Local Maa{\ss} forms and Eichler--Selberg type relations for negative weight vector-valued mock modular forms

By comparing two different evaluations of a modified (\`{a} la Borcherds) higher Siegel theta lift on even lattices of signature $(r,s)$, we prove Eichler--Selberg type relations for a wide class of negative weight vector-valued mock modular forms. In doing so, we detail several properties of the lift, as well as showing that it produces an infinite family of local (and locally harmonic) Maa{\ss} forms on Grassmanians in certain signatures.


Introduction
Theta lifts have a storied history in the literature, receiving a vast amount of attention in the past few decades with applications throughout mathematics.In this paper, we are concerned with generalizations of the Siegel theta lift originally studied by Borcherds in the celebrated paper [3].The classical Siegel lift maps half-integral weight modular forms to those of integral weight, and has seen a wide number of important applications.For example, in arithmetic geometry [15,23], deep results in number theory [12], fundamental work of Bruinier and Funke [10], among many others.
More recently, Bruinier and Schwagenscheidt [13] investigated the Siegel theta lift on Lorentzian lattices (that is, even lattices of signature (1, n)), and in doing so provided a construction of recurrence relations for mock modular forms of weight 3  2 , as well as commenting as to how one could provide a similar structure for those of weight 1  2 , thereby including Ramanujan's classical mock theta functions.
In the last few years, several authors have also considered so-called "higher" Siegel theta lifts of the shape (k : where an iterated version of the Maaß raising operator R κ := 2i ∂ ∂τ + κ v , f is weight k − 2j harmonic Maaß form, and Θ L is the standard Siegel theta function associated to an even lattice L of signature (1, n).Here and throughout, τ = u + iv ∈ H and z ∈ Gr(L), the Grassmanian of L. Furthermore, •, • denotes the natural bilinear pairing.For example, they were considered by Bruinier and Ono (for k = 0, j = 1) in the influential work [11], by Bruinier, Ehlen, and Yang in in the breakthrough paper [9] in relation to the Gross-Zagier conjecture, and by Alfes-Neumann, Bringmann, Schwagenscheidt and the first author in [1] for n = 2 and generic j.
In [33], Mertens investigated the classical Hurwitz class numbers, denoted by H(n) for n ∈ N. Using techniques in (scalar-valued) mock modular forms, he gave an infinite family of class number relations for odd n, two of which are s∈Z where λ k (n) = 1 2 d|n min(d, n d ) k and σ k is the usual k-th power divisor function.Because of their close similarity to the classical formula of Kronecker [29] and Hurwitz [24,25] s∈Z and those arising from the Eichler-Selberg trace formula, Mertens referred to the relationships (1.1) as Eichler-Selberg type relations.More generally, let [•, •] ν denote the ν-th Rankin-Cohen bracket (see Section 2).In general, the Rankin-Cohen bracket [f, g] is a mixed mock modular form of degree ν.It is of inherent interest to determine its natural completion, say Λ, to a holomorphic modular form.Then following Mertens [34], we say that a (mock-) modular form f satisfies an Eichler-Selberg type relation if there exists some holomorphic modular form g and some form Λ such that [f, g] ν + Λ is a holomorphic modular form.In the influential paper [34], Mertens showed the beautiful result that all mock-modular forms of weight 3  2 with holomorphic shadow satisfy Eichler-Selberg type relations, using the powerful theory of holomorphic projection and the Serre-Stark theorem stating that unary theta series form a basis for the spaces of holomorphic modular forms of the dual weight 1  2 . 1 In particular, Mertens explicitly describes the form Λ which completes the Rankin-Cohen brackets.
Following previous examples, to demonstrate the statement, let H denote the generating function of Hurwitz class numbers, let ϑ = n∈Z q n 2 , where τ ∈ H, and q n = e 2πinτ throughout.Then Mertens' results show that [34, pp. 377] is a holomorphic modular form of weight 2ν + 2 for all ν ≥ 1, and a quasimodular form of weight 2 if ν = 0.In [32], the first author combined techniques of [1,13] during a further investigation of the higher Siegel lift on Lorentzian lattices.This lift was shown to be central in producing certain Eichler-Selberg type relations in the vector-valued case, providing an analogue of the scalar-valued weight 3 2 case of Mertens.We remark that the shape of the form Λ in the case of signature (1, 1) is very close to that of Mertens (see [32,Theorem 1.1]), though we do not recall it here to save on complicated definitions in the introduction.
In the current paper, we develop the theory for even generic signature (r, s) lattices L and more general modified Siegel theta functions as in Borcherds [3], and consider the lift 1 Mertens also provides results for mock theta functions in weight 1 2 , but since there is no analogue of Serre-Stark in the dual weight 3  2 this is a real restriction.
where Θ L is a modified Siegel theta function as in Borcherds [3], essentially obtained by including a certain polynomial p in the summand of the usual vector-valued Siegel theta function.We require p to be homogenous and spherical of degree d + ∈ N 0 in the first r variables, and d − ∈ N 0 in the last s variables (see Section 2.6 for precise definitions).Here, ψ is an isometry which in turn defines z -see Section 2.6.Modifying the theta function in this way preserves modular properties of Θ L , while allowing us to obtain different weights of output functions.Furthermore, since the case j = 0 is well-understood in the literature, we assume throughout that j > 0. We remark that the signature (1, 2) with j = 0 case has also been studied in [17,18].
In particular, we evaluate the higher lift in the now-standard ways of unfolding in Theorem 3.2, as well as recognizing it as a constant term in the Fourier expansion of the Rankin-Cohen bracket of a holomorphic modular form and a theta function (up to a boundary integral that vanishes for a certain class of input functions) in Theorem 3.3.For the second of these theorems, we use that at special points w, one may define positve-and negative-definite sublattices P := L ∩ w and N := L ∩ w ⊥ .In the simplest case, which we assume for the introduction, we have that L = P ⊕ N .Then the theta series splits as Θ L = Θ P ⊗ Θ N , where Θ P is a positive definite theta series, and Θ N a negative definite one.Then we let G + P be the holomorphic part of a preimage of Θ P under ξ κ := 2iv κ ∂ ∂τ .For the sake of simplicity, we assume that G + P + g in the statement of Theorem 1.1 is bounded at i∞ in the introduction; we overcome this assumption in Theorem 3.4 and offer a precise relation there.Following the ideas of [32], by comparing these two evaluations of our lift and invoking Serre duality, we obtain the following theorem.
Theorem 1.1.Let L be an even lattice of signature (r, s), with associated Weil representation ρ L .Let g be any holomorphic vector-valued modular form of weight 2 − r 2 + d + for ρ L .Suppose that G + P + g is bounded at i∞.Then G + P + g satisfies an explicit Eichler-Selberg type relation.In particular, the form Λ is explicitly determined.
The concept of so-called locally harmonic Maaß forms was introduced by Bringmann, Kane, and Kohnen in [5].These are functions that behave like classical harmonic Maaß forms, except for an exceptional set of density zero, where they have jump singularities.Since their inception, locally harmonic Maaß forms have seen applications throughout number theory, for example in relation to central values of L-functions of elliptic curves [22], as well as traces of cycle integrals and periods of meromorphic modular forms [1,31] among many others.Examples of such locally harmonic Maaß forms are usually achieved in the literature via similar theta lift machinery to that studied here.In addition to the direction of Theorem 1.1, we also discuss the action of the Laplace-Beltrami operator on the lift Ψ reg j in Theorem 4.2.In doing so, we prove the following theorem, thereby providing an infinite family of local Maaß forms (and locally harmonic Maaß forms) in signatures (2, s).To state the result, we let F m,k−2j,s be a Maaß-Poincaré series as defined in Section 2.4.Theorem 1.2.Let L be an even isotropic lattice of signature (2, s), and assume that the polynomial p inside the Siegel theta function is constant.Then the lift We provide an example of an input function to our lift.To this end, we specialize our setting to signature (1,2), in which case vector-valued modular forms can be identified with the usual scalarvalued framework on the complex upper half plane, and in particular Gr(L) ∼ = H.(We explain the required choices in Section 5.) In 1975, Cohen [16] defined the generalized class numbers where D = D 0 j 2 , as well as their generating functions Here However, evaluating our lift requires negative weight, and a non-constant principal part of the input function.To overcome both obstructions, we let be the unique weakly holomorphic modular form of weight −2ℓ for SL 2 (Z) with such a Fourier expansion, an explicit description of f −2ℓ,N was given by Duke, Jenkins [20], and by Duke, Imamoḡlu, Tóth [19, Theorem 1].Our machinery now enables us to obtain Eichler-Selberg type relations for the weakly holomorphic function f −2ℓ,N (τ )H ℓ (τ ) along the lines of [16,Section 6], as well as the following variant of Theorem 1.2.
Theorem 1.3.Assuming that the polynomial p inside the Siegel theta function is constant, the lift Ψ reg j (f −2ℓ,N H ℓ , z) is a local Maaß form on H for every j ∈ N, ℓ ∈ N \ {1}, and −m ≤ N ∈ N with exceptional set given by the net of Heegner geodesics

Remarks.
(1) Theorem 1.3 generalizes immediately to any weakly holomorphic modular form g. The exceptional set is given by the union of geodesics of discriminant D > 0, for which the coefficient of g at q −D is non-zero.(2) Recently, Wagner [37] constructed a pullback of H ℓ under the ξ-operator, namely a harmonic Maaß form An explicit definition of H ℓ can be found in [37, eq. (1.5), (1.6)].However, H ℓ is a harmonic Maaß form with non-cuspidal image under ξ, and we restrict ourselves to a more restricitve growth condition in Section 2.3 to ensure convergence of our lift.It would be interesting to investigate different regularizations of our lift, and in particular lift the function H ℓ .
The paper is organized as follows.We establish the overall framework in Section 2. Section 3 is devoted to two evaluations of our theta lift, and to the proof of Theorem 1.1.In Section 4, we compute the action of the Laplace-Beltrami operator on our theta lift and prove Theorem 1.2.Lastly, Section 5 offers more details on the specialization to signature (1, 2), a proof of Theorem 1.3, and an indication on Eichler-Selberg type relations for Cohen-Eisenstein series at the very end.
, which is generated by the pairs where we fix a suitable branch of the complex square root throughout.Furthermore, we define Γ ∞ as the subgroup generated by T .We let L be an even lattice of signature (r, s), and Q be a quadratic form on L with associated bilinear form (•, •) Q .Moreover, we denote the dual lattice of L by L ′ , and the group ring of has a standard basis, whose elements will be called e µ for µ ∈ L ′ /L.We recall that there is a natural bilinear form Equipped with this structure, the Weil representation ρ L of Γ associated to L is defined on the generators by where we stipulate e(x) := e 2πix throughout.We let L − := (L, −Q), and call ρ L − the dual Weil representation of L.

2.2.
The generalized upper half plane and the invariant Laplacian.We follow the introduction in [8, Sections 3.2, 4.1], and let the signature of L be (2, s) here.We assume that L is isotropic, i.e. that it contains a non-trivial vector x of norm 0, and by rescaling we may assume that it is primitive, that is if x = cy for some y ∈ L and c ∈ Z then c = ±1.Note that for s ≥ 3 all lattices contain such an isotropic vector (see [3,Section 8]).
Let z ∈ L be a primitive norm 0 vector, and Each z j has a real part x j and a imaginary part y j , and we note that ℓ .This gives rise to the generalized upper half plane it can be shown that the invariant Laplacian on H ℓ has the coordinate representation [35] Ω := ℓ µ,ν=1 The modular transformation in this setting is captured by the slash-operator , which leads to vector-valued Maaß forms as follows [10].
Then f is a Maaß form of weight κ with respect to ρ L if it satisfies the following three conditions.
(2) The function f is an eigenfunction of the weight κ hyperbolic Laplace operator, which is explicitly given by (3) There exists a polynomial2 in q denoted by P f : as v → ∞ for some ε > 0.
We call f a harmonic Maaß form if the eigenvalue equals 0.
We write H κ,L for the vector space of harmonic Maaß forms of weight κ with respect to ρ L , and M !κ,L ⊆ H κ,L for the subspace of weakly holomorphic vector valued modular forms.The subspace S !κ,L ⊆ M !κ,L collects all forms that vanish at all cusps, and such forms are referred to as weakly holomorphic cusp forms.
Bruinier and Funke [10] proved that a harmonic Maaß form f of weight κ = 1 decomposes as a sum f = f + + f − of a holomorphic and a non-holomorphic part, whose Fourier expansions are of the shape where Γ(t, x) := ∞ x u t−1 e −u du denotes the incomplete Gamma function.Harmonic Maaß forms can be inspected via the action of various differential operators.We require the antiholomorphic operator as well as the Maaß raising and lowering operators The operator ξ κ defines a surjective map from H κ,L to S !2−κ,L − [10].In particular, it intertwines with the slash operator introduced above, and the space M !κ,L is precisely the kernel of ξ κ when restricted to H κ,L .Hence, every f ∈ H κ,L has a cuspidal shadow in our case.
The operators R κ and L κ increase and decrease the weight κ by 2 respectively, but do not preserve the eigenvalue under ∆ κ .For any n ∈ N 0 , we let L κ be the iterated Maaß raising and lowering operators, which increase or decrease the weight κ by 2n.
Remark.If one relaxes the growth condition (iii) to linear exponential growth, that is f (τ ) ∈ O (e εv ) as v → ∞ for some ε > 0, then f − is permitted to have an additional (constant) term of the form c − f (µ, 0)v 1−κ e µ .In this case, ξ κ maps such a form to a weakly holomorphic modular form instead of a weakly holomorphic cusp form.

Local Maaß forms.
Locally harmonic Maaß forms were introduced by Bringmann, Kane, Kohnen [5] for negative weights, and independently by Hövel [26] for weight 0. We generalize the exposition due to Bringmann, Kane, Kohnen here, and provide a definition in our setting on Grassmannians and for arbitrary eigenvalues.To this end, we continue to work in the setting of Section 2.2, namely we let the signature of L be (2, s) here.
which satisfies the following four properties: Paralleling the definition of harmonic Maaß forms, we call a local Maaß form locally harmonic if the eigenvalue from the second condition is 0.
The authors of [28] prove that F µ,m,κ converges absolutely, and that it defines a weakly holomorphic modular form of weight κ for ρ L .In addition, they computed the Fourier expansion of F µ,m,κ , which is of the shape The Fourier coefficients c(n) can be found in [28, Theorem 3.2] explicitly.
Furthermore, the Maaß Poincaré series have the following useful property thanks to their simple principal part.
2 N, and principal part Then, we have Additionally, we require the following computational lemma, which is taken from [1, Lemma 2.1], and follows inductively from [9, Proposition 3.4].

Lemma 2.2. For any
2.5.Restriction, trace maps, and Rankin-Cohen brackets.As before, we fix an even lattice L. We let A κ,L be the space of smooth functions f : H → C[L ′ /L], which are invariant under the weight κ slash operator with respect to the representation ρ L .Moreover, let K ⊆ L be a finite index sublattice.Hence, we have L ′ ⊆ K ′ , and thus This induces a map , and µ is a fixed preimage of μ in L ′ /K, we define The following lemma may be found in [14, Section 3].
Lemma 2.3.In the notation above, there are two natural maps and letting n ∈ N 0 , we define the tensor product of f and g as well as the n-th Rankin-Cohen bracket of f and g as where f (r) and g (s) are usual higher derivatives of f and g.Then we have the following vector-valued analogue of [9, Proposition 3.6].
Finally, we have the following lemma, which can be verified straightforwardly (see [1, Proof of Theorem 4.1]).Lemma 2.5.Let h be a smooth function, g be holomorphic, and κ, ℓ ∈ R. Then it holds that 2.6.Theta functions and special points.We fix an even lattice L of signature (r, s), and extend the quadratic form on L to L ⊗ R in the natural way.We denote the orthogonal projection of λ ∈ L + µ onto the linear subspaces spanned by z and its orthogonal complement with respect to (•, •) Q by λ z and λ z ⊥ respectively.In other words, we have Let Gr(L) be the Grassmannian of r-dimensional subspaces of L ⊗ R. Let Z ⊆ Gr(L) be the set of all such subspaces on which Q is positive definite.One can endow Z with the structure of a smooth manifold.Let p r : R r,0 → C, and p s : R 0,s → C be spherical polynomials, which are homogeneous of degree d + , d − ∈ N 0 respectively.Define p := p r ⊗ p s , and let ψ : L ⊗ R → R r,s be an isometry.We set For a positive-definite lattice (K, Q) of rank n, and a homogeneous spherical polynomial p of degree d, we define the usual theta function where ψ K is the isometry associated to K. It is a holomorphic modular form of weight n 2 + d for ρ K .If the isometry is trivial, we write Θ K (τ, p).
Following Borcherds [3] and Hövel [26], we define the general Siegel theta function as follows. 3 Definition.Let τ ∈ H and assume the notation above.Then we put One can check that the function Θ L converges absolutely on H × Z.The following result is [26, Satz 1.55], which follows directly from [3, Theorem 4.1].Lemma 2.6.Let (γ, φ) ∈ Γ.Then we have

Thus, we define
The following terminology is borrowed from [13].
Definition.An element w ∈ Gr(L) is called a special point if it is defined over Q that is w ∈ L⊗Q.
We observe that if w is a special point, then w ⊥ is a special point as well.This yields the splitting which in turn yields the positive and negative definite lattices Clearly, P ⊕ N is a sublattice of L of finite index, and according to Lemma 2.3, the theta functions associated to both lattices are related by and let ψ P , ψ N be the restrictions of ψ onto P , N respectively.Consequently, we have the splitting at a special point w, which can be verified straightforwardly.Furthermore, we observe that Θ P (τ, ψ P , p r ) is holomorphic and of weight r 2 + d + as a function of τ , while v 3 In fact, Borcherds considered a slightly more general theta function, where the polynomial p does not necessarily vanish under ∆κ.For us however, this more general case would not yield spherical theta functions as we desire.
for every weakly holomorphic modular form f of weight 2 − κ for ρ L .

The theta lift
We consider the theta lift Ψ reg j (f, z) and evaluate it in two different ways.Using Serre duality goes back to Borcherds [4].
3.1.Evaluation in terms of 2 F 1 .We begin by evaluating the higher modified lift as a series involving Gauss hypergeometric functions as follows.

Evaluating the theta lift of Maaß-Poincaré series for general spectral parameters. Let s ∈ C be such that
converges absolutely, that is Re(s) > 1 − κ 2 .Theorem 3.1.We have Remark.Choosing the homogeneous polynomial in the theta kernel function to be the constant function 1 and computing the action of R j k−2j on F m,k−2j,s by Lemma 2.2, this result becomes [8, Theorem 2.14].
Proof.We summarize the argument from [8, Theorem 2.14] for convenience of the reader.We need to evaluate Consequently, we compute the action of the raising operator first, and have by Lemma 2.2.Secondly, we insert the definitions of both functions, and unfold the integral, obtaining Third, we compute the integral over u using that e(w) = e(−w), and that Hence, we obtain The integral is a Laplace transform.Using that m 2m + We recall and rewrite the argument of the hypergeometric function to Thus, we arrive at Combining the previous result with Lemma 2.1 yields the following consequence.
Corollary 3.2.Let j ∈ N 0 , and f ∈ H k−2j,L .Assume that k − 2j < 0. Then we have Proof.Since the weight of f is negative, we have according to Lemma 2.1, and we observe that the term corresponding to m = 0 will vanish due to c + f (h, 0) = 0 by our more restrictive growth condition on Maaß forms.Consequently, we have We insert the spectral parameter s = 1 − k−2j 2 into Theorem 3.1, which yields the claim.

Evaluation in terms of the constant term in a Fourier expansion.
Next we determine the lift as a constant term in a Fourier expansion plus a certain boundary integral that vanishes for a certain class of input function.
and w be a special point, and G + P be the holomorphic part of a preimage of Θ P under ξ 2−( r 2 +d + ) .Then we have Remark.In general, the coefficients of G + P are expected to be transcendental.However, in weight 1 2 and 3 2 the function G + P may be chosen to have rational coefficients -a situation which is expected to also hold for ξ-preimages of CM modular forms.It is therefore expected that one obtains rationality (up to powers of π) of the modified higher lift only in these cases, and stipulating that f is weakly holomorphic meaning that the final integral vanishes.
By a slight abuse of notation, we write Θ L (τ, w, p) for the theta function evaluated at an isometry ψ that produces a special point w.
Proof of Theorem 3.3.We restrict to special points w ∈ Gr(L).This enables us to write w, p) .Next, we use that the raising and lowering operator are adjoint to each other (cf.[8, Lemma 4.2]), which gives We observe that the boundary terms disappear in the same fashion as during the proof of [8,Lemma 4.4].Next, we rewrite and recall that Consequently, we obtain by Lemma 2.5.In particular we note that v k+ s We apply the computation of the Rankin-Cohen brackets given in Lemma 2.4 noting that L ℓ Θ N − = 0, and that it suffices to deal with the holomorphic part G + P of G P (both by virtue of holomorphicity in computing the Rankin-Cohen bracket).Thus, Hence, the theta lift becomes The last step is to apply Stokes' theorem, compare the proof of [8,Lemma 4.2] for example, which yields utilizing again that boundary terms vanish.We observe that the left integral can be regarded as the Fourier coefficient of index 0 in the Fourier expansion of the integrand, see the bottom of page 14 in [13].This proves the claim.
We end this section by noting that to obtain recurrence relations, as in [13], one would need to compute the Fourier expansion of the lift.In general, this is a lengthy but straightforward process, and since we do not require it in this paper we omit the details.In essence, one follows the calculations of Borcherds [3] by using Lemma 2.2.A resulting technicality is to then take care of the different spectral parameter.One may overcome this by relating the coefficients of Maaß-Poincaré series to those with other spectral parameters, again using the action of the iterated Maaß raising operator as in Lemma 2.2.

Eichler-Selberg type relations.
We now prove a refined version of Theorem 1.1.To this end, we define the function for j > 0. We write and furthermore define Since one may add any weakly holomorphic modular form of appropriate weight for ρ L to G + P , Theorem 1.1 follows directly from the following result (noting that the linear combination of Maaß Poincaré series may change).Theorem 3.4.Let L be an even lattice of signature (r, s), let p be as before, and w be a special point defined by the isometry ψ.Let j > 0 and k be such that 2j + 2 − k > 2. Then the function is a holomorphic vector-valued modular form of weight 2j + 2 − k for ρ L . Remarks.
(1) This provides the general vector-valued analogue, assuming that the lattice is chosen such that 2j + 2 − k > 2, of Mertens' scalar-valued results in weight 1 2 and 3 2 [34].(2) Note that the slight correction of G + P by Poincaré series was missing in [32].(3) In certain cases the hypergeometric function may be simplified (for example, the n = 1 case as in [13,32], which yields a form very similar to Mertens' scalar-valued result).It appears to be possible that one should be able to prove the same results via holomorphic projection acting on vector-valued modular forms (see [27]) in much the same way as Mertens' original scalar valued proofs in [34].
Proof of Theorem 3.4.Let f be a weakly holomorphic form of weight k−2j with Fourier coefficients c + f .By construction, the form G + P is holomorphic at i∞, and hence contains only the Fourier coefficients of non-positive index of f .We note that L k−2j f = 0, and subtract the resulting expressions of the lift from Corollary 3.2 and Theorem 3.3.We obtain The Rankin-Cohen bracket is bilinear, and a linear combination of vector-valued Poincaré series is modular itself.We apply Proposition 2.7 and the claim follows.
In a similar way to [34, Corollary 5.4], we obtain the following structural corollary by rewriting Theorem 3.4, keeping the same notation as throughout this paper.

The action of the Laplace-Beltrami operator
In this section, we prove Theorem 1.3.To this end, we compute the action of the Laplace-Beltrami operator on the lift, and show that for certain spectral parameters, we obtain a local Maaß form.We recall that the signature of L is assumed to be (2, s) here.
Remark.As remarked in the introduction, a refinement of this result for a particular non-constant polynomial p is done by Zemel [ which collects the singularities of Ψ reg j (F m,k−2j,s , Z) as a function of Z.We apply the previous proposition to our theta lift, which yields a variant of [8,Theorem 4.6].
Proof.First, we note that because all partial derivatives with respect to Z converge locally uniformly in Z as T → ∞ (see [8, p. 99]).By the previous proposition, we infer that Lastly, we observe that ∆ k and R j k−2j commute by virtue of Lemma 2.2.Namely, we have and this establishes the claim by rewriting We end this section by proving Theorem 1.2.
Proof of Theorem 1.2.By Theorem 4.2, the lift is an eigenfunction of the Laplace-Beltrami operator with the quoted eigenvalue.Since Ψ reg j (F m,k−2j,s , Z) is an eigenfunction of an elliptic differential operator, it is real-analytic in Gr(L) outside of H(m).The other conditions for the lift to be a vector-valued local Maaß form can be easily seen by applying the proof of [6, Theorem 1.1] mutatis mutandis.When s = k 2 or s = k 2 − 1 we obtain locally harmonic Maaß forms.

Cohen-Eisenstein series
We specialize the framework from Section 2 following [13, Section 4.4] (or [36, Section 2.2]).We fix the signature (1, 2) as mentioned in the introduction, and the rational quadratic space V := X = ( We remark that both are invariant under modular substutions.By a result from Eichler and Zagier [21,Theorem 5.4], the space of vector-valued modular forms of weight k for ρ L is isomorphic to the space M + k (Γ 0 (4)) of scalar-valued modular forms satisfying the Kohnen plus space condition via the map f 0 (τ )e 0 + f 1 (τ )e 1 → f 0 (4τ ) + f 1 (4τ ).This enables us to use scalar-valued forms as inputs for our theta lift.
Proof of Theorem 1.3.As outlined in the introduction, the function f (τ ) := f −2ℓ,N (τ )H ℓ (τ ) is of weight −ℓ − 1 2 < 0 for Γ 0 (4), has non-constant principal part at the cusp i∞, and its image under ξ is trivial, hence in particular cuspidal.This enables us to apply Corollary 3.2 to f .To this end, we have the parameters Rewriting those yields and the hypergeometric function from Theorem 3.1 becomes This proves the Theorem.
where Q D denotes the set of integral binary quadratic forms of discriminant D. After evaluating the hypergeometric function as in the previous proof, one may follow our proof of Theorem 3.4, namely subtract the two evaluations of Ψ reg j (f, z) from each other, and apply Serre duality to the resulting expression.Computing the principal part of G + P in addition, this yields the desired result.However, we do not pursue this here explicitly as the resulting expression is rather lengthy.
there exists a neighborhood of Z, in which f is real analytic and an eigenfunction of Ω. (iii) We have f ((iε, 0, . . ., 0) t + f Z − (iε, 0, . . ., 0) t for every Z ∈ X. (iv) The function f is of at most polynomial growth towards all cusps.

Proposition 4 . 1 (
[8, Proposition 4.5]).Suppose that the polynomial p inside the Siegel theta function is constant.Then, the Siegel theta function Θ L (τ, Z, 1) considered as a function on H×H ℓ satisfies the differential equation
, ζ refers to the Riemann zeta function, L(s, χ) to the Dirichlet L-function twisted by a Dirichlet character χ, and µ is the Möbius function.The functions H ℓ are known as Cohen-Eisenstein series today, and can be viewed as half integral weight analogues of the classical integral weight Eisenstein series.Note that the numbers H(2, n) are precisely the Hurwitz class numbers introduced above, and H 2 = H.Cohen proved that H ℓ ∈ M ℓ− 1 [30,Serre duality.The following result can be found in[30, Proposition 2.5]for instance.