Meromorphic modular forms with rational cycle integrals

We study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas for the cycle integrals of suitable linear combinations of these meromorphic modular forms.


Introduction
One of the fundamental results in the classical theory of modular forms is the fact that the vector spaces of modular forms are spanned by forms with rational Fourier coefficients. Besides that, there are other natural rational structures on these spaces, for example coming from the rationality of periods or cycle integrals of modular forms. This was first shown by Kohnen and Zagier in [20], where they proved the rationality of the even periods of the cusp forms of weight 2k for Γ(1) = SL 2 (Z), for k ≥ 2 and all discriminants D > 0. Here the sum runs over the set Q D of all integral binary quadratic forms of discriminant D. These cusp forms were introduced by Zagier while investigating the Doi-Naganuma lift in [24], and they played a prominent role in the explicit description of the Shimura-Shintani correspondence in [19]. The aforementioned rationality result of Kohnen and Zagier was generalized to Fuchsian groups of the first kind by Katok [18]. Periods and cycle integrals of other types of modular forms, such as weakly holomorphic modular forms, harmonic Maass forms, or meromorphic modular forms, have been the object of active research over the last years, see for example [3,4,9,10,13].
If we allow negative discriminants D < 0 in (1.1) and restrict the summation to positive definite forms Q ∈ Q D , then we obtain meromorphic modular forms f k,D of weight 2k for Γ (1) with poles of order k at the CM points of discriminant D. These forms recently attracted some attention, starting with the work of Bengoechea [2] on the rationality properties of their Fourier coefficients. Their regularized inner products and connections to locally harmonic Maass forms were investigated by Bringmann, Kane, and von Pippich [8] and the first author [21]. Furthermore, Zemel [27] used them to prove a higher-dimensional analogue of the Gross-Kohnen-Zagier theorem [16], which hints at a deeper geometric meaning of the meromorphic Date: July 10, 2019. The work of the first author is supported by ERC starting grant H2020 ERC StG #640159. The second author is supported by the SFB-TRR 191 'Symplectic Structures in Geometry, Algebra and Dynamics', funded by the DFG. f k,D . Recently, Alfes-Neumann, Bringmann, and the second author in [1] established modularity properties of the generating series of traces of cycle integrals of f k,D and used this to show the rationality of suitable linear combinations of these traces.
It is natural to ask whether the individual cycle integrals of the meromorphic modular forms f k,D for D < 0 have nice rationality properties too. For an indefinite integral binary quadratic form A = [a, b, c] of non-square discriminant the cycle integral of f k,D along the closed geodesic corresponding to A is defined by where S A := {z ∈ H : a|z| 2 + bRe(z) + c = 0} is a semi-circle centered at the real line and Γ(1) A denotes the stabilizer of A in Γ (1). Note that, due to the modularity of f k,D , the cycle integral depends only on the Γ(1)-equivalence class of A. If f k,D has a pole on S A , the cycle integral can be defined as a Cauchy principal value, see Section 3.5. Numerical integration yields the following approximations for k ∈ {2, 4, 6} and D = −3. It seems that the cycle integrals of f 2,−3 and f 4,−3 are integers, but there is little reason to believe that the cycle integrals of f 6,−3 are rational numbers.
The main aim of the present work is to investigate the rationality of the cycle integrals of f k,D for D < 0. As we will see, the failure of rationality of these cycle integrals is due to the existence of cusp forms of weight 2k. In particular, we have to take certain linear combinations of cycle integrals of a fixed f k,D or a fixed cycle integral of linear combinations of forms f k,D to obtain convenient rationality results. We also treat forms of higher level Γ 0 (N ), as well as the case k = 1. We remark that our results generalize the rationality results of [1] in several aspects, using a very different proof.

Statement of results
Let N and k be positive integers and let Γ = Γ 0 (N ). For any D ∈ Z the group Γ acts on the set Q D of (positive definite if D < 0) integral binary quadratic forms Q = [a, b, c] of discriminant D = b 2 − 4ac with N | a, with finitely many orbits if D = 0. We write [Q 0 ] for the Γ-class of Q 0 ∈ Q D . For D = 0 and k ≥ 2 we define the associated function For k = 1 the function f 1,Q 0 (z) is defined using Hecke's trick, see Section 3.3. Throughout, we let A ∈ Q D denote an indefinite quadratic form of non-square discriminant D > 0, and P ∈ Q d a positive definite quadratic form of discriminant d < 0. Then f k,A is a cusp form of weight 2k for Γ, and f k,P is a meromorphic modular form of weight 2k for Γ which has poles of order k at the CM points τ Q ∈ H (defined by Q(τ Q , 1) = 0) for Q ∈ [P ].
Our explicit formulas for the cycle integrals of f k,P will be given in terms of the following function. For k ≥ 2, an indefinite quadratic form A ∈ Q D of non-square discriminant D > 0, and τ ∈ H not lying on any of the semi-circles S Q for Q ∈ [A], we define the function where the zeta function ζ Γ,A (s) is defined in (3.4), P k−1 denotes the usual Legendre polynomial, and Int(S Q ) denotes the bounded component of H \ S Q . For k = 1 the function P 1,A (τ ) is defined analogously, but the first line has to be omitted. If τ ∈ H does lie on one of the semi-circles S Q for Q ∈ [A], we define the value of P k,A at τ by the average value Note that the sum in the second line of (2.1) is finite and P k,A (τ ) has discontinuities along the semi-circles S Q for Q ∈ [A]. From the properties of ζ Γ,A (s) given in Section 3.4 it easily follows that the special values |d| k−1 2 P k,A (τ P ) at CM points τ P ∈ H associated to positive definite forms P ∈ Q d are rational numbers.
Our first rationality result concerns linear combinations of cycle integrals of a fixed f k,P .
Theorem 2.1. Let Q be a finite family of indefinite quadratic forms of non-square discriminants and a A ∈ Z for A ∈ Q such that A∈Q a A f k,A = 0 in S 2k (Γ). Furthermore, let P ∈ Q d be a positive definite quadratic form of discriminant d. Then we have the formula where Γ P is the stabilizer of P in Γ/{±1}. In particular, this linear combination of cycle integrals is a rational number whose denominator is bounded only in k and N .
We would like to emphasize that the formula on the right-hand side can be evaluated exactly, giving the precise rational value of the linear combination of cycle integrals on the left-hand side.
The proof of Theorem 2.1 uses the fact that the cycle integral C(f k,P , A) equals the special value at the CM point τ P of (the iterated derivative of) a so-called locally harmonic Maass form F 1−k,A (τ ), see Corollary 4.3. This function was introduced by Bringmann, Kane, and Kohnen in [6]. They showed that F 1−k,A can be decomposed into a sum of a certain local polynomial (whose iterated derivative is P k,A ), and holomorphic and non-holomorphic Eichler integrals of the cusp form f k,A . Taking suitable linear combinations as in the theorem, one can achieve that the Eichler integrals cancel out, which yields the formula in Theorem 2.1. We refer to Section 5 for the details of the proof.
Example 2.2. Let N = 1. Since there are no non-trivial cusp forms of weight less than 12 or weight 14 for Γ(1), the functions f k,A for k ≤ 5 and k = 7 vanish identically for every indefinite quadratic form A. Thus it follows from Theorem 2.1 that the cycle integrals C(f k,P , A) are rational for k ≤ 5 and k = 7 for every choice of P and A. This explains the rationality of the cycle integrals of f 2,−3 and f 4,−3 that we observed in the introduction. In contrast, we have seen in the introduction that the cycle integrals C(f 6,−3 , A) do not seem to be rational. This corresponds to the fact that f 6,A is a cusp form of weight 12 which does usually not vanish identically. However, using results of [25], one can prove the relations  Next, we consider cycle integrals of certain linear combinations of forms f k,P over a single geodesic. Following [15], we call a sequence λ = For a meromorphic function f on H which transforms like a modular form of weight 2k for Γ we define its Hecke translate corresponding to a relation λ by where T m denotes the usual m-th Hecke operator of level N , see (6.1). Bengoechea [2] showed that the Fourier coefficients of f k,P |T λ are algebraic multiples of π k−1 for every positive definite quadratic form P and every relation λ for S 2k (Γ). We obtain a rationality result for the cycle integrals of f k,P |T λ . Theorem 2.3. Let A ∈ Q D be an indefinite quadratic form of non-square discriminant D. Furthermore, let P ∈ Q d be a positive definite quadratic form of discriminant d and let λ = (λ m ) be a relation for S 2k (Γ). Then we have the formula In particular, the cycle integrals of f k,P |T λ are rational numbers whose denominators are bounded only in k and N .
The idea of the proof is similar as for Theorem 2.1. See Section 6 for the details. has rational cycle integrals. Here we used the action of T p on f k,D as stated in [2]. Indeed, we have Similarly, for f 6,−7 = f 6, [1,1,2] , we have 2728656 7282240 17047968 15937488 26668656 Finally, we consider cycle integrals of linear combinations of the forms f k,D and their twisted analogs f k,∆,δ , which we define now. For simplicity, we now assume that N is odd and square-free. Let k ≥ 1, let ∆ be a discriminant with (−1) k ∆ > 0, and let δ be a fundamental discriminant with (−1) k δ < 0, such that δ is a square modulo 4N . Let χ δ be the generalized genus character on Q ∆δ as defined in [16]. For k ≥ 1 we define the twisted function Then f k,∆,δ is a meromorphic modular form of weight 2k for Γ. Suppose that is a weakly holomorphic modular form of weight 3 2 − k for Γ 0 (4N ) satisfying the Kohnen plus space condition, such that the Fourier coefficients c F (m) are rational for all m < 0. We will show in Proposition 7.1 that the Fourier coefficients of the meromorphic modular form are algebraic multiples of π k−1 . Based on extensive numerical experiments, we arrived at the following conjecture. It seems that the methods used to prove Theorem 2.1 and Theorem 2.3 are not suitable to prove the conjecture. In particular, numerical computations suggest that the cycle integrals of the function in (2.2) cannot be expressed in a simple way in terms of the functions P k,A . However, we are able to prove the conjecture in the case k = 1, using different methods.
The proof relies on the fact that the function πif 1,∆,δ (z)dz is the canonical differential of the third kind for its residue divisor on the compactified modular curve X 0 (N ). Together with a rationality criterion of Scholl [23] for such differentials we obtain Theorem 2.6. We refer to Section 7 for the proof. We remark that, unfortunately, the proof does not yield finite rational formulas for the cycle integrals of the linear combination (2.2).
Example 2.7. We give some numerical examples of Conjecture 2.5. Let N = 1. If k is odd, we can pick δ = 1, such that there is no twist. The first odd k for which there are nontrivial weight 2k cusp forms is k = 9 with S 18 = C∆E 6 . The space S 18 is isomorphic to the Kohnen plus space of weight 9 + 1 2 under the Shimura correspondence, and the latter space is spanned by the cusp form This implies that there is a weakly holomorphic modular form with principal part q −4 + 2q −3 + O(1) in the Kohnen plus space of weight 3 2 − 9. In this case, Conjecture 2.5 predicts that the linear combination g := f 9,−4 + 2f 9,−3 has rational cycle integrals. One can easily see that, since k is odd, we have C(g, A) = 0 whenever the form A is Γ(1)-equivalent to −A. But for quadratic forms that are not equivalent to their negatives, we obtain numerically: C(g, A) 3343284 235476 4350060 116285048 255683332 254947680 If k is even, we have to introduce a twist, since δ < 0. Here we consider k = 6 and δ = −3. The weight 6 + 1 2 cusp form corresponding to ∆ ∈ S 12 under the Shimura correspondence is given by q − 56q 4 + 120q 5 − 240q 8 + 9q 9 + O(q 12 ), so for example the functions should have rational cycle integrals. Indeed, it is easy to check that the function g 1 coincides with f 6,−3 |T λ from Example 2.4, so it does have rational cycle integrals by Theorem 2.3. In constrast, g 2 cannot be obtained by acting with Hecke operators, since −15 is squarefree. However, for this function we obtain numerically: The work is organized as follows. In Section 3 we introduce the necessary functions and notation. Then we relate the cycle integrals C(f k,P , A) to locally harmonic Maass forms in Section 4. The proofs of Theorems 2.1, 2.3, and 2.6 are given in the remaining sections.

Preliminaries
3.1. Weight 2 Eisenstein series. For z = x + iy ∈ H we define the quasimodular weight 2 Eisenstein series for Γ = Γ 0 (N ) associated to the cusp i∞ by the conditionally convergent series The function E 2,Γ has a non-holomorphic modular completion that has constant term 1 at i∞ and 0 at all other cusps. The following lemma expresses E * 2,Γ in terms of the Eisenstein series for the full modular group and follows from eq. (9) on p. 546 of [16].
where σ denotes the divisor sum function and we set σ(x) which has weight 2k in z for Γ and weight 0 in τ for Γ. Furthermore, it is meromorphic as a function of z, and an eigenfunction of the invariant Laplace operator ∆ 0 with eigenvalue k(1 − k) as a function of τ for τ not lying in the Γ-orbit of z. The series does not converge for k = 1. However, we can apply Hecke's trick as in [5] and define for Re(s) > 0 One can show that H 1,s (z, τ ) has an analytic continuation H * 1 (z, τ ) to s = 0. It is not meromorphic in z anymore, but the function is a meromorphic modular form of weight 2 in z and a harmonic Maass form of weight 0 in τ for Γ. The function z → H 1 (z, τ ) has a simple pole when z is Γ-conjugate to τ .
Similarly, we define for k ≥ 2 and ℓ ∈ Z the function It has weight 2k in z and weight −2ℓ in τ for Γ, and it also behaves nicely under the raising and lowering operators which raise and lower the weight of an automorphic form of weight κ by 2, respectively. The following lemma can be checked by a direct computation.
We are particularly interested in the function H k,k−1 (z, τ ) (with H 1,0 (z, τ ) := H 1 (z, τ )), which has weight 2k in z and 2 − 2k in τ . It is meromorphic in z and harmonic in τ for τ not lying in the Γ-orbit of z, and as a function of τ it is bounded at the cusps (and vanishes at i∞ if k = 1, compare Lemma 5.4 in [5]). Furthermore, by Lemma 3.2 it is related to H k (z, τ ) by is an iterated version of the raising operator.
3.3. Modular forms associated to quadratic forms. In the introduction we defined the function f k,Q 0 associated to an integral binary quadratic form Q 0 of discriminant D = 0 for k ≥ 2. We briefly explain the definition for k = 1. For s ∈ C with Re(s) > 1 we consider the series It converges absolutely and has a holomorphic continuation to s = 0. If Q 0 = A is indefinite and D not a square, then is a cusp form of weight 2 for Γ (see [16], p. 517). If Q 0 = P is positive definite, then it follows from the following lemma and (3.1) that is a meromorphic modular form of weight 2 for Γ.
Proof. We have the formula Hence, for k ≥ 2 we get For k = 1 we can show in the same way that H 1,s (z, τ P ) is a multiple of f 1,P,s (z) and then use analytic continuation.
We will also need the Fourier expansion of f k,P . The proof of the following formula is analogous to the proof of Proposition 2.2 from [2] (correcting a sign error), but additionally uses (3.3) and Lemma 3.1 in case that k = 1.
Proposition 3.4. For k ≥ 2 and z ∈ H with y > |d|/2 we have the Fourier expansion with the usual I-Bessel function and the exponential sum For k = 1 the formula is analogous, but we have to add dn N to c f 1,P (n), and we get a constant term c f 1,P (0) = − 2 |Γ P | . 3.4. Zeta functions associated to indefinite quadratic forms. Let A ∈ Q D be an indefinite quadratic form of non-square discriminant D > 0. We define the associated zeta function compare [26], Proposition 3 (i). We first express ζ Γ,A (s) in terms of zeta functions ζ Γ(1),A d (s) for level 1 and suitable quadratic forms A d .
Proposition 3.6. The expression is rational for any k ≥ 2, any non-square discriminant D > 0, and N ∈ N.
Proof. We set ζ A (s) := ζ(2s)ζ Γ(1),A (s). It is well known that for k ≥ 2 and any non-square discriminant D > 0 we have the functional equation compare [20], p. 230. Furthermore, ζ A (1 − k) is rational by Theorem 8 in [20]. Dividing by on both sides, we see that is rational, too. Using Lemma 3.5 we obtain the result for all N ∈ N.
Remark 3.7. It follows from the explicit formula in Theorem 8 of [20] that the denominator of ζ A (1 − k) is bounded by a constant only depending on k, but not on A. This formula can also be used to evaluate D k− 1 2 ζ Γ,A (k) + (−1) k ζ Γ,−A (k) explicitly as a rational number.
Finally, we relate the expression from Proposition 3.6 to the cycle integrals of the Eisenstein series E 2k,Γ of weight 2k for the cusp i∞ of Γ, which is normalized such that its Fourier expansion at i∞ has constant term 1. The following result can be proven by a similar computation as on pp. 240-241 of [20].
Proposition 3.8. Let k ≥ 2 and let A ∈ Q D be an indefinite quadratic form of non-square discriminant D > 0. Then Although we will not use this formula in the proofs of our main results, we decided to include it since it gives an interesting interpretation of the expression from Proposition 3.6. If some poles of f do lie on S A , we modify S A by circumventing these poles and all of their Γ-translates on small arcs of radius ε > 0 above and below the poles. Thereby we obtain two paths S + A,ε and S − A,ε and corresponding geodesics c + A,ε and c − A,ε which avoid the poles of f . We define the regularized cycle integral of f along c A by the Cauchy principal value Note that, since f is meromorphic, the integrals on the right-hand side are actually independent of ε for ε > 0 small enough, so the limit exists.
3.6. Maass Poincaré series. Throughout this section we let N be odd and square-free. One can construct harmonic Maass form of half-integral weight as special values of Maass Poincaré series, see [22], for example. In this way, one obtains for every integer k ≥ 1 and n < 0 with (−1) k n ≡ 0, 3 (mod 4) a harmonic Maass form P 3 2 −k,n (τ ) of weight 3 2 − k for Γ 0 (4N ) which satisfies the Kohnen plus space condition, whose Fourier expansion at i∞ starts with q −|n| + O(1), and which is bounded at the other cusps.
The holomorphic part of P 3 2 −k,n has a Fourier expansion of the shape whose coefficients of positive index are given as follows.
Theorem 3.9 (Theorem 2.1 in [22]). Let n < 0 and m > 0 with (−1) k n, (−1) k m ≡ 0, 3 (mod 4). Then with the half-integral weight Kloosterman sum Let ∆, δ ∈ Z be discriminants and assume that δ is fundamental. For a, n ∈ Z we consider the Salié sum where χ δ is the generalized genus character of Q ∆δ as defined in [16]. It is related to the half-integral weight Kloosterman sum by the following formula.
Proposition 3.10 (Proposition 3 in [13]). Let ∆, δ ∈ Z be discriminants and assume that δ is fundamental. Then for a, n ∈ Z we have the identity

Locally harmonic Maass forms
The key to proving Theorems 2.1 and 2.3 is relating the cycle integrals of f k,P to certain locally harmonic Maass forms introduced by Bringman, Kane, and Kohnen in [6] and Hövel in [17]. Namely, for k ≥ 2, τ = u + iv ∈ H, and an indefinite quadratic form A ∈ Q D of non-square discriminant D > 0, these are defined by the series is a special value of the incomplete β-function. For k = 1 one can define a weight 0 analogue F 0,A of (4.1) using the Hecke trick as in [7,14]. By adding a suitable constant we can normalize F 0,A such that it vanishes at i∞. The function F 1−k,A transforms like a modular form of weight 2 − 2k for Γ, is harmonic on H \ Q∈[A] S Q and bounded at the cusps, and has discontinuities along the semi-circles S Q for Q ∈ [A]. Its value at a point τ lying on S Q for Q ∈ [A] is given by the average value Furthermore, outside the singularities F 1−k,A is related to the cusp form f k,A ∈ S 2k (Γ) by the differential equations where ξ κ := 2iv κ ∂ ∂τ and D := 1 2πi ∂ ∂τ . Note that our normalization of f k,A differs from the one used in [6], which explains the different constants in the above differential equations.
Recall that the non-holomorphic and holomorphic Eichler integrals of a cusp form f = n≥1 c f (n)q n ∈ S 2k (Γ) are defined by They satisfy The following decomposition of F 1−k,A was derived by Bringmann, Kane and Kohnen for N = 1 and k ≥ 2 in [6], but the same methods work for all N ≥ 1 and k = 1 (see also [14] for k = 1).
where P 1−k,A (τ ) is locally a polynomial of degree at most 2k − 2. More precisely, it is a polynomial on each connected component of H \ Q∈[A] S Q , which is given by where c 1 (A) := 0 and The main goal of this section is to show that F 1−k,A (τ ) can be written as a cycle integral of Petersson's Poincaré series H k,k−1 (z, τ ).
Theorem 4.2. We have If τ lies on a semi-circle S Q for Q ∈ [A] the left-hand side has to interpreted as the average value (4.2), and the cycle integral on the right-hand side is defined as the Cauchy principal value (3.5).
Before we come to the proof of the theorem we state an important corollary, which immediately follows from Theorem 4.2 together with Lemma 3.3 and the identity (3.2).
where R k−1 2−2k denotes the iterated raising operator defined in Section 3.2. Note that a harmonic function on H which transforms like a modular form of weight 2 − 2k and is bounded at the cusps has to be a constant (and therefore vanishes if k > 1). Hence, in order to prove Theorem 4.2 in the case that τ does not lie on S Q for Q ∈ [A] it suffices to show that both sides in the theorem have the same singularities on H and are bounded at the cusps (and vanish at i∞ if k = 1).
We say that a function f has a singularity of type g at a point τ 0 if there exists a neighbourhood U of τ 0 such that f and g are defined on a dense subset of U and f − g can be extended to a harmonic function on U . For example, Theorem 4.1 shows that the function F 1−k,A (τ ) has a singularity of type at each point τ 0 ∈ H, which easily follows from the fact that τ ∈ Int(S Q ) is equivalent to sgn(a) sgn(Q τ ) < 0.
To determine the singularities, we keep τ 0 ∈ H fixed and consider the function Note that the sum over Q ∈ [A] with τ 0 ∈ S Q is finite, and the group Γ Q is infinite cyclic. It is not hard to show that the series converges absolutely and locally uniformly for all k ≥ 1, and is meromorphic in z and harmonic in τ for τ not lying in the Γ-orbit of z. We split the cycle integral into The function is harmonic in a neighborhood of τ 0 . For the second summand we compute for any τ / Note that the integrand is meromorphic in z. The integral is oriented counterclockwise if a > 0 and clockwise if a < 0. We complete S Q to a closed path by adding the horizontal line connecting the two real endpoints w < w ′ of S Q . The function is harmonic on H, so it does not contribute to the singularity. From the residue theorem we obtain that the integral over the closed path equals 0 if τ / ∈ Int(S Q ) and 2πi sgn(a) Res . This yields the claimed singularity.
Proof of Theorem 4.2. By what we have said above, Theorem 4.2 for τ not lying on S Q for any Q ∈ [A] follows from the above lemma. By a similiar idea as in the proof of the lemma above we find that for τ lying on a semi-circle S Q for Q ∈ [A] we have where the cycle integral integral on the left-hand side is defined as the Cauchy principal value (3.5). This implies that Theorem 4.2 is also true for τ lying on S Q for some Q ∈ [A].

The Proof of Theorem 2.1
By Corollary 4.3 we have the identity Let Q be a finite family of indefinite quadratic forms A ∈ Q D of non-square discriminants D A > 0, and let a A ∈ Z for A ∈ Q such that A∈Q a A f k,A = 0. If we multiply (5.1) by a A and sum over A ∈ Q, and then plug in the splitting of F 1−k,A from Theorem 4.1, we see that the Eichler integrals f * k,A and E f k,A cancel out due to the assumption A∈Q a A f k,A = 0. Hence we obtain where P 1−k,A is the local polynomial defined in Theorem 4.1. The action of the iterated raising operator on P 1−k,A has been computed in Lemmas 5.3 and 5.4 of [1], and is given as follows.
where P k,A (τ ) is the function defined in (2.1).
We arrive at which is the formula from Theorem 2.1. Finally, we show that the right-hand side is rational. Proof. If P = [a, b, c] with a > 0, then τ P is given by In particular, √ |d| Im(τ P ) and |d|Q τ P are rational. We have seen in Proposition 3.6 that is a rational number for k ≥ 2 (and this expression does not occur in P k,A for k = 1). Moreover, the Legendre poynomial P k−1 is odd if k is even and even if k is odd. Hence is rational. Combining all these facts we see that |d| In order to show Theorem 2.3 we would like to use the splitting of F 1−k,A from Theorem 4.1 and get rid of the Eichler integrals by taking suitable linear combinations. To this end, the following well-known lemma is useful.
Since the innermost sum is just the n-th coefficient of the cusp form f |T m , the sum over n vanishes by the definition of a relation for S 2k (Γ).
An important ingredient in the proof of Theorem 2.3 is the fact that Petersson's Poincaré series H k (z, τ ) behaves nicely under the action of Hecke operators.
Proof. For k ≥ 2 we plug in the definition of H k (z, τ ) and T m and write Now a short calculation gives Since M ′ also runs through M m (N ) we obtain the stated identity for k ≥ 2. For k = 1 and Re(s) > 0 we compute analogously Using the well-known fact that E * 2,Γ (z)| z T m = σ(m)E * 2,Γ (z) = |Γ\M m (N )|E * 2,Γ (z) = mE * 2,Γ (z)| τ T m and analytic continuation we also obtain the result for k = 1.
We now come to the proof of Theorem 2.3. Using Lemmas 3.3 and 6.2 we compute By (3.2) and Theorem 4.2 we obtain Since every coset in Γ\M m (N ) is represented by a matrix M with M i∞ = i∞, we have for any f ∈ S 2k (Γ) This implies It follows from Lemma 6.1 that m>0 λ m f k,A |T m = 0, and therefore The expression R k−1 2−2k (P 1−k,A ) can be rewritten using Lemma 5.1. We plug in the definition of T m and choose as a system of representatives for Γ\M m (N ) the matrices α β 0 δ with α, β, δ ∈ Z, α > 0, αδ = m, and β (mod δ). This yields the formula in Theorem 2.3. Note that ατ P +β δ is a CM point of discriminant δ 2 d. Hence Lemma 5.2 implies that the expression |d| k−1 2 P k,A ατ P + β δ is rational. This finishes the proof of Theorem 2.3.

The Proof of Theorem 2.6
Throughout this section we assume that N is odd and square-free. Furthermore, we let ∆ be a discriminant with (−1) k ∆ > 0 and δ a fundamental discriminant with (−1) k δ < 0 such that δ is a square modulo 4N . Finally, let F (τ ) = m≫−∞ c F (m)q m be a weakly holomorphic modular form of weight 3 2 − k for Γ 0 (4N ) in the Kohnen plus space with rational coefficients c F (m) for m < 0. We first show that the Fourier coefficients of the meromorphic modular form (2.2) are algebraic multiples of π k−1 .
Proposition 7.1. For k ≥ 1 the meromorphic modular form has rational Fourier coefficients.
For the proof we write the coefficients of f k,∆,δ as linear combinations of coefficients of half-integral weight Maass Poincaré series.
Proof. This identity follows from a straightforward calculation using Proposition 3.4, Theorem 3.9 and Proposition 3.10. It could alternatively be derived from the fact that f k,∆,δ is a theta lift of P 3 2 −k,−|∆| , compare [8,27].
Proof of Proposition 7.1. Looking at the formula for c f k,∆,δ (n) in Lemma 7.2, we see that the second summand on the right-hand side is rational if k = δ = 1 and vanishes otherwise. It remains to show that the coefficients of If k > 1, we have F = F since there are no holomorphic modular forms of negative weight. In particular, since the space of weakly holomorphic modular forms of weight 3 2 − k has a basis consisting of forms with rational coefficients and the principal part of F is rational, we find that all coefficients of F are rational for k > 1. However, for k = 1 the functions F and F may differ by a holomorphic modular form. Note that every P 3 2 −k,m is orthogonal to cusp forms with respect to the regularized Petersson inner product and has rational principal part. Hence the same is true for F . It now follows from Proposition 3.2 in [12] that all Fourier coefficients of F are rational. Now we see that (7.1) equals n≥1 n 2k−1 m|n δ m m −k c F n 2 |δ| m 2 q n , which has rational Fourier coefficients. This finishes the proof.
We now proceed to the proof of Theorem 2.6. For the rest of this section we let k = 1 and δ > 0 a fundamental discriminant which is a square modulo 4N . We can assume without loss of generality that the coefficients c F (∆) for ∆ < 0 are integers. We consider the differential η δ (F ) := πi ∆<0 c F (∆)f 1,∆,δ (z)dz on X 0 (N ). For P ∈ Q ∆δ we have Res z=τ P (f 1,∆,δ (z)) = χ δ (P ) πi , so η δ (F ) has simple poles with integral residues. In particular, η δ (F ) is a differential of the third kind on X 0 (N ). Following [11], we define the twisted Heegner divisor By [11], Lemma 5.1, y δ (F ) is defined over Q( √ δ). Note that y δ (F ) is precisely the residue divisor of η δ (F ) on X 0 (N ). Moreover, we have the following result. Lemma 7.3. The differential η δ (F ) is the canonical differential of the third kind for y δ (F ), i.e., the unique differential of the third kind with residue divisor y δ (F ) such that if γ is any cycle of the form c A which does not meet any poles of η δ (F ). It is well-known that the group H 1 (X 0 (N ) \ y δ (F ), Z) is generated by these cycles, which yields the result.
The crucial ingredient for the proof of Theorem 2.6 is the following rationality result of Scholl [23] for differentials of the third kind (see also Theorem 3.3 of [11]). Theorem 7.4 (Scholl). Let D be a divisor of degree 0 on X 0 (N ) defined over a number field F . Let η D be the canonical differential of the third kind associated to D and write η D = 2πif dz. If all the Fourier coefficients of f are contained in F , then some non-zero multiple of D is a principal divisor.
It follows from Theorem 7.1 that the Fourier coefficients of 1 2πi η δ (F ) are contained in Q( √ δ), which is also the field of definition of the divisor y δ (F ). In particular, the above criterion of Scholl implies that some non-zero multiple of y δ (F ), say m · y δ (F ) for some m ∈ Z, is the divisor of a meromorphic function g on X 0 (N ).