Cycle integrals of meromorphic modular forms and Siegel theta functions

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.


Introduction
Let Q d be the set of all (positive definite if d < 0) integral binary quadratic forms of discriminant d.For k ∈ N with k ≥ 2 we consider the functions which transform like modular forms of weight 2k for Γ = PSL 2 (Z).These functions were first studied by Zagier [12] for d > 0, in which case they are cusp forms, and by Bengoechea for d < 0, in which case they are meromorphic modular forms.We obtain a refinement f k,P of f k,d by summing only over quadratic forms in the equivalence class of a fixed form P .
In [1] we computed the cycle integrals of the meromorphic modular forms f k,P associated with positive definite quadratic forms P .These cycle integrals are defined by 1   C A (f k,P ) = Γ A \S A f k,P (z)A(z, 1) k−1 dz, where A = [a, b, c] is an indefinite integral binary quadratic form of positive non-square discriminant D > 0, Γ A denotes the stabilizer of A, and S A = {z ∈ H : a|z| 2 + bℜ(z) + c = 0} is the geodesic corresponding to A. We let C D (f k,P ) be the D-th trace of cycle integrals, that is, the sum of C A (f k,P ) over all classes A ∈ Q D /Γ, and we let M !3/2−k denote the space of weakly holomorphic modular forms of weight 3/2 − k for Γ 0 (4) satisfying Kohnen's plus space condition a g (n) = 0 unless (−1) k+1 n ≡ 0, 1 (mod 4).Then the main result of [1] is as follows.
Theorem 1.1 (Theorem 1.1 in [1]).Let k ≥ 2 be even and let g ∈ M !3/2−k .Suppose that the coefficients a g (−D) are rational for D > 0 and vanish if D is a square.Then the linear combinations of traces of cycle integrals D>0 a g (−D)C D (f k,P ) of the individual meromomorphic modular forms f k,P are rational. 1If f k,P has poles on SA then the cycle integral can be defined using the Cauchy principal value as in [11]. 1 In [11] we gave some refinements of this result and made a conjecture concerning the rationality of the individual cycle integrals of linear combinations of various f k,d 's, which can be viewed as a dual version of Theorem 1.1 for odd k.Our goal is to prove this conjecture.Theorem 1.2 (Conjecture 2.5 in [11]).Let k ≥ 3 be odd and let g ∈ M !3/2−k .Suppose that the coefficients a g (d) for d < 0 are rational.Then the individual cycle integrals of linear combinations of the meromorphic f k,d are rational.
In [11] this result was stated as a conjecture for higher level, even and odd k, and twisted versions of f k,d .Our arguments also work in this general setup, see Section 5.
The proof of Theorem 1.2 follows a method of Bruinier, Ehlen, and Yang (see the proof of [8,Theorem 5.4]).We use the well-known fact that f k,d can be written as a regularized theta lift.Then we interchange the cycle integral with the regularized integral, which leads us to study the cycle integrals of the Siegel theta function Θ L (τ, z) associated with an even lattice L of signature (1,2).The evaluation of these cycle integrals is the main novelty of this work.Let us describe the result in more detail.
The indefinite quadratic form A defines a vector of negative length in the lattice L, and hence yields sublattices I = L ∩ (QA) ⊥ and N = L ∩ (QA) of signature (1, 1) and (0, 1), respectively.Hecke [9] constructed an indefinite theta series ϑ I (τ ) associated with the lattice I, which is a cusp form of weight 1 for the Weil representation ρ I , see (3.3).Moreover, there is a classical holomorphic weight 3/2 theta series Θ 3/2,N (τ ) associated to the unary lattice N , see (3.1).
Theorem 1.3.Suppose that L = I ⊕ N .Then we have Here • = means equality up to an explicit non-zero constant factor.For the precise version of this result we refer to Theorem 3.1.The idea of the proof is that Θ L (τ, z) essentially splits as a product of two Siegel theta functions Θ I (τ, z) and Θ N (τ ) along the geodesic S A , and the cycle integral of Θ I (τ, z) yields the Hecke theta series ϑ I (τ ).
Combining the above evaluation of the cycle integral of the Siegel theta function with the theta lift realization of f k,d , we obtain a representation of C A (f k,d ) as a regularized Petersson inner product, which can in turn be evaluated using the theory of harmonic Maass forms.This gives an explicit formula for the cycle integrals appearing in Theorem 1.2 (see Theorem 4.1), from which their rationality can easily be deduced.
This paper is organized as follows.In Section 2 we recall the theta lift realization of f k,d .In Section 3 we evaluate the cycle integrals of the Siegel theta function, which is the technical heart of this work.In Section 4 we put everything together and deduce an explicit rational formula for the cycle integrals of f k,d , which also implies Theorem 1.2.Finally, in Section 5 we sketch the proof of a higher level version of Theorem 1.2.
Acknowledments.The author was supported by SNF project PZ00P2 202210.

Meromorphic modular forms as theta lifts
In this section we recall the realization of the meromophic modular form f k,d as a theta lift of a harmonic Maass form, following [3].We consider the even lattice with the quadratic form q(X) = det(X).It has signature (1, 2), and its dual lattice is given The group Γ = PSL 2 (Z) acts on L via g.X = gXg −1 .We can view L ′ as the set of integral binary quadratic forms [a, b, c], where −4q(X) = b 2 − 4ac corresponds to the discriminant.Let Gr(L) be the Grassmannian of positive definite lines in V (R) = L ⊗ R. We can identify Gr(L) with H by sending z = x + iy ∈ H to the positive line generated by This bijection is compatible with the actions of PSL 2 (R), in the sense that g.X(z) = X(gz) for any g ∈ PSL 2 (R).We also consider the vectors which, together with X(z), form an orthogonal basis of V (R).For each fixed z = x + iy ∈ H we define the polynomials on V (R).They satisfy the invariances for each g = a b c d ∈ PSL 2 (R).For brevity we write X z and X z ⊥ for the orthogonal projection of X to the positive line generated by X(z) and to the negative plane X(z) ⊥ spanned by U 1 (z), U 2 (z), respectively.We have the useful relations Since L has signature (1, 2), the Siegel theta function transforms like a modular form of weight −1/2 in τ for the Weil representation ρ L associated with L.Moreover, it is Γ-invariant in z.Here e X = e X+L are the standard basis vectors of the group ring (2.3) Let A k,L denote the space of all C[L ′ /L]-valued functions on H which transform like modular forms of weight k for ρ L .If M ⊂ L is a sublattice of finite index, we have natural maps For a smooth modular form f of weight 1/2 for ρ L the regularized theta lift is defined by where F is a fundamental domain for Γ\H, the dot denotes the natural bilinear pairing on C[L ′ /L], and dµ(τ ) is the invariant measure.The integral is regularized as explained in [7].
We let P 3/2−k,d (τ ) be the unique harmonic Maass form of weight 3/2 − k for the dual Weil representation ρ L with principal part q d e d +O(1).Proposition 2.1 (Proposition 4.1 in [3]).For odd k ≥ 3 we have

Cycle integrals of Siegel theta functions
In this section we compute the cycle integrals of the derivative of the Siegel theta function Θ L (τ, z).Throughout this section we let be an indefinite integral binary quadratic form of positive non-square discriminant D. We let S A = {z ∈ H : a|z| 2 + bx + c = 0} be the corresponding geodesic semi-circle in H. Changing A to −A only changes the sign of the cycle integral, so we will assume a > 0 from now on.Then S A is oriented counter-clockwise.Since A is as a vector of negative norm in L ′ , we can define two sublattices I = L ∩ (QA) ⊥ , N = L ∩ (QA), of signature (1, 1) and (0, 1), respectively.Note that I is anisotropic since A has non-square discriminant, and that N − = (N, −q) is a positive definite one-dimensional lattice.We define the weight 3/2 unary theta function corresponding to N (or rather N − ) by be the real endpoint of the geodesic S A , and consider the vector We define the Hecke theta series associated with the signature (1, 1) lattice I by It is a cusp form of weight 1 for ρ I , which was first constructed by Hecke [9] as a theta lift.
Theorem 3.1.We have with the map f L defined in (2.4).
Proof.Note that I ⊕ N is a sublattice of finite index in L, and we have so we can assume L = I ⊕ N for now.We first split R 0 Θ I⊕N along the geodesic S A into tensor products of theta functions associated with I and N .We can write X ∈ (I ⊕ N ) ′ uniquely as X = Y + W with Y ∈ I ′ and W ∈ N ′ .Note that W = (W,A) (A,A) A. Hence, for z ∈ S A we have Plugging this into the explicit formula (2.3) for R 0 Θ I⊕N , we obtain for z ∈ S A the splitting with the Siegel theta functions and the holomorphic weight 1/2 and weight 3/2 theta functions associated with N .It remains to compute the cycle integrals of Θ * I (τ, z) and Q A (z) −1 Θ I (τ, z).Note that the geodesic S A can be identified with the Grassmannian Gr(I) of positive lines in I ⊗ R. Hence, the cycle integrals can be computed as done by Hecke [9].We give the details for Θ I (τ, z).
Recall that we assume a > 0. The two real endpoints w < w ′ of S A are given by w maps 0 to w and i∞ to w ′ and hence maps the imaginary axis (oriented from i∞ to 0) to S A (oriented counter-clockwise).Note that σ −1 .A = [0, − √ D, 0].Modding out the stabilizer Γ A = Γ I in the sum over Y , and using the parametrization z = σ.it of S A , we obtain where λ ′ is the conjugate of λ in Q( √ D).Using (2.1) and (2.2) we find Hence, using q(Y z )τ + q(Y z ⊥ )τ = q(Y )τ − 2ivq(Y z ⊥ ), we get A standard computation 2 shows that the integral is given by 2), we get sgn(λ) = sgn(Y, Y 0 ).Summarizing, we obtain The computation of the cycle integral of Θ * I (τ, z) is very similar.Using we see that we in fact have C A (Θ * I ) = 0.This finishes the proof.Remark 3.2.One can compute the cycle integrals of R 0 Θ L (τ, z) along infinite geodesics (if D is a square) in a similar way.In this case, the lattice I is isotropic, and ϑ I has to be replaced with a weight 1 Hecke Eisenstein series.

Cycle integrals of meromorphic modular forms: The proof of Theorem 1.2
We are now ready to give an explicit evaluation of the cycle integrals appearing in Theorem 1.2.As before, we let A ∈ L ′ be an indefinite integral binary quadratic form of non-square discriminant D = −4q(A) > 0 with a > 0, and we let I = L ∩ (QA) ⊥ and N = L ∩ (QA) be the corresponding indefinite and negative definite sublattices of L.Moreover, we let ϑ I be the Hecke theta series of weight 1 for ρ I defined in (3.3) and Θ 3/2,N the holomorphic theta function of weight 3/2 for ρ N as in (3.1).
We choose a harmonic Maass form Θ 3/2,N of weight 1/2 for ρ N with 2 For example, one can use the K-Bessel function and we let Θ + 3/2,N be its holomorphic part.Moreover, we require the Rankin-Cohen bracket which maps modular forms f of weight κ and g of weight ℓ to a form of weight κ + ℓ + 2n.
Proof.We follow the idea of the proof of [8,Theorem 5.4].For brevity we put f k,g = d<0 a g (d)f k,d .Using Proposition 2.1 we can write f k,g as a theta lift, with the constant c k = (2 k+1 π (k+1)/2 (k − 1)!) −1 .Now we take the cycle integral C A on both sides.As a preliminary step, we will reduce the power of the outer raising operator [7,Theorem 4.6].Hence, a repeated application of [5,Theorem 1.1] gives Interchanging the cycle integral and the regularized integral gives Now we plug in the evaluation of the cycle integral from Theorem 3.1 to get Here we used that the maps f L and g I⊕N are adjoint to each other.Next, we use the "self-adjointness" of the raising operator to compute In the last step we used that Θ 3/2,N is holomorphic.By [8, Proposition 3.6] we have Note that the constants in front simplify to (−1) . Now a standard application of Stokes' Theorem as in [8,Theorem 5.4] finishes the proof.
We remark that there is a similar formula for the linear combination of cycle integrals of f k,P from Theorem 1.1, see [2,Theorem 1.1].We can now prove Theorem 1.2.
Proof of Theorem 1.2.We show that the right-hand side in Theorem 4.1 is rational.Note that g and ϑ I have rational Fourier coefficients.Moreover, by [10, Theorem 1.1] we can choose Θ 3/2,N such that its holomorphic part has rational Fourier coefficients.Here we use that the lattice N is of the form (Z, q(x) = −Dr 2 x 2 ) for some rational number r.It remains to note that the Rankin-Cohen brackets preserve the rationality of the Fourier coefficients.

Generalization to higher level
In [11, Conjecture 2.5], Theorem 1.2 was stated as a conjecture for level Γ 0 (N ) and twisted versions of f k,d .Here we sketch the proof of this general conjecture.
Let N ∈ N. We let ∆ ∈ Z be a fundamental discriminant (possibly 1) and ρ ∈ Z/2N Z such that ∆ ≡ ρ 2 (mod 2N ).Moreover, we let d < 0 be a negative integer and r ∈ Z/2N Z with d ≡ sgn(∆)r 2 (mod 2N ).We consider the set Q N,d|∆|,rρ of (not necessarily positive definite) integral binary quadratic forms Q = [aN, b, c] of discriminant d|∆| with b ≡ rρ (mod 2N ).We let χ ∆ be the usual generalized genus character on Q N,d|∆|,rρ .For k ∈ N with k ≥ 2 we define the function where we put sgn(Q) = sgn(a).It defines a meromorphic modular form of weight 2k for Γ 0 (N ).For N = 1 and odd k we recover f k,d as f k,d,d,1,1 .We consider the signature (1, 2) lattice with the quadratic form q N (X) = N det(X).We have L ′ N /L N ∼ = Z/2N Z, so we will write e r with r ∈ Z/2N Z for the standard basis elements of C[L ′ N /L N ].Note that the elements of L ′ N correspond to binary quadratic forms [aN, b, c], with the discriminant being given by −4N q N (X).We let ρ L N be equal to ρ L N if ∆ > 0, and to ρ L N if ∆ < 0.
We have the following higher level version of Theorem 1.2.Proof.This can be proved analogously to Theorem 1.2, so we only give a sketch.First, by [3,Proposition 4.1] the meromorphic modular form f k,d,r,∆,ρ can be written as a twisted theta lift on the lattice L N .We can reduce to the non-twisted case using an intertwining operator for the involved Weil representations as explained in [4].Then it remains to evaluate the cycle integrals of certain Siegel theta functions associated with a general even lattice L of signature (1,2).However, note that we did not use the precise shape of L in the proof of Theorem 3.1, so the arguments work exactly the same for arbitrary L, at least for odd k.For even k the theta lift involves a Siegel theta function of the shape p z (X)e (q(X z )τ + q(X z ⊥ )τ ) e X .
A computation similar to the proof of Theorem 3.1 shows that we have with the holomorphic weight 1/2 theta function Θ 1/2,N (τ ) = X∈N ′ e(−q(X)τ ) e X .Apart from this, the proofs for even and odd k are analogous.
Finally, one might speculate that the method from Theorem 3.1 could be used to compute cycle integrals of Siegel theta functions (and the corresponding theta lifts) on lattices of other signatures, e.g.signature (1, n) or (2, n).We hope to come back to this problem in the future.

. 4 )
These maps are adjoint with respect to the bilinear pairings on C[L ′ /L] and C[M ′ /M ].Moreover, the Siegel theta functions for M and L are related by Θ L (τ, z) = Θ M (τ, z) L .

. 1 )
It is a cusp form of weight 3/2 for the dual Weil representation ρ N .Letw = −b− √ D 2a

Theorem 4 . 1 .
Let k ≥ 3 be odd and let g ∈ M !3/2−k,ρ L be a weakly holomorphic modular form of weight 3/2 − k for ρ L .Then we have

Theorem 5 . 1 .
Let k ∈ N with k ≥ 2 and let g be a weakly holomorphic modular form of weight 3/2−k for ρ L N .Suppose that the Fourier coefficients a g (d, r) for d < 0 and r ∈ Z/2N Z are rational.Then the cycle integralsC A   r∈Z/2N Z d<0 a g (d, r)f k,d,r,∆,ρ