z) := e((λ, Z)) = e 2πi(λ,Z) .
Definition 3.2.We say that a modular form f is a holomorphic modular form if f is holomorphic and for all 0 dimensional cusps z ∈ Iso 0 (L) we have a z (λ) = 0 for λ / ∈ C. We write M κ (Γ) for the space of modular forms.
Remark 3.3.For l ≥ 4 the weight κ = l 2 − 1 is called singular weight.By the theory of singular weights, it is the smallest positive weight such that there are holomorphic modular forms for Γ(L).Moreover, if is a holomorphic modular form of singular weight, then a z (λ) = 0 implies λ ∈ C. Definition 3.4.For t > 0 we define the Siegel domain S t as the set of Z = X +iY ∈ H l satisfying We have the following Proposition 3.5 ([Bru02, Proposition 4.10]).Let Γ ⊆ Γ(L) be a subgroup of finite index.
(i) For any t > 0 and any is finite.(ii) There exists a t > 0 and finitely many g 1 , . . ., g n ∈ O + (V ) such that for The invariant volume element on H l is given by dXdY q(Y ) l .
The previous proposition means in particular, that for a Γ-invariant measurable function F : is finite.We will need the following estimates.

Siegel Operator
For an isotropic line I ⊆ V (R) generated by some isotropic vector λ ∈ I the corresponding point [λ] is in the closure of K + in N .To see this, take two sequences x n , y n ∈ V (R) with positive norm which are orthogonal and converge to λ.Then Proof.Let I ⊆ V (R) be an isotropic plane.Take a basis z, d of I and consider z, d isotropic such that (z, z) = (d, d) = 1 and all other products vanish.We will use the shorthand notation (z 1 , z 2 , z 3 , z 4 ) for z 1 z + z 2 z + z 3 d + z 4 d.Then the elements of I have the form (0, z 2 , 0, z 4 ).Assume that this is not a multiple of a real point, i.e. it is not a special boundary points.Then z 2 = 0 = z 4 and we can normalize it such that z 4 = 1, i.e. we have a point of the form (0, τ, 0, 1) for some τ ∈ C \ R. making suitable choices of the basis and K + we can assume that [1, 1, i, i] ∈ K + .Then we have an embedding In the projective space we have The set of all boundary points represented by elements in By the theory of Baily-Borel (see [BB66], [BJ06]), there is a topology on H * l such that for congruence subgroups Γ ⊆ O + (V ) the quotient X Γ = H * l /Γ carries the structure of a projective variety, which contains H l /Γ as a Zariski open subvariety.
Let now I ⊆ L ⊗ Q be an isotropic plane.Let z ∈ L ∩ I be primitive and and that we write Assume that F has a Fourier expansion of the form then the Siegel operator exists and we have The value in the 0-dimensional cusp z is given by the constant term of the Fourier expansion, i.e.
for arbitrary Z ∈ H l .
Let now F : K+ → C be a holomorphic modular form of weight κ for some congruence subgroup Γ.Then its restriction to the boundary component F z | I : H → C is a holomorphic modular form of weight κ for some subgroup of SL 2 (Q).One easily sees that for the constant Fourier coefficient we have a −z (0) = (−1) κ a z (0).Definition 4.5.We call a holomorphic modular form F : H l → C of weight κ for some congruence subgroup Γ a cusp form if F | I vanishes identically for all boundary components I.We say that F is an Eisenstein series on the boundary if F | I is an Eisenstein series for all boundary components I.We write S κ (Γ) for the space of cusp forms and M ∂ Eis κ (Γ) for the space of holomorphic modular forms that are Eisenstein series on the boundary.In particular we have Remark 4.6.By the theory of singular weights, there are no cusp forms of singular weight.In particular, a holomorphic modular form of singular weight is completely determined by its values at the boundary and a holomorphic modular form of singular weight that is an Eisenstein series on the boundary is completely determined by its values in the 0-dimensional cusps.
Remark 4.7.If a holomorphic modular form F is an Eisenstein series on the boundary, then its restrictions to the boundary are fully determined by the values in the 0-dimensional cusps, i.e. the constant Fourier coefficients.Definition 4.8.For Γ = Γ(L) recall the map We write M π κ (Γ(L)) for the subspace of M ∂ Eis κ (Γ(L)) that consists of holomorphic modular forms whose values in the 0-dimensional cusps only depend on their image in L ′ /L.In particular, if δ = −δ ∈ L ′ /L, the value in the 0-dimensional cusps corresponding to δ vanish if κ is odd.We have We want to mention the following result which is sometimes called Eichler criterion.

Differential Operators
Let M be a hermitian manifold and E a holomorphic vector bundle.We write E(E) = E(M, E) for the space of (global) sections of E and, more generally, for U ⊆ M open, we write E(U, E) for the space of sections over U. Assume now that E carries an hermitian metric.Then the hermitian metric on E induces a hermitian metric on E-valued differential forms A * (M, E) locally given by on decomposable forms, where the first factor is the hermitian metric on differential forms coming from the hermitian metric on M and the second factor is the hermitian metric on the vector bundle E.Moreover, the hermitian metric on E together with the Hodge- * -operator * : The evaluation map E ⊗ E * → C induces a wedge product ∧ which satisfies where vol is the volume form induced from the hermitian metric on M. Moreover we have a hermitian inner product on E-valued differential forms given by be the Laplace operator on sections of E. We will need the following Theorem 5.1 ([Che73, Section 3, Example (B)]).Let M be a complete connected hermitian manifold and let E be an hermitian vector bundle.If u, v are smooth square integrable sections of E such that ∆ E u, ∆ E v are also square integrable, then Let now L be an even lattice of signature (2, l).For z ∈ Iso 0 (L) and The Kähler form is given by where h(Z) = h(Y ) = (h ij ) 1≤i,j≤l is the associated hermitian form given by and its inverse is Its determinant is det(h) = 1 2 l q(Y ) l and hence the volume form is given by We write and similarly for dz i .Then we have Now the Hodge- * -operator is defined by the equality and thus we have Modular forms of weight κ form a line bundle L κ which carries a hermitian metric given by the Petersson metric, i.e.F (Z)G(Z)q(Y ) κ on the fiber.The dual bundle L * κ can be identified using the hermitian metric with the line bundle L −κ of modular forms of weight −κ and the mapping F (Z) → q(Y ) κ F (Z) defines an anti-linear bundle isomorphism.This gives us the Hodge- * -operator * κ : The weight κ Laplace operator is then given by Theorem 5.2.For a modular form F of weight κ the weight κ Laplace operator is given by Proof.We have Applying * −κ ∂ yields A short calculation yields Thus, for i = 1 we have and similarly for i > 1 and hence Remark 5.3.There is an ad hoc definition of the weight (m, n) Laplace operator given by [Zem15], [Zem17] which coincides with 4Ω κ for m = κ, n = 0.In particular Ω κ commutes with the weight κ slash operator and satisfies

Siegel Theta Function
Let p be a polynomial on R (b + ,b − ) which is homogeneous of degree κ in the positive definite variables and independent negative definite variables.For an isometry ν : L⊗R → R (b + ,b − ) we write ν + and ν − for the inverse image of R (b + ,0) and R (0,b − ) .For an element λ ∈ L ⊗ R we write λ ν ± for the projection of λ onto ν ± .The positive definite majorant associated to ν is then given by q ν (λ) = q(λ ν + ) − q(λ ν − ).For γ ∈ L ′ /L, τ ∈ H and an isometry ν : L ⊗ R → R (b + ,b − ) we define the Siegel theta function × e(τ q((λ where ∆ is the usual Laplace operator on R b + +b − and α, β ∈ L ⊗ R. Moreover we define For α = β = 0 we write and Using Poisson summation one obtains In particular, for α = β = 0, the theta function Θ L (τ, ν, p) has weight [Bor98] that the theta function can be written as a Poincaré series.We will indicate the construction.Write Iso 0 (L) for the primitive isotropic elements of L and let z ∈ Iso 0 (L), z ′ ∈ L ′ with (z, z ′ ) = 1.Let N z be the level of z and define the lattice

Borcherds shows in
. For a vector λ ∈ L ⊗ R we write λ K for its orthogonal projection to K ⊗ R, which is given by

Consider the sublattice
and the projection This projection induces a surjective map L ′ 0 /L → K ′ /K which we also denote by π and we have we write ω ± for the orthogonal complement of z ν ± in ν ± .This yields a decomposition and for λ ∈ L ⊗ R we write λ ω ± for the corresponding projections of λ onto ω ± .Additionally the map can be seen to be an isometry K ⊗R → R (b + −1,b − −1) by restriction.For a polynomial p on R (b + ,b − ) as above we now define the homogeneous polynomials p ω,h of degree κ − h in the positive definite variables by We have the following We will need the following e aλ (abq(aλ)) e aγ (abq(aγ))e amz Nz = (ρ K (M)e γ )e amz Nz .
Multiplying by e − mn Nz and summing over m ∈ Z/N z Z yields the result.Let now c = 0. Again, Shintani's formula yields where Now the last sum vanishes by orthogonality of characters unless in which case it sums to c.This yields We now multiply this by e − mn Nz and sum over m ∈ Z/N z Z to obtain The latter sum again vanishes unless in which case it is equal to N z .In particular we have (z, β + ncz ′ ) ≡ 0 mod N z and thus β + ncz ′ ∈ L ′ 0 /L.But every element in L ′ 0 /L can be written as α + mz Nz for some α ∈ K ′ /K and m ∈ Z/N z Z, which shows that (β + ncz ′ , z Nz ) = 0.But this implies k = 0. Hence we obtain which shows the claim.
Theorem 6.4.We have Proof.We have We make the change γ → γ + cz ′ and sum over coprime c, d to obtain The elements γ ∈ L ′ 0 /L are represented by γ + mz Nz for γ ∈ K ′ /K, m ∈ z Z. Hence we obtain using the transformation formula for the theta function where M = ( * * c d ) ∈ SL 2 (Z).This yields Let now L be an even lattice of signature (2, l) with l ≡ 0 mod 2. Let κ = l 2 −1+k and p(x 1 , x 2 ) = (x 1 + ix 2 ) κ .Recall the identification ν : 0) and by abuse of notation we will write ν Z L for every isometry which equals ν Z L on ν(Z L ).For λ ∈ V (R) we write λ Z L and λ Z ⊥ L for the projection of λ to ν(Z L ) and ν(Z L ) ⊥ .Then q(λ) = q(λ Z L ) + q(λ Z ⊥ L ) and we denote by q Z L (λ) = q(λ Z L ) − q(λ Z ⊥ L ) the positive definite majorant.Now, the map is well-defined, since p only depends on the positive definite variables.Thus we define and we expand Θ K (τ, Y /|Y |, p Y,0 ) in the first summand with respect to d, d ′ , D to obtain Observe that the second and third summand can be rewritten using the weight k slash operator | k,L .We want to get bounds for Θ L (τ, Z) and Ω k Θ L (τ, Z).According to [Zem15, Proposition 2.5] we have where Since the Laplace operator commutes with the slash operator, we only have to find bounds for f, ∆ k f and g, ∆ k g.We will need the following elementary lemma.
(i) The function x n exp(−ax 2 ) has a maximum given by n x has a minimum on x > 0 given by 2 √ bc We start with a bound for Observe that the absolute value of f and ∆ k f can be bounded by finite sums of the form We have Lemma 6.6.Let Assume that v ∈ R >0 , y > C > 0 and n ≥ 1.Then we have the bound where the implied constant is independent of v, y, n.In particular, we obtain with Proof.We split up the exponential term and use Lemma 6.5 to obtain This yields where we have used Lemma 6.5 again.
Next consider The theta function is given by The terms are given by a finite sum of constants times terms of the form Again, we see that the absolute value of g h and ∆ k g h can be bounded by a finite sum of terms of the form Lemma 6.7.Let g h,i,j,l (v, Y, n) denote the function where the implied constant is independent of v, Y, n.In particular, we have and and by Lemma 3.6 the exponential term can be bounded by ) .Again we split up the exponential term Lemma 6.5 to obtain We obtain where we have used Lemma 6.5 again.Now use and q(Y ) > t −4 on every Siegel domain S t to obtain y 1 for all Z ∈ S t and τ ∈ H with Im(τ ) > C and some s ∈ R. The constant only depends on the constant C. Similarly, the function Proof.This is now a direct consequence of the previous two Lemmas together with the fact that the Laplace operator ∆ k commutes with the slash operator | k,L .Corollary 6.9.Both, the theta function Θ L and Ω κ Θ L are square-integrable for l ≥ 3 and κ = l 2 − 1 + k > 0. Proof.The square is bounded by q(Y ) 1−κ and hence we have to show that St q(Y ) dXdY q(Y ) l < ∞.
By Lemma 3.7 this is the case if l ≥ 3.
For a vector-valued modular form f : Here, the regularization is defined by the constant term at t = 0 of the Laurent expansion of lim where F T is a truncated fundamental domain.For the lift of the vector-valued nonholomorphic Eisenstein series E k,β (τ, s) we write Φ k,β (Z, s).We have the following Theorem 7.1 ([Kie21, Theorem 8.1]).The theta lift is equal to where k δβ ∈ Z/N δ Z with β = k δβ δ (and the corresponding summands vanish if such a k δβ does not exist), N λ is the level of λ and N δ is the order of δ.
Hence, instead of evaluating the Eisenstein series E κ,λ at s = 0, we will instead consider the theta lifts Φ k,β .Their Fourier expansion is given by Theorem 7.3 ([Kie21, Theorem 8.6]).Let z ∈ Iso 0 (L) of level N z and let z ′ ∈ L ′ with (z, z ′ ) = 1.The theta lift Φ k,β (Z, s) has the Fourier expansion in the 0dimensional cusp z given by where

Theta Lifts at Harmonic Points
We will now consider the theta lifts at their harmonic points s = 0 and for k = 0 at s = 1.We set . if y > 0 and the left hand side vanishes otherwise.
Proof.This is done in [Bor98,Theorem 14.3].We use the formula and observe that the terms with m = 0, j = 0 cancel out according to [Bor98, Corollary 14.2].Hence we obtain Now the sum over h vanishes for y < 0 and is equal to 2 κ for y > 0, which shows the result.Now we can calculate the Fourier coefficient for q(λ) ≥ 0 at s = 0.
We now plug in s = 0.According to [BK03] we have and use the formula [EMOT54, p. 313, 6.
Definition 8.3.Define the holomorphic part Φ + k,β (Z) of the theta lift by Remark 8.4.Assume that E k,v (τ ) := E k,v (τ, 0) is holomorphic.Then Φ − v (Z) vanishes identically and hence Φ v (Z) = Φ + v (Z) is a holomorphic modular form.See also [Bor98,Theorem 14.3].Write M Φ κ (Γ(L)) for the space of holomorphic modular forms of weight κ that are given as a theta lift Φ v (Z) for some v ∈ Iso(C[L ′ /L]).Definition 8.5.Define the holomorphic boundary part of the theta lift Φ ∂+ k,β (Z) to be Proposition 8.6.We have In particular, for an isotropic plane I ⊆ L ⊗ Q with I = z, d , we have, writing Nz , (if such a decomposition exists it is unique and if it does not exist then Φ ∂+ k,β | I vanishes identically) which is the holomorphic part of an Eisenstein series on the boundary component I (see [DS05], [Miy06]).Here for c ∈ Z/N λ Z, b ∈ Z/N z Z we have the divisor sum where the sum is over positive and negative divisors.Observe that the constant term only depends on the image of z Nz in L ′ /L.Proof.For λ ∈ K primitive let N λ be its level.Then we can rewrite the second summand as Summing over positive and negative divisors in the divisor sum we can rewrite the holomorphic boundary part as Let now I ⊆ L ⊗ Q be an isotropic plane with I = z, d , d ∈ Iso 0 (K) and consider the Siegel operator corresponding to this plane.Then Nz (if such a decomposition exists it is unique and if it does not exist then Φ ∂+ k,β | I vanishes identically), we obtain which shows the result.
Proposition 8.7.Assume that the map π is surjective.Then the theta lift is injective.
Proof.Assume that there is some v ∈ Iso(C[L ′ /L]) with Φ v (Z) = 0. Then in particular, Φ ∂+ v (Z) = 0 for every 0-dimensional cusp z, i.e. for all 0-dimensional cusps z.We want to show that E k,v (τ, 0) = 0, which is equivalent to v + (−1) κ v * = 0. Therefore, assume there is some δ ∈ Iso(L ′ /L) with v δ = −(−1) κ v −δ .By surjectivity of π there is a 0-dimensional cusp z corresponding to δ. Choose such δ with minimal order.Then by assumption the value in the But of course this is also true for the 0-dimensional cusps corresponding to cδ for c ∈ (Z/N z Z) × , i.e. we have Rewrite this using For a character χ : (Z/N z Z) × → C × consider now Since L(χ, κ) = 0 we have Of course, if π is not surjective, then the theta lift is not injective, i.e. the converse of the previous theorem is also true.Proposition 8.8.For κ even and every δ ∈ L ′ /L isotropic there is a theta lift Φ k,v for some v ∈ Iso(C[L ′ /L]) such that the holomorphic part vanishes in all 0dimensional cusps except for the 0-dimensional cusps corresponding to ±δ.For κ odd this is true if δ = −δ.Moreover, in this case we have Φ k,v = G κ,δ .
Proof.If δ generates a maximal cyclic isotropic subgroup, then the holomorphic part of Φ cδ , c ∈ (Z/N δ Z) × vanishes in every 0-dimensional cusp except for the 0dimensional cusps corresponding to the generators of δ .In the 0-dimensional cusps corresponding to bδ for b ∈ (Z/N δ Z) × the value is given by i.e. the value in the 0-dimensional cusps vanishes except for the 0-dimensional cusps corresponding to δ, where the value is 1.Moreover, using Theorem 7.1 shows that this linear combination is in fact given by G κ,δ .Now do induction over the maximal length of chains of cyclic isotropic subgroups containing δ.
This means in particular that for F ∈ M π κ (Γ(L)) there is a theta lift Φ v for some v ∈ Iso(C[L ′ /L]) whose holomorphic boundary part is given by the boundary part of F (observe that for κ odd the values in 0-dimensional cusps corresponding to δ ∈ Iso(L ′ /L) with δ = −δ must be zero).
Theorem 8.9.Let k > 2 and thus κ > l 2 + 1.Then Φ k,β (Z) = Φ + k,β (Z) is a holomorphic modular form of weight κ which is an Eisenstein series on the boundary.In particular we have M π κ (Γ(L)) = S κ (Γ(L)) + M Φ κ (Γ(L)) in this case.Proof.Using that the coefficients c k,β (γ, n, 0) vanish for n ≤ 0 one obtains the result using Theorem 7. We obtain the Fourier expansion of the Eisenstein series E κ,v (Z) and that the Eisenstein series is injective, then we obtain all holomorphic orthogonal Eisenstein series as a lift of vector-valued Eisenstein series E k,v (τ ).
In the 0-dimensional cusp z we have the expansion In particular, every invariant vector yields a holomorphic modular form of singular weight which is an Eisenstein series on the boundary.If κ = 1, l = 4 we obtain the additional summand res s=1 Φ K k,β (Y /|Y |, s), which can be shown to be a constant.Proof.One observes that the terms with q(λ) = 0 are holomorphic in s = 1 and hence their residue vanishes (this follows from the corresponding result for vectorvalued Eisenstein series).The calculation for the other Fourier coefficients is analogous to the case for s = 0.The question if this yields all holomorphic modular forms of singular weight which are Eisenstein series on the boundary will be answered in the next section.We want to mention that we can also construct these in a different way.For an invariant where E k (τ, s) is the usual suitably normalized scalar-valued Eisenstein series for SL 2 (Z).Now the left-hand-side is holomorphic in s = 0 and equal to a multiple of v and the lift of E k (τ, s)v is holomorphic in s = 0 and yields a holomorphic modular form for s = 0 as in the case k > 2.

Lifting Holomorphic Orthogonal Modular Forms
Let L be an even lattice of signature (2, l), l ≥ 3.For z ∈ Iso 0 (L) and z , where σ λ K λ acts via translation and Γ(σ λ K λ ) via multiplication on H l = K z ⊗R+iC.A fundamental domain is given by F = F 1 +iF 2 , where F 1 is a fundamental domain of the action σ λ K λ on K z ⊗R and F 2 is a fundamental domain of the action Γ(σ λ K λ ) on C. By abuse of notation we will write σ λ K λ \K z ⊗ R + iΓ(σ λ K λ )\C.Moreover, recall the map π L : Γ(L)\ Iso 0 (L ′ ) → Iso(L ′ /L).Let F : H l → C be a modular form of weight κ.We define its theta lift to be (if it exists) q(Y ) l .Proposition 6.8 and Lemma 3.7 show that the theta lift exists for holomorphic modular forms of singular weight (in fact the lift exists for weight κ < l − 1 and for arbitrary weight if F is a cusp form).A straight forward calculation yields Lemma 9.1.Let F : H l → C be a holomorphic modular form of singular weight κ where the left hand side denotes the Petersson inner product on invariant vectors and the right hand side denotes the Petersson inner product on holomorphic modular forms of singular weight.In particular, the theta lifts are adjoint to eachother.
Proof.Let Γ ⊆ Γ(L) be a finite index subgroup which acts freely.Then Γ\H l is a complete connected hermitian manifold.By Corollary 6.9, Θ L and Ω κ Θ L are square-integrable.Using the assumption on F and Ω κ F , we can apply Theorem 5.1 to square-integrable sections of the hermitian line bundle of modular forms of weight κ.This yields, using Ω κ Θ L = ∆ 0 Θ L and Theorem 5.1, The other assertions follow immediately.
Hence we obtain In particular, this is independent of v. Hence Φ * (τ, F ) is bounded and since it is harmonic of weight 0 it is an invariant vector.
Remark 9.4.Observe that Φ * (τ, F ) only depends on the values in the 0-dimensional cusps.In particular it vanishes on functions that are zero in every 0-dimensional cusp.
Remark 9.5.Let F : H l → C be a cusp form of arbitrary weight.Then Φ * (τ, F ) exists and one can show as above that it is harmonic.Moreover, the same calculation as in the previous proof shows that the constant Fourier coefficient vanishes.Hence, Φ * (τ, F ) decays exponentially and hence is square-integrable and harmonic of weight k.Thus, Φ * (τ, F ) is a holomorphic cusp form.This reproduces the result of [Oda78].
Proof.If Φ * (τ, F ) vanishes, then the coefficient of e γ , given by δ∈L ′ /L q(δ)=0 γ=k δ δ ζ k δ + (l − κ)a F,δ (0), vanishes for all isotropic γ ∈ L ′ /L.In particular, if γ ∈ L ′ /L is isotropic with maximal order such that a F,γ (0) = 0, then for k ′ ∈ (Z/N γ Z) × the coefficient of e k ′ γ k∈(Z/Nγ Z) × Moreover we write •, • for the standard inner product on C[L ′ /L] which is anti-linear in the second variable.Write Iso(L ′ /L) ⊆ L ′ /L for the set of isotropic elements and denote the subspace of vectors that are supported on isotropic elements by Iso(C[L ′ /L]).Moreover we introduce the notation e γ (τ ) := e(τ )e γ = e 2πiτ e γ .Definition 2.2.The Weil representation is the unitary representation ρ L of Mp 2 (Z) on C[L ′ /L] defined by ρ L (T )e γ := e γ (q(γ)) and ρ L (S)e γ := √ i b − −b + L ′ /L δ∈L ′ /L e δ (−(γ, δ)).The Weil representation factors through a finite quotient of Mp 2 (Z).The space of invariant vectors under the Weil representation is denoted by Inv(C[L ′ /L]).A short calculation using orthogonality of characters shows for example ρ L (Z)e γ = i b − −b + e −γ .For β, γ ∈ L ′ /L we define the coefficients ρ β,γ (M, φ) := ρ L (M, φ)e γ , e β and ρ −1 β,γ (M, φ) := ρ −1 L (M, φ)e γ , e β .Theorem 2.3 ([Shi75, Proposition 1.6]).For M ∈ SL 2 (Z) the coefficient ρ β,γ ( M ) is given by √ i (b − −b + )(1−sgn(d)) δ β,aγ e(abq(β)) Definition 4.1.A boundary point of K + of the form [λ] ∈ N which is represented by a real isotropic line is called special boundary point.A set consisting of one special boundary points is called 0-dimensional boundary component.A non-special boundary point is called generic boundary point.Let now I ⊆ V (R) be an isotropic plane and consider the set of all boundary points which can be represented by elements of I ⊗ C. Definition 4.2.For an totally isotropic plane I ⊆ V (R) the set of all generic boundary points which can be represented by an element of I ⊗ C is called 1dimensional boundary component attached to I. By a boundary component we mean a 0-dimensional or 1-dimensional boundary component.Lemma 4.3.The 1-dimensional boundary components are isomorphic to usual upper half-planes.Moreover, there is a bijective correspondence between boundary components and non-zero isotropic subspaces of V (R).
)d and let d 3 , . . ., d l be a basis of D. Then we obviously have I = z, d .Recall the orthogonal upper half plane corresponding to z we define the Siegel operator corresponding to the boundary component I as (if it exists) Theorem 6.1 ([Bor98, Theorem 4.1]).For (M, φ) ∈ Mp 2 (Z), M = ( a b c d ) we have Θ L (Mτ, aα + bβ, cα + dβ, ν, p) = φ(τ The function x n exp(−ax) has a maximum on x ≥ 0 given by n ae n .(iii) The function bx + c 3 and Lemma 8.2 together with [Bor98, Theorem 10.3] for the term Φ K k,β (Y /|Y |).Of course, this reproduces the result of [Bor98, Theorem 14.3].