On the rationality of cycle integrals of meromorphic modular forms
Claudia Alfes-Neumann, Kathrin Bringmann, Markus Schwagenscheidt
aa r X i v : . [ m a t h . N T ] M a y ON THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHICMODULAR FORMS
CLAUDIA ALFES-NEUMANN, KATHRIN BRINGMANN, AND MARKUS SCHWAGENSCHEIDT
Abstract.
We derive finite rational formulas for the traces of cycle integrals of certainmeromorphic modular forms. Moreover, we prove the modularity of a completion of thegenerating function of such traces. The theoretical framework for these results is an extensionof the Shintani theta lift to meromorphic modular forms of positive even weight. Introduction and statement of results
Rationality of traces of cycle integrals of meromorphic cusp forms.
Let k be apositive even integer. While considering the Doi-Naganuma lift, Zagier [23] encountered thefunctions f k,d ( z ) := | d | k +12 π X Q∈Q d Q ( z, − k , where Q d denotes the set of all integral binary quadratic forms Q = [ a, b, c ] of discriminant d = b − ac . These are holomorphic cusp forms of weight 2 k for Γ := SL ( Z ) if d >
0, andmeromorphic cusp forms of weight 2 k for Γ if d <
0, i.e., they are meromorphic modularforms which decay like cusp forms towards i ∞ .Kohnen and Zagier [16] showed that certain simple linear combinations of the cycle integrals Z c Q f k,d ( z ) Q ( z, k − dz of the cusp forms f k,d for d > c Q := Γ Q \ C Q is the image in Γ \ H of thegeodesic C Q := (cid:8) z ∈ H : a | z | + bx + c = 0 (cid:9) ( z = x + iy )associated to Q = [ a, b, c ] ∈ Q D with D >
0. Complementing these results, we presentrational formulas for the tracestr f k, A ( D ) := X Q ∈Q D / Γ Z c Q f k, A ( z ) Q ( z, k − dz of cycle integrals of the refined functions f k, A ( z ) := | d | k +12 π X Q ∈A Q ( z, − k , (1.1) Date : May 8, 2020.The research of the second author is supported by the Alfried Krupp Prize for Young University Teachersof the Krupp foundation and the second and third author are supported by the SFB-TRR 191 “SymplecticStructures in Geometry, Algebra and Dynamics”, funded by the DFG. where
A ∈ Q d / Γ is a fixed equivalence class of quadratic forms of discriminant d <
0. Thepoles of f k, A lie at the CM points z Q ∈ H for Q ∈ A , which are characterized by Q ( z Q ,
1) = 0.We assume that they do not lie on any of the geodesics C Q for Q ∈ Q D . Let z A := x A + iy A ∈ H denote a fixed CM point z Q for some Q ∈ A . We obtain the following rationality result forthe traces of f k, A . Theorem 1.1.
Let F be a weakly holomorphic modular form of weight − k for Γ (4) satisfying the Kohnen plus space condition. Suppose that the Fourier coefficients a F ( − D ) vanish for all D > which are squares and that a F ( − D ) is rational for D > . Moreover,assume that z A does not lie on any of the geodesics C Q for Q ∈ Q D for any D > for which a F ( − D ) = 0 . Then the linear combinations X D> a F ( − D ) tr f k, A ( D ) are rational. We compute some numerical values of the above traces in Example 1.4 below. Theorem 1.1follows from the following explicit formulas for the traces.
Theorem 1.2.
Assume the hypotheses of Theorem 1.1. Then we have the formula X D> a F ( − D ) tr f k, A ( D ) = p | d | (cid:12)(cid:12) Γ z A (cid:12)(cid:12) X D> a F ( − D ) c k ( D ) y k − A + 4 (cid:16) i √ D (cid:17) k − X Q =[ a,b,c ] ∈Q D Q z A > >a P k − (cid:18) iQ z A √ D (cid:19) , where P ℓ is the ℓ -th Legendre polynomial, | Γ z A | is the order of the stabilizer of z A in Γ :=Γ / {± } , and Q z := 1 y (cid:0) a | z | + bx + c (cid:1) for Q = [ a, b, c ] . The constant c k ( D ) is given by c k ( D ) := D k − ζ ( k )2 k − (2 k − ζ (2 k ) L D ( k ) X m | f µ ( m ) (cid:18) D m (cid:19) m − k σ − k (cid:18) fm (cid:19) . (1.2) Here we write D = D f with a fundamental discriminant D , µ is the M¨obius function, σ κ ( n ) := P d | n d κ is the κ -th divisor sum, ζ is the Riemann zeta function, ( D · ) is the Kro-necker symbol, and L D ( s ) is the associated Dirichlet L -function. In particular, c k ( D ) isrational. Remark 1.3.
For
D > Q = [ a, b, c ] ∈ Q D the geodesic C Q isa semi-circle centered at the real line. The condition Q z A > > a means that C Q is orientedclockwise and that z A lies in the interior of the bounded component of H \ C Q . Since, forfixed non-square D >
0, every point z ∈ H lies in the interior of the bounded component of H \ C Q for only finitely many Q ∈ Q D , the sum over Q ∈ Q D in Theorem 1.2 is finite.The proof of Theorem 1.2 relies on the Fourier expansion of a certain theta lift of themeromorphic modular forms f k, A , which is explained in Section 1.3. An outline of the proofof Theorem 1.2 can be found in Section 1.4, and the full proof is given in Section 5.To illustrate Theorem 1.1 and Theorem 1.2, we treat two examples in low weights. N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 3
Example 1.4.
We consider the cases k ∈ { , } , since then the space S k of cusp forms ofweight 2 k for Γ is trivial. By the Shimura correspondence, the space of cusp forms of weight k + for Γ (4) in the Kohnen plus space is isomorphic to S k , and hence trivial as well. Thisimplies that for every discriminant D > F of weight − k for Γ (4) satisfying the Kohnen plus space condition such that a F ( − D ) = 1and a F ( ℓ ) = 0 for − D = ℓ <
0. Suppose that
D > z A does not lie on any of the geodesics C Q for Q ∈ Q D . Using that P ( x ) = x and P ( x ) = x − x we obtain the special casestr f , A ( D ) = p | d | (cid:12)(cid:12) Γ z A (cid:12)(cid:12) c ( D ) y A − X Q =[ a,b,c ] ∈Q D Q z A > >a Q z A , tr f , A ( D ) = p | d | (cid:12)(cid:12) Γ z A (cid:12)(cid:12) c ( D ) y A − X Q =[ a,b,c ] ∈Q D Q z A > >a (cid:0) Q z A + 3 DQ z A (cid:1) . In the following table, we give some numerical values for the D -th trace of f , [1 , , and f , [1 , , .To clear the denominators, we display the values of tr f , [1 , , ( D ) and 3 tr f , [1 , , ( D ). D f , [1 , , ( D ) 4 8 12 28 24 20 32 20 40 64 44 64 763 tr f , [1 , , ( D ) 20 48 92 452 320 340 576 260 880 1664 1596 1920 2612We left out those discriminants D which are squares, and D = 12 and D = 28 since inthese cases the CM point z A lies on one of the geodesics of discriminant D . We explain thenumerical evaluation in Section 6. We checked the above values using Sage, by computingthe cycle integrals using numerical integration. As the values in the table above suggest, for k ∈ { , } the numbers | Γ z A | · | d | k − · tr f k, A ( D ) are always even integers (for any A ), whichis not hard to show using the formula (5.8) for c k ( D ).1.2. The regularized Shintani theta lift of a meromorphic cusp form.
We now de-scribe the theoretical foundation of our work. The classical Shimura-Shintani correspondenceestablishes a Hecke equivariant isomorphism between the spaces of cusp forms of half-integralweight k + and even integral weight 2 k , with k ∈ N := { , , , . . . } . Soon after its discoveryby Shimura [20], this correspondence was realized by Niwa [17] and Shintani [21] as a thetalift, that is, as an integral constructed from a theta kernel in two variables. The classicalShintani theta lift for cusp forms was recently generalized to weakly holomorphic modularforms by Guerzhoy, Kane, and the second author [4], to harmonic Maass forms by the firstand the third author [2], and to differentials of the third kind by Bruinier, Funke, Imamoglu,and Li [11]. Extending the results of [11], we also include meromorphic cusp forms of arbi-trary positive even weight with poles of arbitrary order in the upper half-plane in the Shintanitheta lift.For k ∈ N we let S k denote the space of meromorphic cusp forms of weight 2 k for Γ.Every meromorphic modular form can be written as a sum of a meromorphic cusp form, aweakly holomorphic modular form, and, if k = 1, a multiple of j ′ /j , with j the usual modular CLAUDIA ALFES-NEUMANN, KATHRIN BRINGMANN, AND MARKUS SCHWAGENSCHEIDT j -invariant. Since the Shintani theta lifts of weakly holomorphic modular forms and of themeromorphic modular form j ′ /j have already been determined in [2, 11], we restrict ourattention to theta lifts of meromorphic cusp forms.For k ∈ N and a fundamental discriminant ∆ ∈ Z satisfying ( − k ∆ > k, ∆ ( z, τ )denote the Shintani theta function defined in (2.1). The function Θ k, ∆ ( − z, τ ) is real-analyticin both variables and transforms like a modular form of weight 2 k in z for Γ and of weight k + in τ for Γ (4). We define the regularized Shintani theta lift of f ∈ S k byΦ k, ∆ ( f, τ ) := D f, Θ k, ∆ ( · , τ ) E reg , (1.3)where the regularized inner product is defined in (2.3). The regularized integral in (1.3) existsby the following theorem. Theorem 1.5.
For f ∈ S k the Shintani theta lift Φ k, ∆ ( f, τ ) is a real-analytic function on H that transforms like a modular form of weight k + for Γ (4) and satisfies the Kohnen plusspace condition. More generally, Proposition 3.2 shows that the regularized inner product in (1.3) convergeseven if we replace Θ k, ∆ ( · , τ ) by any real-analytic function g which transforms like a modularform of weight 2 k and is of moderate growth at i ∞ .1.3. The Fourier expansion of the Shintani theta lift.
One of the main results of thispaper is the Fourier expansion of the Shintani theta lift Φ k, ∆ ( f, τ ) of f ∈ S k . It turns out thatΦ k, ∆ ( f, τ ) yields a completion of the generating function of twisted traces of cycle integralstr f, ∆ ( D ) := X Q ∈Q | ∆ | D / Γ χ ∆ ( Q ) Z reg c Q f ( z ) Q ( z, k − dz, where χ ∆ is the usual genus character as defined on page 238 of [15], and the cycle integralshave to be regularized as explained in Section 2.3 if poles of f lie on the geodesic c Q .To describe the non-holomorphic part of the Shintani theta lift, we define for z ∈ H , v > Q = [ a, b, c ] the function φ Q ( z, v ) := p | ∆ | Q ( z, k − sgn( Q z ) − erf √ πv Q z p | ∆ | !! , (1.4)where erf( x ) := √ π R x e − t dt is the error function and sgn(0) := 0. Note that z φ Q ( z, v ) isreal-analytic up to a jump singularity along C Q if Q has positive discriminant. More generally,for any n ∈ N and z ∈ H , v >
0, we consider the function R n − k,z ( φ Q ( z, v )):= p | ∆ | sgn( Q z ) R n − k,z (cid:16) Q ( z, k − (cid:17) − R n − k,z Q ( z, k − erf √ πv Q z p | ∆ | !!! , where R nκ := R κ +2 n − ◦ · · · ◦ R κ is an iterated version of the Maass raising operator R κ :=2 i ∂∂z + κy . Furthermore, we define for n ∈ N and z, τ = u + iv ∈ H the “theta function” R n − k,z (cid:0) θ ∗ − k, ∆ ( z, τ ) (cid:1) := X D ∈ Z X Q ∈Q | ∆ | D χ ∆ ( Q ) R n − k,z ( φ Q ( z, v )) e πiDτ . We are now ready to state the Fourier expansion of the Shintani theta lift.
N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 5
Theorem 1.6.
Let f ∈ S k . Then the Fourier expansion of the Shintani theta lift of f isgiven by Φ k, ∆ ( f, τ ) = p | ∆ | X D> tr f, ∆ ( D ) e πiDτ + ( − − k π X ̺ ∈ Γ \ H (cid:12)(cid:12) Γ ̺ (cid:12)(cid:12) X n ≥ c f,̺ ( − n ) Im( ̺ ) n − k ( n − h R n − − k,z (cid:0) θ ∗ − k, ∆ ( z, τ ) (cid:1)i z = ̺ , where c f,̺ ( ℓ ) denotes the ℓ -th coefficient in the elliptic expansion (3.1) of f at ̺ ∈ H and Γ ̺ is the stabilizer of ̺ in Γ = Γ / {± } . Remark 1.7. (1) The sum over ̺ ∈ Γ \ H only runs over the finitely many poles of f modulo Γ.(2) Using a vector-valued setting as in [2, 11], the methods of the present paper can be appliedto compute the Shintani theta lift of meromorphic cusp forms for Γ ( N ).(3) Let ∆ = 1 and k ∈ N even. The authors of [8] constructed a function b Ψ( z, τ ) on H × H that is real-analytic in both variables and transforms like a modular form of weight2 − k in z for Γ and of weight k + in τ for Γ (4). Comparing their constructionwith Theorem 1.6, it is not hard to see that the Shintani theta lift Φ k, ( f, τ ) of f ∈ S k is, up to addition of a holomorphic cusp form, given by a linear combination of thefunctions [ R n − − k,z ( b Ψ( z, τ ))] z = ̺ , with ̺ running over the poles of f in Γ \ H and n ∈ N with c f,̺ ( − n ) = 0.1.4. Outline of the proof of Theorem 1.2.
We now briefly explain how the above resultsabout the Shintani theta lift of meromorphic modular forms imply the formula in Theorem 1.2.For the details of the proof we refer to Section 5. We let k be even and ∆ = 1, and drop ∆from the notation. For simplicity, we assume S k = { } , that is, k ∈ { , } .The proof of Theorem 1.2 uses the theory of locally harmonic Maass forms introduced byKane, Kohnen and the second author in [5], together with an intimate interplay between theShintani theta lift of meromorphic cusp forms studied in this work and another theta liftinvestigated by Kane, Viazovska, and the second author in [7]. For a harmonic Maass form τ g ( τ ) of weight − k for Γ (4) satisfying the Kohnen plus space condition this theta liftis defined by a regularized integralΦ ∗ k ( g, z ) := Z regΓ (4) \ H g ( τ )Θ ∗ k ( z, τ ) dudvv , where Θ ∗ k ( z, τ ) is a theta function which has weight k − in τ and weight 2 − k in z (see(2.2) for its definition), and the regularized integral is defined as in (5.3) below. In [7], theauthors showed that for D > D -th weakly holomorphic Poincar´e series P − k,D ( τ ) = q − D + O (1) of weight − k is a multiple of a so-called locally harmonic Maassform F − k,D ( z ), that is, Φ ∗ k (cid:16) P − k,D , z (cid:17) • = F − k,D ( z ) , (1.5)where the symbol • = means equality up to some non-zero constant factor. The function F − k,D , which was first studied in [5], transforms like a modular form of weight 2 − k for Γand is harmonic on H up to jump singularities along the geodesics C Q for Q ∈ Q D . Moreover, CLAUDIA ALFES-NEUMANN, KATHRIN BRINGMANN, AND MARKUS SCHWAGENSCHEIDT it turns out that in the case S k = { } the locally harmonic Maass form F − k,D is in factlocally a polynomial, which has been explicitly determined in [5].On the other hand, let us consider the Shintani theta lift Φ k ( f k, A , τ ) of the meromorphiccusp form f k, A for a class A ∈ Q d / Γ with d <
0. A short calculation using the Fourier expan-sion from Theorem 1.6 shows that the lowering operator L := − iv ∂∂τ acts on Φ k ( f k, A , τ )by L (Φ k ( f k, A , τ )) • = h R k − − k,z (Θ ∗ k ( z, τ )) i z = z A , (1.6)where z A is a fixed CM point z Q for some Q ∈ A . We apply the iterated raising operator to(1.5) and plug in (1.6) to obtain Z regΓ (4) \ H P − k,D ( τ ) L (Φ k ( f k, A , τ )) dudvv • = h R k − − k ( F − k,D ( z )) i z = z A . The integral on the left-hand side can now be evaluated by a standard argument using Stokes’Theorem. It turns out that it essentially equals the D -th Fourier coefficient of Φ k ( f k, A , τ ),which is the D -th trace of cycle integrals of f k, A . Hence we arrive attr f k, A ( D ) • = h R k − − k ( F − k,D ( z )) i z = z A . The action of the iterated raising operator on the locally polynomial function F − k,D caneasily be calculated, which yields the formula in Theorem 1.2.1.5. Organization of the paper.
We start with a section on the necessary preliminariesabout theta functions, regularized inner products, and cycle integrals of meromorphic cuspforms. In the remaining part of this work, we give the proofs of the above results. Toprove Theorem 1.5 we derive elliptic expansions of real-analytic functions on H and studyregularized inner products of meromorphic and real-analytic modular forms in Section 3. Thecomputation of the Fourier expansion from Theorem 1.6 is explained in Section 4. In Section 5,we give the proof of Theorem 1.1 and Theorem 1.2. Finally, in Section 6 we provide somedetails on the numerical evaluation of the rational formulas for the traces from Theorem 1.2. Acknowledgements
We thank Jan Bruinier for helpful discussions and Stephan Ehlen, Jens Funke, PavelGuerzhoy, Chris Jennings-Shaffer, Ken Ono, and Shaul Zemel for comments on an earlierversion of this paper. Moreover, we thank the referee for helpful comments.2.
Preliminaries
Theta functions.
For k ∈ N and ∆ ∈ Z a fundamental discriminant satisfying ( − k ∆ >
0, the Shintani theta function is defined asΘ k, ∆ ( z, τ ) := y − k v X D ∈ Z X Q ∈Q | ∆ | D χ ∆ ( Q ) Q ( z, k e − πv Q z | ∆ | e πiDτ . (2.1)Note that Θ k, ∆ ( z, τ ) would vanish identically if ( − k ∆ <
0. The function Θ k, ∆ ( − z, τ ) isreal-analytic in both variables and transforms like a modular form of weight 2 k in z for Γ andweight k + in τ for Γ (4) (see Proposition 3.2 of [7]). Moreover, as a function of z it hasmoderate growth at i ∞ (see Proposition 4.2 of [2]). N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 7
We also consider the Millson theta functionΘ ∗ k, ∆ ( z, τ ) := v X D ∈ Z X Q ∈Q | ∆ | D χ ∆ ( Q ) Q z Q ( z, k − e − πv Q z | ∆ | e πiDτ , (2.2)which transforms like a modular form of weight 2 − k in z for Γ and weight k − in τ forΓ (4) (see Proposition 3.2 of [7]).2.2. Regularized inner products.
Next, we describe the regularized inner product in (1.3),which was first introduced by Petersson [19] and later rediscovered and extended by Harveyand Moore [14], Borcherds [3], Bruinier [10], and others. We denote by [ ̺ ] , . . . , [ ̺ r ] ∈ Γ \ H the equivalence classes of all of the poles of f on H and we choose a fundamental domain F ∗ for Γ \ H such that ̺ ℓ ∈ Γ ̺ ℓ F ∗ for all 1 ≤ ℓ ≤ r . For any ̺ ∈ H and ε > ε -ball around ̺ , B ε ( ̺ ) := { z ∈ H : | X ̺ ( z ) | < ε } , X ̺ ( z ) := z − ̺z − ̺ . Let g : H → C be real-analytic and assume that g transforms like a modular form of weight 2 k for Γ and is of moderate (i.e., polynomial) growth at i ∞ . We define the regularized Peterssoninner product of f and g by h f, g i reg := lim ε ,...,ε r → Z F ∗ \ S rℓ =1 B εℓ ( ̺ ℓ ) f ( z ) g ( z ) y k dxdyy . (2.3)We see in Proposition 3.2 that the regularized inner product exists. Remark 2.1.
Similar regularized inner products in the case that both f and g are mero-morphic cusp forms or weakly holomorphic modular forms have recently been studied, forexample, in [6, 24].2.3. Regularized cycle integrals of meromorphic cusp forms.
Let
D > Q = [ a, b, c ] ∈ Q D . The associated geodesic C Q is a semi-circle centered at the real line if a = 0, and a vertical line if a = 0. We orient it counterclockwise if a > − cb to i ∞ if a = 0 and b >
0. If poles of f lie on C Q , then we modify it as follows. For every pole ̺ of f lying on C Q choose ε > f lie on B ε ( ̺ ).We denote by C ± Q,ε the path that agrees with C Q outside of every such ball but circumventsevery pole ̺ of f along the boundary arc of B ε ( ̺ ) that lies in the connected component of H \ C Q with ± Q z >
0. Moreover c Q := Γ Q \ C Q is the image of C Q in the modular curve Γ \ H and c ± Q,ε := Γ Q \ C ± Q,ε .We define the regularized geodesic cycle integral of f ∈ S k along c Q by Z reg c Q f ( z ) Q ( z, k − dz := 12 lim ε → Z c + Q,ε f ( z ) Q ( z, k − dz + Z c − Q,ε f ( z ) Q ( z, k − dz ! . This is sometimes also called the Cauchy principal value of the geodesic cycle integral, seeSection 2.4 of [11]. Furthermore, if no pole of f lies on c Q , then the above definition agreeswith the usual definition of geodesic cycle integrals. CLAUDIA ALFES-NEUMANN, KATHRIN BRINGMANN, AND MARKUS SCHWAGENSCHEIDT Proof of Theorem 1.5
A meromorphic function f : H → C has an elliptic expansion near ̺ ∈ H of the shape f ( z ) = ( z − ̺ ) − k X n ≫−∞ c f,̺ ( n ) X n̺ ( z ) , (3.1)with coefficients c f,̺ ( n ) ∈ C . For a proof see Proposition 17 in Zagier’s part of [12], note thatthe required modularity of f in the cited proposition is actually not necessary.In order to prove Theorem 1.5, we would like to plug in elliptic expansions of f andΘ k, ∆ ( z, τ ) near the poles of f . Note that z Θ k, ∆ ( z, τ ) is real-analytic. The shape ofelliptic expansions of real-analytic functions are described in the following lemma. Lemma 3.1.
Let κ ∈ Z , ̺ ∈ H , and g : H → C be real-analytic near ̺ . Then g has an ellipticexpansion of the shape (near ̺ ) g ( z ) = ( z − ̺ ) − κ X n ∈ Z c g,̺ ( | X ̺ ( z ) | , n ) X n̺ ( z ) , (3.2) with coefficients c g,̺ ( r, n ) ∈ C , which are analytic as functions of the real variable r . Near r = 0 we have the Taylor expansion c g,̺ ( r, n ) = X m ≥ max { , − n } a g,̺,n ( m ) r m , (3.3) with coefficients a g,̺,n ( m ) ∈ C given in (3.6) . The constant term c g,̺ (0 , n ) is given by c g,̺ (0 , n ) = ( n ! (2 i ) κ Im( ̺ ) n + κ R nκ ( g )( ̺ ) if n ≥ , if n < . (3.4) Proof.
We generalize the proof of Proposition 17 in Zagier’s part in [12]. Expanding as aTaylor series we obtain g ( ̺ + W ) = X a,b ≥ (cid:20) ∂ b ∂z b ∂ a ∂z a g ( z ) (cid:21) z = ̺ W a a ! W b b ! . Writing ̺ − ̺w − w = ̺ + i̺ w − w ( ̺ := Im( ̺ )) we obtain, for | w | sufficiently small, the formula(1 − w ) − κ g (cid:18) ̺ − ̺w − w (cid:19) = (1 − w ) − κ X a,b ≥ (cid:20) ∂ b ∂z b ∂ a ∂z a g ( z ) (cid:21) z = ̺ (cid:16) i̺ w − w (cid:17) a a ! (cid:16) − i̺ w − w (cid:17) b b ! . (3.5)Expanding (1 − w ) − κ − a and (1 − w ) − b using the Binomial Theorem, the right-hand side of(3.5) becomes X a,b,j,ℓ ≥ (cid:18) κ + a + j − j (cid:19)(cid:18) b + ℓ − ℓ (cid:19) (cid:20) ∂ b ∂z b ∂ a ∂z a g ( z ) (cid:21) z = ̺ (2 i̺ ) a a ! ( − i̺ ) b b ! w a + j − b − ℓ | w | b +2 ℓ . We reorder the summation by setting n = a + j − b − ℓ and m = b + ℓ . Plugging in w = X ̺ ( z )and using the formulas 1 − w = 2 i̺ z − ̺ , ̺ − ̺w − w = z, N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 9 we obtain the elliptic expansion g ( z ) = ( z − ̺ ) − κ X n ∈ Z X m ≥ max { , − n } a g,̺,n ( m ) | X ̺ ( z ) | m X n̺ ( z )with coefficients a g,̺,n ( m ):= (2 i̺ ) κ X ≤ a ≤ m + n ≤ b ≤ m (cid:18) κ + m + n − m + n − a (cid:19)(cid:18) m − m − b (cid:19) (cid:20) ∂ a ∂z a ∂ b ∂z b g ( z ) (cid:21) z = ̺ (2 i̺ ) a a ! ( − i̺ ) b b ! . (3.6)The formula for c g,̺ (0 , n ) = a g,̺,n (0) follows from (3.6) and (56) in Zagier’s part in [12]. (cid:3) Using the elliptic expansion of a real-analytic function given in Lemma 3.1, we can nowprove that the regularized Petersson inner product of a meromorphic cusp form and a real-analytic modular form exists.
Proposition 3.2.
The regularized Petersson inner product, defined in (2.3) , exists.Proof.
We divide F ∗ into a compact domain containing all of the poles ̺ , . . . , ̺ r of f , and aremaining set on which f is holomorphic. The integral over the second set converges since f decays like a cusp form towards i ∞ and g is of moderate growth at i ∞ . Hence it suffices toshow that for every pole ̺ := ̺ ℓ of f and δ > ε → Z A δε ( ̺ ) ∩F ∗ f ( z ) g ( z ) y k dxdyy , A δε ( ̺ ) := B δ ( ̺ ) \ B ε ( ̺ ) , exists. In order to obtain an integral over the whole annulus A δε ( ̺ ), we recall that, by (2a.15)of [18], we have the disjoint union B ε ( ̺ ) = ˙ [ M ∈ Γ ̺ M ( B ε ( ̺ ) ∩ F ∗ ) . (3.7)Therefore we can write, using the modularity of f and g , Z A δε ( ̺ ) ∩F ∗ f ( z ) g ( z ) y k dxdyy = 1 (cid:12)(cid:12) Γ ̺ (cid:12)(cid:12) Z A δε ( ̺ ) f ( z ) g ( z ) y k dxdyy . (3.8)We plug in the elliptic expansions of f from (3.1) and g from (3.2) with κ = 2 k to rewritethe right-hand side of (3.8) as1 (cid:12)(cid:12) Γ ̺ (cid:12)(cid:12) Z A δε ( ̺ ) X n ≫−∞ m ∈ Z c f,̺ ( n ) c g,̺ ( | X ̺ ( z ) | , m ) X n̺ ( z ) X m̺ ( z ) y k | z − ̺ | k dxdyy . We next make the change of variables X ̺ ( z ) = e iϑ r with 0 < ϑ < π and ε < r < δ . Then ashort calculation shows that y k | z − ̺ | k = (cid:18) − r ̺ (cid:19) k , dxdyy = 4 r (1 − r ) dϑdr, where again ̺ = Im( ̺ ). Thus4(4 ̺ ) k (cid:12)(cid:12) Γ ̺ (cid:12)(cid:12) Z δε Z π X n ≫−∞ m ∈ Z c f,̺ ( n ) c g,̺ ( r, m ) e iϑ ( n − m ) r m + n +1 (cid:0) − r (cid:1) k − dϑdr. (3.9)The integral over ϑ vanishes unless m = n , in which case it equals 2 π , thus (3.9) becomes8 π (4 ̺ ) k (cid:12)(cid:12) Γ ̺ (cid:12)(cid:12) Z δε X n ≫−∞ c f,̺ ( n ) c g,̺ ( r, n ) r n +1 (cid:0) − r (cid:1) k − dr. The limit as ε → n ≥ Z δ c g,̺ ( r, n ) r n +1 (cid:0) − r (cid:1) k − dr exists for the finitely many n < c f,̺ ( n ) = 0. Indeed, the integral exists since for n < c g,̺ ( r, n ) = O ( r | n | ) by (3.3). This finishes the proof. (cid:3) We are now ready to prove Theorem 1.5.
Proof of Theorem 1.5.
Noting that g ( z ) = Θ k, ∆ ( z, τ ) is real-analytic in z and of moderategrowth at i ∞ , Proposition 3.2 shows that the regularized Shintani theta lift (1.3) exists andhence transforms like a modular form of weight k + for Γ (4). The fact that the Shintanitheta lift is real-analytic follows from its Fourier expansion given in Theorem 1.6. This finishesthe proof of Theorem 1.5. (cid:3) Proof of Theorem 1.6
In this section we compute the Fourier expansion of the Shintani theta lift and therebyprove Theorem 1.6.4.1.
Preliminary computations.
We start with a useful formula which can be viewed asa generalization of the Residue Theorem.
Lemma 4.1.
Let ̺ ∈ H , let f : H → C be meromorphic near ̺ , and let g : H → C bereal-analytic near ̺ . Then we have the formula lim ε → Z ∂B ε ( ̺ ) f ( z ) g ( z ) dz = π Im( ̺ ) X n< c f,̺ ( n ) c g,̺ (0 , − n − , where c f,̺ ( n ) and c g,̺ ( r, n ) are the coefficients of the elliptic expansions (3.1) of f and (3.2) of g at ̺ (with κ = 2 − k ).Proof. For ε > f and of g toobtain that Z ∂B ε ( ̺ ) f ( z ) g ( z ) dz = Z ∂B ε ( ̺ ) ( z − ̺ ) − X n ≫−∞ m ∈ Z c f,̺ ( n ) c g,̺ ( | X ̺ ( z ) | , m ) X m + n̺ ( z ) dz = X n ≫−∞ m ∈ Z c f,̺ ( n ) c g,̺ ( ε, m ) Z ∂B ε ( ̺ ) ( z − ̺ ) m + n ( z − ̺ ) m + n +2 dz. N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 11
In the last step we use that | X ̺ ( z ) | = ε on ∂B ε ( ̺ ). By the Residue Theorem the last integralvanishes unless m = − n −
1, in which case it equals πi̺ − ̺ . Taking the limit ε → c g,̺ (0 , − n −
1) = 0 for n ≥ (cid:3) To ease notation, define ϕ Q ( z, v ) := y − k v Q ( z, k e − πv Q z | ∆ | , so that the Shintani theta function can be written asΘ k, ∆ ( z, τ ) = X D ∈ Z X Q ∈Q | ∆ | D χ ∆ ( Q ) ϕ Q ( z, v ) e πiDτ . A short calculation, using that ∂∂z Q z = − y i Q ( z, Lemma 4.2.
For z ∈ H with Q z = 0 and all v > the function φ Q , defined in (1.4) , satisfies L z ( φ Q ( z, v )) = ϕ Q ( z, v ) y k . Remark 4.3.
The function φ Q ( z, v ) was used in [2] to compute the Shintani theta lift ofharmonic Maass forms and in [11] for k = 1 to compute the Shintani theta lift of meromorphicmodular forms of weight two that are holomorphic at the cusps and have at most simple polesin H .Using Lemma 4.2, we see that the D -th Fourier coefficient of the regularized Shintani thetalift (1.3) of f ∈ S k with poles at [ ̺ ] , . . . , [ ̺ r ] ∈ Γ \ H is given bylim ε ,...,ε r → Z F ∗ \ S rℓ =1 B εℓ ( ̺ ℓ ) f ( z ) L z X Q ∈Q | ∆ | D χ ∆ ( Q ) φ Q ( z, v ) dxdyy . (4.1)We compute the Fourier coefficients for D ≤ D >
The coefficients of index D ≤ . For D ≤ Q ∈ Q | ∆ | D the function z φ Q ( z, v )is real-analytic. Using Stokes’ Theorem, we find that (4.1) equals − lim ε ,...,ε r → Z ∂ ( F ∗ \ S rℓ =1 B εℓ ( ̺ ℓ ) ) f ( z ) X Q ∈Q | ∆ | D χ ∆ ( Q ) φ Q ( z, v ) dz. (4.2)Here we also use the fact that f decays like a cusp form towards i ∞ . If ε , . . . , ε r aresufficiently small, then the boundary of F ∗ \ S rℓ =1 B ε ℓ ( ̺ ℓ ) consists of a disjoint union of theboundary arcs − ∂B ε ℓ ( ̺ ℓ ) ∩ F ∗ for 1 ≤ ℓ ≤ r and further remaining boundary pieces, whichcome in Γ-equivalent pairs with opposite orientation and hence cancel out in the integral dueto the modularity of the integrand. Therefore, (4.2) equals X ≤ ℓ ≤ r X Q ∈Q | ∆ | D χ ∆ ( Q ) lim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) ∩F ∗ f ( z ) φ Q ( z, v ) dz. (4.3)Using the disjoint union (3.7) again, we can rewrite (4.3) as X ≤ ℓ ≤ r (cid:12)(cid:12) Γ ̺ ℓ (cid:12)(cid:12) X Q ∈Q | ∆ | D χ ∆ ( Q ) lim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) f ( z ) φ Q ( z, v ) dz. Since φ Q ( z, v ) is real-analytic at z = ̺ ℓ , Lemma 4.1 combined with (3.4) yields thatlim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) f ( z ) φ Q ( z, v ) dz = ( − − k π X n ≥ c f,̺ ℓ ( − n ) Im( ̺ ℓ ) n − k ( n − h R n − − k,z ( φ Q ( z, v )) i z = ̺ ℓ . (4.4)We obtain the coefficients of index D ≤ The coefficients of index
D > . We first assume that the poles of f do not lie on anyof the geodesics C Q for Q ∈ Q | ∆ | D . Since ϕ Q ( z, v ) has a jump singularity along the geodesic C Q , one cannot directly apply Stokes’ Theorem. Hence, for δ > Q ∈ Q | ∆ | D wefirst cut out a δ -tube U δ ( C Q ) := { z ∈ H : | Q z | < δ } around C Q from F ∗ . Applying Stokes’ Theorem, we can rewrite (4.1) as X Q ∈Q | ∆ | D χ ∆ ( Q ) lim δ → Z ∂U δ ( C Q ) ∩F ∗ f ( z ) φ Q ( z, v ) dz + X ≤ ℓ ≤ r (cid:12)(cid:12) Γ ̺ ℓ (cid:12)(cid:12) X Q ∈Q | ∆ | D χ ∆ ( Q ) lim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) f ( z ) φ Q ( z, v ) dz. (4.5)Since z φ Q ( z, v ) is real-analytic at every pole of f by assumption, the integral in the secondterm can be evaluated using Lemma 4.1 as in (4.4) above. The first term in (4.5) is computedin the following proposition. Proposition 4.4.
Assume that no pole of f lies on any geodesic C Q for Q ∈ Q | ∆ | D . Thenwe have X Q ∈Q | ∆ | D χ ∆ ( Q ) lim δ → Z ∂U δ ( C Q ) ∩F ∗ f ( z ) φ Q ( z, v ) dz = p | ∆ | f, ∆ ( D ) . (4.6) Proof.
By definition (1.4) the left-hand side of (4.6) equals p | ∆ | X Q ∈Q | ∆ | D χ ∆ ( Q ) × lim δ → Z ∂U δ ( C Q ) ∩F ∗ f ( z ) Q ( z, k − sgn( Q z ) − erf √ πv Q z p | ∆ | !! dz. (4.7)The path ∂U δ ( C Q ) ∩ F ∗ can be divided into two paths which are distinguished by the signof Q z . The orientation of C Q is given such that the connected component of H \ C Q with Q z > ∂U δ ( C Q ) ∩ F ∗ which lies in thecomponent of H \ C Q with Q z > p | ∆ | X Q ∈Q | ∆ | D χ ∆ ( Q ) Z C Q ∩F ∗ f ( z ) Q ( z, k − dz. (4.8) N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 13
We split Q ∈ Q | ∆ | D as Q ◦ M with Q ∈ Q | ∆ | D / Γ and M ∈ Γ Q \ Γ, and rewrite (4.8) as p | ∆ | X Q ∈Q | ∆ | D / Γ X M ∈ Γ Q \ Γ χ ∆ ( Q ◦ M ) Z C Q ◦ M ∩F ∗ f ( z )( Q ◦ M )( z, k − dz = p | ∆ | X Q ∈Q | ∆ | D / Γ χ ∆ ( Q ) X M ∈ Γ Q \ Γ Z C Q ∩ M F ∗ f ( z ) Q ( z, k − dz = p | ∆ | X Q ∈Q | ∆ | D / Γ χ ∆ ( Q ) Z Γ Q \ C Q f ( z ) Q ( z, k − dz. For the first equality we use that χ ∆ ( Q ◦ M ) = χ ∆ ( Q ), ( Q ◦ M )( z,
1) = Q ( z, | − M , and C Q ◦ M = M − C Q . We obtain the formula as stated in the proposition. (cid:3) We next consider the case that the poles of f do lie on a geodesic C Q for some Q ∈ Q | ∆ | D .By similar arguments as above we find that (4.1) for D > p | ∆ | X Q ∈Q | ∆ | D ̺ ,...,̺ r C Q χ ∆ ( Q ) Z C Q ∩F ∗ f ( z ) Q ( z, k − dz + X Q ∈Q | ∆ | D ̺ ,...,̺ r / ∈ C Q χ ∆ ( Q ) X ≤ ℓ ≤ r (cid:12)(cid:12) Γ ̺ ℓ (cid:12)(cid:12) lim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) f ( z ) φ Q ( z, v ) dz + X Q ∈Q | ∆ | D ̺ ℓ ∈ C Q for some ℓ χ ∆ ( Q ) lim ε ,...,ε r → lim δ → Z ∂ ( U δ ( C Q ) ∪ S rℓ =1 B εℓ ( ̺ ℓ ) ) ∩F ∗ f ( z ) φ Q ( z, v ) dz. (4.9)As in (4.4), the second term of (4.9) can again be computed using Lemma 4.1. By pluggingin the definition (1.4) of φ Q ( z, v ), we can rewrite the third term of (4.9) as p | ∆ | X Q ∈Q | ∆ | D ̺ ℓ ∈ C Q for some ℓ χ ∆ ( Q ) lim ε → Z C + Q,ε ∩F ∗ f ( z ) Q ( z, k − dz + Z C − Q,ε ∩F ∗ f ( z ) Q ( z, k − dz ! − p | ∆ | X Q ∈Q | ∆ | D ̺ ℓ ∈ C Q for some ℓ χ ∆ ( Q ) X ≤ ℓ ≤ r̺ ℓ ∈ C Q (cid:12)(cid:12) Γ ̺ ℓ (cid:12)(cid:12) lim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) f ( z ) Q ( z, k − erf √ πv Q z p | ∆ | ! dz + p | ∆ | X Q ∈Q | ∆ | D ̺ ℓ ∈ C Q for some ℓ χ ∆ ( Q ) X ≤ ℓ ≤ r̺ ℓ / ∈ C Q (cid:12)(cid:12) Γ ̺ ℓ (cid:12)(cid:12) lim ε ℓ → Z ∂B εℓ ( ̺ ℓ ) f ( z ) φ Q ( z, v ) dz, (4.10)where C ± Q,ε is the modified geodesic defined in Section 2.3. The first term of (4.10) combineswith the first term of (4.9) to the D -th ∆-twisted trace of the regularized cycle integrals of f . Since the integrands in the second and third line of (4.10) are real-analytic near ̺ ℓ , the integrals can again be evaluated using Lemma 4.1. We obtain that the second and third termof (4.10) combine to( − − k π X Q ∈Q | ∆ | D ̺ ℓ ∈ C Q for some ℓ χ ∆ ( Q ) X ≤ ℓ ≤ r (cid:12)(cid:12) Γ ̺ (cid:12)(cid:12) X n< c f,̺ ℓ ( n ) Im( ̺ ℓ ) − n − k ( − n − h R − n − − k,z ( φ Q ( z, v )) i z = ̺ ℓ . This finishes the proof of Theorem 1.6.5.
Proofs of Theorem 1.1 and Theorem 1.2
Throughout this section we let ∆ = 1 and drop it from the notation. Furthermore, let d <
A ∈ Q d / Γ be a fixed class of quadratic forms.We consider the Shintani theta lift of the meromorphic cusp form f k, A . A short calculationverifies the formula Q ( z,
1) = p | d | z Q ) ( z − z Q ) X z Q ( z )for every Q ∈ Q d . In particular, the only poles of f k, A in H lie at the CM points z Q for Q ∈ A , and the elliptic expansion of f k, A at z Q has the shape f k, A ( z ) = ( z − z Q ) − k p | d | π (2 Im ( z Q )) k X − kz Q ( z ) + O (1) ! . Hence, by Theorem 1.6 the Shintani theta lift of f k, A is given byΦ k ( f k, A , τ ) = 12 X D> tr f k, A ( D ) e πiDτ − − k p | d | ( k − (cid:12)(cid:12) Γ z A (cid:12)(cid:12) h R k − − k,z (cid:0) θ ∗ − k ( z, τ ) (cid:1)i z = z A . (5.1)Furthermore, a short calculation yields L (Φ k ( f k, A , τ )) = 2 − k p | d | ( k − (cid:12)(cid:12) Γ z A (cid:12)(cid:12) h R k − − k,z (Θ ∗ k ( z, τ )) i z = z A . (5.2)Following [5], we define the function F − k,D ( z ) := D − k (cid:0) k − k − (cid:1) π X Q ∈Q D sgn( Q z ) Q ( z, k − ψ (cid:18) Dy | Q ( z, | (cid:19) , where ψ ( v ):= β ( v ; k − , ) is a special value of the incomplete β -function. The function F − k,D is a locally harmonic Maass form (in the sense of [5]) of weight 2 − k for Γ with jumpsingularities along the geodesics C Q for Q ∈ Q D . By Theorem 1.2 of [5] the function F − k,D maps to a constant multiple of f k,D under ξ − k . Proposition 5.1.
Under the assumptions of Theorem 1.1, we have X D> a F ( − D ) tr f k, A ( D ) = 2 k p | d | ( k − (cid:12)(cid:12) Γ z A (cid:12)(cid:12) X D> a F ( − D ) D k − R k − − k ( F − k,D ) ( z A ) . Proof.
Let P − k,D be the unique harmonic Maass form of weight − k for Γ (4) satisfying theKohnen plus space condition which maps to a cusp form of weight k + under the ξ -operatorand which has a Fourier expansion of the form e − πiDτ + O (1) at i ∞ (see [9] for its explicit N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 15 construction as a Poincar´e series). Then we obtain that F = P D> a F ( − D ) P − k,D . ByTheorem 1.3 (2) of [7], we have2 k − D k − F − k,D ( z ) = lim T →∞ Z F T (4) P − k,D ( τ )Θ ∗ k ( z, τ ) dudvv , (5.3)where F T (4) := [ M ∈ Γ (4) \ Γ M F T , F T := (cid:26) τ = u + iv ∈ H : | u | ≤ , | τ | ≥ , v ≤ T (cid:27) , is a truncated fundamental domain for Γ (4) \ H . Note that the normalization of the functions P − k,D and F − k,D in [7] differs from the normalization used in this paper, which explainsthe different constants when comparing equation (5.3) to Theorem 1.3 (2) in [7].Now we apply the iterated raising operator and plug z = z A into (5.3). A standardargument involving the dominated convergence theorem shows that, for z A not lying on anyof the geodesics C Q for Q ∈ Q D , we have2 k − D k − R k − − k ( F − k,D ) ( z A ) = lim T →∞ Z F T (4) P − k,D ( τ ) h R k − − k,z (Θ ∗ k ( z, τ )) i z = z A dudvv . Using (5.2), Stokes’ Theorem in the form given in Lemma 2 in [13], and the fact that F = P D> a F ( − D ) P − k,D is holomorphic on H , we obtain2 k − p | d | ( k − (cid:12)(cid:12) Γ z A (cid:12)(cid:12) X D> a F ( − D ) D k − R k − − k ( F − k,D ) ( z A )= 23 lim T →∞ Z F T (4) F ( τ ) ξ k + (Φ k ( f k, A , τ )) v − k dudvv = 23 lim T →∞ Z + iT − + iT (cid:18) F ( τ )Φ k ( f k, A , τ ) + 12 F e ( τ )Φ e k ( f k, A , τ ) + 12 F o ( τ )Φ o k ( f k, A , τ ) (cid:19) dτ. (5.4)Here we set, for f ( τ ) = P D ∈ Z a ( v, D ) e πiDτ , f e ( τ ) := X D ∈ Z D ≡ a (cid:16) v , D (cid:17) e πiD τ , f o ( τ ) := X D ∈ Z D ≡ a (cid:16) v , D (cid:17) e πi D e πiD τ . The integral in (5.4) picks out the constant term in the Fourier expansion of the integrand.Using (5.1) we get that (5.4) equals12 X D> a F ( − D ) tr f k, A ( D ) − lim T →∞ − k p | d | k − (cid:12)(cid:12) Γ z A (cid:12)(cid:12)X D ∈ Z a F ( − D ) X Q ∈Q D h R k − − k,z ( φ Q ( z, T )) i z = z A − lim T →∞ − k p | d | k − (cid:12)(cid:12) Γ z A (cid:12)(cid:12)X D ∈ Z a F ( − D ) X Q ∈Q D h R k − − k,z (cid:0) φ Q (cid:0) z, T (cid:1)(cid:1)i z = z A . (5.5)One can show that h R k − − k,z ( φ Q ( z, T )) i z = z A = P (cid:16) √ T (cid:17) erfc (cid:16) √ πT | Q z A | (cid:17) + P (cid:16) √ T (cid:17) e − πT Q z A , where P and P are polynomials. For D ≤ Q ∈ Q D \{ [0 , , } we have Q z > z ∈ H , for Q = [0 , ,
0] we have φ Q ( z, v ) = 0 by definition, and for those Q ∈ Q D with D > appearing above (i.e., a F ( − D ) = 0) we have Q z A > z A does not lie on any geodesic C Q for Q ∈ Q D with a F ( − D ) = 0, by assumption. Since erfc( C √ T ) and e − CT for C > T → ∞ , the limits on the right-hand side of (5.5) vanish, and weobtain the formula stated in the proposition. (cid:3) By Theorem 7.1 in [5], the function F − k,D is locally a polynomial if S k = { } . Moregenerally, by taking suitable linear combinations, we get the following result. Lemma 5.2.
Assuming the conditions as in Theorem 1.1, we have X D> a F ( − D ) D k − F − k,D ( z )= X D> a F ( − D ) − c k ( D )2 k (cid:0) k − k − (cid:1) + 2 − k X Q =[ a,b,c ] ∈Q D Q z > >a Q ( z, k − , (5.6) where c k ( D ) is the constant defined in (1.2) .Proof. Recall that the non-holomorphic and holomorphic Eichler integrals of a cusp form f ( τ ) = P n ≥ a f ( n ) q n ∈ S k are defined by f ∗ ( z ) := ( − i ) − k Z i ∞− z f ( − τ )( τ + z ) k − dτ, E f ( z ) := X n ≥ a f ( n ) n k − q n . By Theorem 7.1 of [5] the locally harmonic Maass form F − k,D ( z ) decomposes as a sum F − k,D ( z ) = D − k +12 (cid:0) k − k − (cid:1) f ∗ k,D ( z ) + D − k +12 ( k − (4 π ) k − E f k,D ( z ) + P − k,D ( z ) , (5.7)where P − k,D ( z ) := − c k ( D )2 k (cid:0) k − k − (cid:1) + 2 − k X Q =[ a,b,c ] ∈Q D Q z > >a Q ( z, k − is locally a polynomial. Note that the constant c ∞ defined in (7.3) of [5] is missing a factor2, which is corrected in the above formula.By Theorem 1.1 (2) of [7], we have2 k − (cid:0) k − k − (cid:1) D k − f k,D ( z ) = Φ ∗ k (cid:16) ξ − k (cid:16) P − k,D (cid:17) , z (cid:17) , where Φ ∗ k ( f, z ) := 16 lim T →∞ Z F T (4) f ( τ )Θ k ( z, τ ) v k + dudvv is the Shimura theta lift of a cusp form f of weight k + for Γ (4). Using that F = P D> a F ( − D ) P − k,D is holomorphic on H , we find that2 k − (cid:0) k − k − (cid:1) X D> a F ( − D ) D k − f k,D ( z ) = Φ ∗ k ξ − k X D> a F ( − D ) P − k,D ! , z ! = 0 , N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 17 which implies that the analogous linear combinations of the non-holomorphic and holomorphicEichler integrals f ∗ k,D and E f k,D in the decomposition (5.7) vanish. This yields the statedformula. (cid:3) In view of Proposition 5.1, we need to apply the iterated raising operator R k − − k to theright-hand side of (5.6). The following lemma is well-known and not hard to prove. Lemma 5.3.
For k ∈ N we have R k − − k (1) = ( − k +1 ( k − (cid:18) k − k − (cid:19) y − k . Next, we compute the iterated raising operator applied to Q ( z, k − . Lemma 5.4.
For k ∈ N and Q ∈ Q D with D > we have R k − − k (cid:16) Q ( z, k − (cid:17) = (cid:16) i √ D (cid:17) k − ( k − P k − (cid:18) iQ z √ D (cid:19) . Proof.
Since
D > M ∈ SL ( R ) such that Q ◦ M = [0 , √ D, Q ( z, | − M = ( Q ◦ M )( z, , Q Mz = ( Q ◦ M ) z , and the fact that the slash operator commuteswith the raising operator, we can assume without loss of generality that Q = [0 , √ D, Q ( z,
1) = √ Dz and Q z = √ D xy . We rewrite the iterated raising operator using formula (56)in Zagier’s part in [12] to get R k − − k (cid:16) z k − (cid:17) = ( − π ) k − k − X j =0 ( − k +1+ j (cid:18) k − j (cid:19) (2 − k + j ) k − − j (4 πy ) k − − j (2 πi ) j ∂ j ∂z j z k − = k − X j =0 (2 i ) j (cid:18) k − j (cid:19) (2 − k + j ) k − − j y k − − j ( k − k − − j )! z k − − j , where ( a ) m := a ( a + 1) · · · ( a + m −
1) is the Pochhammer symbol. We replace j k − − j and obtain, after some simplification, that this equals(2 i ) k − ( k − k − X j =0 (cid:18) k − j (cid:19)(cid:18) k − jj (cid:19) ixy − ! j = (2 i ) k − ( k − P k − (cid:18) ixy (cid:19) , where we use a formula for the Legendre polynomials which can be obtained by combining(22.3.2) and (22.5.24) in [1]. Recalling that ixy = iQ z √ D finishes the proof. (cid:3) We are now ready to prove Theorem 1.1 and Theorem 1.2.
Proof of Theorem 1.2.
By combining Proposition 5.1, Lemma 5.2, Lemma 5.3, and Lemma 5.4,we obtain the formulas given in Theorem 1.2. (cid:3)
Proof of Theorem 1.1.
From the functional equation of the Dirichlet L -function and its eval-utation at negative integers in terms of Bernoulli polynomials B k ( x ) ∈ Q [ x ] (see § B k ∈ Q , we obtain the rational number c k ( D ) = − f k − D k − (cid:0) kk (cid:1) B k k − (2 k − B k D X ℓ =1 (cid:18) D ℓ (cid:19) B k (cid:18) ℓD (cid:19) X m | f µ ( m ) (cid:18) D m (cid:19) m − k σ − k (cid:18) fm (cid:19) . (5.8) Furthermore, we have that | z A | , x A ∈ Q and y A ∈ p | d | Q , hence Q z A ∈ p | d | Q . This impliesthat P D> a F ( − D ) tr f k, A ( D ) ∈ Q , finishing the proof of Theorem 1.1. (cid:3) Numerical evaluation of tr f k, A ( D )In order to emphasize the explicit nature of the rational formulas given in Theorem 1.2, wegive some details on their numerical evaluation. The constant c k ( D ) can be computed using(5.8). To evaluate the sum over Q = [ a, b, c ] ∈ Q D appearing in the formulas, recall that Q z A > > a implies that the CM point z A = x A + iy A lies in the interior of the boundedcomponent of H \ C Q . This can only happen if x A lies between the two real endpoints of thesemi-circle C Q and if y A is smaller than the radius of C Q . It is not hard to see that for a < | a | x A − √ D ≤ b ≤ | a | x A + √ D, | a | ≤ √ D y A , c := D − b | a | ∈ Z . (6.1)There are only finitely many integral binary quadratic forms [ a, b, c ] with a < Q z A > k = 2 , A = [ A ] with A = [1 , , D = 5. Then we have d = − , z A = − √ = e πi , and | Γ z A | = 3. A short calculation gives that c (5) = 8. Theconditions (6.1) are only satisfied by the four quadratic forms [ − , ± , , [ − , ± , − Q = [ − , − ,
1] satisfies Q z A >
0. Hence the sum in tr f , [ A ] (5) has only one summand,whose value is [ − , − , z A = √ . Altogether, we obtain tr f , [ A ] (5) = 4. References [1] M. Abramowitz and I. A. Stegun,
Handbook of mathematical functions with formulas, graphs, and math-ematical tables , National Bureau of Standards Applied Mathematics Series (1964).[2] C. Alfes-Neumann and M. Schwagenscheidt, Shintani theta lifts of harmonic Maass forms , preprint arXiv:1712.04491 .[3] R. Borcherds,
Automorphic forms with singularities on Grassmannians , Invent. Math. (1998), 491–562.[4] K. Bringmann, P. Guerzhoy, and B. Kane,
On cycle integrals of weakly holomorphic modular forms , Math.Proc. Cambridge Philos. Soc. (2015), 439–449.[5] K. Bringmann, B. Kane, and W. Kohnen,
Locally harmonic Maass forms and the kernel of the Shintanilift , Int. Math. Res. Notices (2015), 3185–3224.[6] K. Bringmann, B. Kane, and A. von Pippich, Regularized inner products of meromorphic modular formsand higher Green’s Functions , to appear in Commun. Contemp. Math. (2018).[7] K. Bringmann, B. Kane, and M. Viazovska,
Theta lifts and local Maass forms , Math. Res. Lett. (2013),213–234.[8] K. Bringmann, B. Kane, and S. Zwegers, On a completed generating function of locally harmonic Maassforms , Compositio Math. (2014), 749–762.[9] K. Bringmann and K. Ono,
Arithmetic properties of coefficients of half–integral weight Maass–Poincar´eseries , Math. Ann. (2007), 591–612.[10] J. Bruinier,
Borcherds products on O(2, l ) and Chern classes of Heegner divisors , Lecture Notes in Math-ematics , Springer-Verlag, Berlin (2002).[11] J. Bruinier, J. Funke, ¨O. Imamo¯glu, and Y. Li, Modularity of generating series of winding numbers ,Research in the Math. Sci. (2018), 23.[12] J. Bruinier, G. van der Geer, G. Harder, and D. Zagier, The 1-2-3 of modular forms , Lectures from theSummer School on Modular Forms and their Applications held in Nordfjordeid, June 2004, Edited byKristian Ranestad, Springer-Verlag, Berlin (2008).
N THE RATIONALITY OF CYCLE INTEGRALS OF MEROMORPHIC MODULAR FORMS 19 [13] W. Duke, ¨O. Imamo¯glu, and ´A. T´oth,
Real quadratic analogues of traces of singular invariants , Int. Math.Res. Notices (2011), 3082–3094.[14] J. Harvey, and G. Moore, Algebras, BPS states, and strings , Nuclear Phys. B (1996), 315–368.[15] W. Kohnen
Fourier coefficients of modular forms of half-integral weight , Math. Ann. (1985), 237–268.[16] W. Kohnen and D. Zagier,
Modular forms with rational periods in “Modular forms”, ed. by R. A. Rankin,Ellis Horwood, (1985), 197–249.[17] S. Niwa.
Modular forms of half integral weight and the integral of certain theta-functions , Nagoya Math.J. (1974), 147–161.[18] H. Petersson, Konstruktion der Modulformen und der zu gewissen Grenzkreisgruppen geh¨origen automor-phen Formen von positiver reeller Dimension und die vollst¨andige Bestimmung ihrer Fourierkoeffzienten ,S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. (1950), 415–474.[19] H. Petersson, ¨Uber automorphe Orthogonalfunktionen und die Konstruktion der automorphen Formen vonpositiver reeller Dimension , Math. Ann. (1954), 33–81.[20] G. Shimura,
On modular forms of half integral weight , Ann. of Math. (2) (1973), 440–481.[21] T. Shintani, On construction of holomorphic cusp forms of half integral weight , Nagoya Math. J. (1975),83–126.[22] D. Zagier, Zetafunktionen und quadratische K¨orper: Eine Einf¨uhrung in die h¨ohere Zahlentheorie .Hochschultext (Berlin), Springer-Verlag (1981).[23] D. Zagier,
Modular forms associated to real quadratic fields . Invent. math. (1) (1975), 1–46.[24] S. Zemel, Regularized pairings of meromorphic modular forms and theta lifts , J. Number Theory (2016), 275–311.
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