A problem of Petersson about weight 0 meromorphic modular forms
aa r X i v : . [ m a t h . N T ] M a r A PROBLEM OF PETERSSON ABOUT WEIGHT 0 MEROMORPHICMODULAR FORMS
KATHRIN BRINGMANN AND BEN KANE
Abstract.
In this paper, we provide an explicit construction of weight 0 meromorphic modularforms. Following work of Petersson, we build these via Poincar´e series. There are two mainaspects of our investigation which differ from his approach. Firstly, the naive definition of thePoincar´e series diverges and one must analytically continue via Hecke’s trick. Hecke’s trick isfurther complicated in our situation by the fact that the Fourier expansion does not convergeeverywhere due to singularities in the upper half-plane so it cannot solely be used to analyticallycontinue the functions. To explain the second difference, we recall that Petersson constructed linearcombinations from a family of meromorphic functions which are modular if a certain principalparts condition is satisfied. In contrast to this, we construct linear combinations from a family ofnon-meromorphic modular forms, known as polar harmonic Maass forms, which are meromorphicwhenever the principal parts condition is satisfied. Introduction and statement of results
A special case of the Riemann–Roch Theorem gives a sufficient and necessary condition for theexistence of meromorphic modular forms with prescribed principal parts. Although this implies theexistence of meromorphic modular forms with certain prescribed principal parts, it unfortunatelyfails to explicitly produce them. Using Poincar´e series, Petersson achieved the goal of an explicitconstruction for negative weight forms in [22, 23], but did not cover the case of weight 0 consideredin this paper. This paper deals with difficulties caused by divergence of the naive Poincar´e seriesand also views the problem from a different perspective than Petersson’s. In particular, the questionis placed into the context of a larger space of non-meromorphic modular forms, allowing the usageof modern techniques to avoid some of the difficulties of Petersson’s method.To give a flavor of the differences between these methods, we delve a little deeper into the historyof Petersson’s related work in [22]. The relevant meromorphic modular forms are constructed viaa family of two-variable meromorphic Poincar´e series if the corresponding group only has one cuspand the weight is negative. The Poincar´e series have positive weight in one variable by construction,but the other variable can be used as a gateway between the space of positive-weight forms andtheir dual negative-weight counterparts. In this way, the existence of meromorphic modular formsimplied by the special case of the Riemann–Roch Theorem considered in [22] may be viewed asa sufficient and necessary condition for certain linear combinations of Poincar´e series to satisfy(negative weight) modularity in the second variable. Following this logic, Satz 3 of [22] providesan explicit version of the existence implied by Riemann–Roch.
Date : September 26, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Meromorphic modular forms, polar harmonic Maass forms, Hecke’s trick.The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of theKrupp foundation and the research leading to these results has received funding from the European Research Councilunder the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 -AQSER. The research of the second author was supported by grant project numbers 27300314 and 17302515 of theResearch Grants Council. etersson then pointed out two remaining tasks: the first one pertains to including generalsubgroups. He later achieved this in Satz 16 of [23], with an explicit representation of the forms givenin (56) of [23]. He then asked whether there is a generalization to weight 0. In this paper, we settlePetersson’s second question by viewing it in a larger space of the so-called polar harmonic Maassforms, generalizations of Bruinier-Funke harmonic Maass forms [7]. These are modular objectswhich are no longer meromorphic but which are instead annihilated by the hyperbolic Laplacian.In this larger space, principal parts may essentially be chosen arbitrarily and the principal partcondition of Riemann–Roch translates into a condition which determines whether a polar harmonicMaass form with a given principal part is meromorphic. The subspace of harmonic Maass formshave appeared in a number of recent applications. For example, Zwegers [28] recognized the mocktheta functions, introduced by Ramanujan in his last letter to Hardy, as “holomorphic parts” ofhalf-integral weight harmonic Maass forms. Generating functions for central values and derivativesof quadratic twists of weight 2 modular L -functions were later proven to be weight 1 / S N a set of inequivalent cusps of Γ ( N ) and for each ̺ ∈ S N , ℓ ̺ is the cuspwidth of ̺ . For z ∈ H , we furthermore denote ω z = ω z ,N := z , where Γ z ,N is the stabilizer groupof z in PSL ( Z ) ∩ Γ ( N ). Throughout, we write z = x + iy , z = z + i z , and τ d = u d + iv d . Thereis a well-known family of polar harmonic Maass forms P ̺ − k,n,N ( z ) with n ∈ − N , each of whichhas principal part e πinz/ℓ ̺ as z approaches ̺ ∈ S N and no other singularities in H ∪ S N . Thusfor an explicit construction of weight 0 forms, it only remains to build forms with singularities inthe upper half-plane. In particular, the main step in this paper is to use Hecke’s trick to explicitlydefine a family of polar harmonic Maass forms Y ,n,N ( z , z ) in (5.9) with principal parts X n z ( z ) at z = z and no other singularities in Γ ( N ) \ H ∪ S N , where X z ( z ) := z − z z − z . For this task, we take inspiration from Petersson in two different directions; firstly, an argumentin [19] is augmented to extend Hecke’s trick to the case when the Poincar´e series have poles, yielding Y , − ,N , and secondly differential operators constructed in [23] are applied to Y , − ,N to constructthe family Y ,n,N . The construction of these Poincar´e series is of independent interest; togetherwith the functions P ̺ ,n,N , they form a basis of the space of polar harmonic Maass forms withexplicit principal parts and we plan to study them further in future research. Hence the principalparts of any linear combination of P ̺ − k,n,N and Y − k,n,N may be quickly determined. Moreover,up to constant functions if k = 1, all weight 2 − k ≤ P ̺ − k,n,N and Y − k,n,N . In this language, the necessary and sufficient condition implied by (a special caseof) Riemann–Roch is equivalent to determining whether a given linear combination of these polarharmonic Poincar´e series is meromorphic. Theorem 1.1.
For Γ ( N ) -inequivalent points τ , . . . , τ r ∈ H and k ≥ , there exists a weight − k meromorphic modular form on Γ ( N ) with principal parts at each cusp ̺ equal to P n< a ̺ ( n ) e πinzℓ̺ nd principal parts in H given by r X d =1 ( z − τ d ) k − X n ≡ k − ω τd ) n< b τ d ( n ) X nτ d ( z ) if and only if, for every cusp form g ∈ S k ( N ) , the constants a ̺ ( n ) and b τ d ( n ) satisfy the principalpart condition (1.1) 12 πi X ̺ ∈S N X n> a ̺ ( − n ) a g,̺ ( n ) + r X d =1 iv d ω τ d X n ≡ − k (mod ω τd ) n> b τ d ( − n ) a g,τ d ( n −
1) = 0 , where a g,̺ ( n ) is the n th Fourier coefficient of g at the cusp ̺ and a g, z ( n ) is the n th coefficient inthe elliptic expansion of g around z . Specifically, the weight − k polar harmonic Maass form (1.2) X ̺ ∈S N X n< a ̺ ( n ) P ̺ − k,n,N ( z ) + r X d =1 X n ≡ k − ω τd ) n< b τ d ( n ) Y − k,n,N ( τ d , z ) is a meromorphic modular form if and only if (1.1) is satisfied.Remarks. (1) Note that (1.1) only has to be checked for dim C S k ( N ) many cusp forms.(2) For genus 0 subgroups there is a simpler direct proof of Theorem 1.1, using explicit basiselements.(3) An alternative approach for constructing basis elements is to average coefficients in the ellipticexpansion of a Maass form. For a good description of such types of Poincar´e series, see [15],while the case of forms with singularities at the cusps [14, 17].By computing the Fourier coefficients of the basis elements, Theorem 1.1 yields the Fourierexpansions of all meromorphic modular forms. Corollary 1.2.
Suppose that τ , . . . , τ r ∈ H are Γ ( N ) -inequivalent and that there exists a mero-morphic modular function f on Γ ( N ) with principal parts at each cusp ̺ equal to P n< a ̺ ( n ) e πinzℓ̺ and principal parts in H given by r X d =1 X n ≡ ω τd ) n< b τ d ( n ) X nτ d ( z ) . The function f has a Fourier expansion which is valid for y sufficiently large (depending on v , . . . , v d ). For m ∈ N , the m th Fourier coefficient of f is given by X ̺ ∈S N X n< a ̺ ( n ) c (cid:16) P ̺ ,n,N , m (cid:17) + r X d =1 X n ≡ ω τd ) n< b τ d ( n ) c ( Y ,n,N ( τ d , · ) , m ) , with c ( g, m ) the m th coefficient of g . The coefficients of f are explicitly given, independent of P ̺ ,n,N and Y ,n,N , in Theorem 6.2.Remarks. (1) While the functions Y − k,n,N are useful for Theorem 1.1, for k > Y − k, − ,N by applying another natural differential operator, known as the raising op-erator, which yield Fourier expansions closely resembling the expansions given by Hardy and amanujan for the reciprocal of the weight 6 Eisenstein series. The authors [5] applied thismethod to obtain the Fourier expansions of negative-weight meromorphic modular forms.(2) The expansions of F at the other cusps can easily be derived from Lemma 5.4 and the definition(5.9) of Y ,n,N .(3) The result for k > m th coefficient of Y − k, − ,N as a constant multiple of the weight 2 k Poincar´eseries with principal part e − πim z at i ∞ . Using this, one can write the Fourier coefficients of agiven weight 2 − k meromorphic modular form as the image of an operator acting on weight2 k meromorphic modular forms. The explicit version of Corollary 1.2 given in Theorem 6.2yields an analogous operator on weight 2 meromorphic modular forms, but we do not work outthe details here.In this paper, we do not extensively investigate the properties of the functions z
7→ Y ,n,N ( z , z ).However, there are a number of properties related to weight 2 meromorphic modular forms whichare worth noting. One can show that z
7→ Y , − ,N ( z , z ) satisfies weight 2 modularity. Taking thetrace over z = τ Q for roots τ Q of inequivalent binary quadratic forms Q of discriminant D < f k,D ( z ) := X Q =[ a,b,c ] b − ac = D Q ( z , − k . The analogous functions f k,D , with D >
0, are weight 2 k cusp forms which were investigated byZagier [26] and played an important role in Kohnen and Zagier’s construction of a kernel function[16] for the Shimura [24] and Shintani [25] lifts. Kohnen and Zagier then used this kernel functionto prove the non-negativity of twisted central L -values of cusp forms. As shown by Bengoechea [3],the f k,D functions, with D <
0, are meromorphic modular forms of weight 2 k with poles of order k at z = τ Q and which decay like cusp forms towards the cusps. The authors [4] proved that innerproducts of these meromorphic modular forms with other meromorphic modular forms lead to anew class of modular objects, the first case of which is a polar harmonic Maass form, and that theyalso appear as theta lifts, which was generalized to vector-valued forms by Zemel [27]. Noting theapplications of the f k,D functions for D >
0, it may be interesting to investigate the properties of Y , − ,N if z is a CM-point. In particular, the periods of these forms are of interest because theyhave geometric applications. This will be studied further in future research.We do however investigate one aspect of the properties in the z -variable here. Recall that if f is a weight 2 meromorphic modular form, then f ( z ) d z is a meromorphic differential, and, in theclassical language, we say that the differential is of the first kind if f is holomorphic, it is of the second kind if f is not holomorphic but the residue vanishes at every pole, and of the third kind ifall of the poles are simple (for further information about the connection between differentials andmeromorphic modular forms, see page 182 of [21]). One can use Y , − ,N to construct differentialsof all three kinds. Theorem 1.3. (1)
As a function of z , Y , − ,N ( z , z ) corresponds to a differential of the third kind. (2) The function z ξ ,z ( Y , − ,N ( z , z )) corresponds to a differential of the first kind (as a functionof z ). In other words, ξ ,z ( Y , − ,N ( z , z )) is a cusp form. (3) The function z D z ( Y , − ,N ( z , z )) corresponds to a differential of the second kind. Although we only investigate the Fourier coefficients of Y ,n,N in the z -variable, the same tech-niques can be applied to compute the Fourier coefficients in the z -variable. Noting the connectionsto differentials given above, it might be interesting to explicitly determine the behavior towardseach cusp in order to compute the differential of the third kind associated with Y , − ,N . nother direction future research may take involves the question of whether Y ,n,N may beconstructed in a similar manner for more general subgroups. The methods in this paper canindeed be extended to obtain more general subgroups. The main difficulty lies in proving analyticcontinuation of the Kloosterman zeta functions for these subgroups. Finally we want to mentionthat the properties as functions of z are of interest.The paper is organized as follows. In Section 2, we give background on polar harmonic Maassforms and in particular harmonic Maass forms. We then determine the shape of the elliptic expan-sions of polar harmonic Maass forms. In Section 3 we use Hecke’s trick together with a splitting of[19] to analytically continue two-variable elliptic Poincar´e series y k − Ψ k,N ( z , z ) to include k = 1.After that, we determine the properties of the analytic continuation as a function of z in Section4. In particular, y Ψ ,N ( z , z ) yields Y , − ,N ( z , z ), up to a constant multiple of the non-holomorphicweight 2 Eisenstein series b E ( z ) and is invariant under the action of Γ ( N ) as functions of z . TheFourier expansions of y Ψ ,N ( z , z ) at each cusp are then computed in Section 5 and an explicitbasis of all polar harmonic Maass forms is constructed. Finally, in Section 6, we extend a pairingof Bruinier and Funke [7] to obtain a pairing between weight 0 polar harmonic Maass forms andweight 2 cusp forms. For a fixed polar harmonic Maass form, this pairing is trivial if and only if thepolar harmonic Maass form is a meromorphic modular form. We conclude the paper by computingthe pairing between (1.2) and every cusp form in S ( N ), yielding Theorems 1.1 and 6.2. Acknowledgement
The authors thank Paul Jenkins, Steffen L¨obrich, Ken Ono, Martin Raum, Olav Richter, LarryRolen, and Shaul Zemel for helpful comments on an earlier version of this paper.2.
Preliminaries
In this section, we define the space of polar harmonic Maass forms and some of its distinguishedsubspaces and then determine the shape of the elliptic expansions of such forms. For backgroundon the well-studied subspace of harmonic Maass forms, which were introduced by Bruinier andFunke, we refer the reader for example to [7, 11, 14, 17].2.1.
Polar harmonic Maass forms.
We are now ready to define the modular objects which arecentral for this paper.
Definition.
For κ ∈ Z < and N ∈ N , a polar harmonic Maass form of weight κ on Γ ( N ) is afunction F : H → C := C ∪ {∞} which is real analytic outside of a discrete set and which satisfiesthe following conditions:(1) For every M = (cid:0) a bc d (cid:1) ∈ Γ ( N ), we have F | κ M = F , where F ( z ) | κ M := j ( M, z ) − κ F ( M z )with j ( M, z ) := cz + d .(2) The function F is annihilated by the weight κ hyperbolic Laplacian ∆ κ := − y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + iκy (cid:18) ∂∂x + i ∂∂y (cid:19) . (3) For all z ∈ H , there exists n ∈ N such that ( z − z ) n F ( z ) is bounded in a neighborhood of z .(4) The function F grows at most linear exponentially towards cusps of Γ ( N ).If one allows in (2) a general eigenvalue under ∆ κ , then one obtains a polar Maass form. Moreover weak Maass forms are polar Maass forms which do not have any singularities in H . e denote the space of all weight κ polar harmonic Maass forms on Γ ( N ) by H κ ( N ). Animportant subspace of H κ ( N ) is obtained by noting that ∆ κ splits as(2.1) ∆ κ = − ξ − κ ◦ ξ κ , where ξ κ := 2 iy κ ∂∂z . If F satisfies weight κ modularity, then ξ κ ( F ) is modular of weight 2 − κ .The kernel of ξ κ is the subspace M κ ( N ) of meromorphic modular forms, while one sees from thedecomposition (2.1) that if F ∈ H κ ( N ), then ξ κ ( F ) ∈ M − κ ( N ). It is thus natural to considerthe subspace H cusp κ ( N ) ⊆ H κ ( N ) consisting of those F for which ξ κ ( F ) is a cusp form. The space H cusp κ ( N ) decomposes into the direct sum of the subspace H cusp κ ( N ) of harmonic Maass formswhich map to cusp forms under ξ κ and the subspace H cusp κ ( N ) of polar harmonic Maass formswhose singularities in H are all poles and which are bounded towards all cusps. In addition to ξ κ , further operators on polar Maass forms appear in another natural splitting ∆ κ = − R κ − ◦ L κ .Here R κ := 2 i ∂∂z + κy − is the Maass raising operator and L κ := − iy ∂∂z is the Maass loweringoperator. The raising (resp. lowering) operator sends weight κ polar Maass forms to weight κ + 2(resp. κ −
2) polar Maass forms with different eigenvalues.Every F ∈ H cusp κ ( N ) has a Fourier expansion around each ̺ ∈ S N given by F ̺ ( z ) = X n ≫−∞ a F,̺ ( n ) e πinzℓ̺ + X n< b F,̺ ( n )Γ (cid:18) − κ, π | n | yℓ ̺ (cid:19) e πinzℓ̺ , where F ̺ := F | − κ M ̺ with M − ̺ ̺ = i ∞ ( M ̺ ∈ SL ( Z )) and Γ( s, y ) := R ∞ y t s − e − t dt is the in-complete gamma function . The first sum is the meromorphic part of F ̺ and the second sum is the non-meromorphic part of F ̺ . We call P n< a F,̺ ( n ) e πinzℓ̺ the principal part of F at ̺ . Furthermore,for each z ∈ H , there exist finitely many c n ∈ C such that, in a neighborhood around z , F ( z ) − ( z − z ) − κ X n< c n X n z ( z ) = O (1) . We call ( z − z ) − κ P n< c n X n z ( z ) the principal part of F at z . Construction of weak Maass forms.
We next recall a well-known construction of Maass–Poincar´e series, which constitute a basis of H cusp κ ( N ). Let Γ ̺ be the stabilizer of ̺ in Γ ( N ).Moreover, for s ∈ C and w ∈ R \ { } , set M κ,s ( w ) := | w | − κ M sgn( w ) κ ,s − ( | w | ) , where M ν,µ is the usual M -Whittaker function. Then let φ κ,s ( z ) := M κ,s (4 πy ) e πix and define for m ∈ − N the Maass Poincar´e series associated to the cusp ̺ P ̺κ,m,N,s ( z ) := X M ∈ Γ ̺ \ Γ ( N ) φ κ,s (cid:16) mℓ ̺ M ̺ M z (cid:17) j ( M ̺ M, z ) κ . Note that this function converges absolutely for σ := Re( s ) > s (1 − s ) + ( κ − κ ) / . We are particularly interested in the harmonic Maass forms P ̺κ,m,N,s arising from s = 1 − κ/
2. Tostate their Fourier expansions, we require some notation. Denote by I ν the ν th I -Bessel function nd define for ̺ = α/γ ∈ S N and n, j ∈ Z , the Kloosterman sum(2.2) K α,γ ( n, j ; c ) := X a (mod ℓ ̺ c ) d (mod c ) ad ≡ c ) c ≡− aαγ (mod N ) e ℓ ̺ c ( nℓ ̺ d + ja )with e ℓ ( x ) := e πixℓ . Finally, δ ̺,µ = 1 if ̺ is equivalent to the cusp µ modulo Γ ( N ) and 0 otherwise. Theorem 2.1.
For every n ∈ − N , the function P ̺κ,n,N,s has an analytic continuation to s = 1 .Moreover, for κ ≤ , P ̺κ,n,N ( z ) := 1(1 − κ )! P ̺κ,n,N, − κ ( z ) ∈ H cusp κ ( N ) . For ̺ = α/γ , the meromorphic part of its Fourier expansion at i ∞ is given by δ ̺, ∞ e πinz + (2 π ) − κ ℓ κ − ̺ | n | − κ X c ≥ K α,γ ( n, c ) c − κ + 2 π (cid:18) | n | ℓ ̺ (cid:19) − κ X j ≥ j κ − X c ≥ K α,γ ( n, j ; c ) c I − κ πc s | n | jℓ ̺ ! e πijz . Its principal part at µ ∈ S N is given by δ µ,̺ e πinzℓ̺ . Remark.
The Fourier expansion of P ̺κ,n,N was explicitly computed in Theorem 1.1 of [6]. Note thatthere are two small typos in the formula in [6]; the condition ( ad, c ) = 1 in (1.11) of [6] should bereplaced by ad ≡ c ) and the power of the cusp width t µ (denoted ℓ ̺ in this paper) in (1.15)should be 1 / − k/ − / − k/ k written as κ here).2.3. Elliptic expansions of Maass forms.
In this section, we determine the elliptic expansionsof polar harmonic Maass forms. We assume throughout that k ∈ N . Proposition 2.2. (1)
Let z ∈ H and assume that F satisfies ∆ − k ( F ) = 0 and that there exists n ∈ N such that r n z ( z ) F ( z ) is bounded in some neighborhood N around z . Then there exist a n , b n ∈ C , suchthat, for z ∈ N , (2.3) F ( z ) = ( z − z ) k − X n ≥− n a n X n z ( z ) − ( z − z ) k − X ≤ n ≤ n b n β (cid:0) − r z ( z ); 2 k − , − n (cid:1) X n z ( z )+ ( z − z ) k − X n ≤− b n β (cid:0) r z ( z ); − n, k − (cid:1) X n z ( z ) , where r z ( z ) := | X z ( z ) | and β ( y ; a, b ) := R y t a − (1 − t ) b − dt is the incomplete beta function. (2) If F ∈ H − k ( N ) , then, for every z ∈ H , the sums in (2.3) only run over those n which satisfy n ≡ k − ω z ) . Furthermore, if F ∈ H cusp2 − k ( N ) , then the second sum in (2.3) vanishes. Proof: (1) The claim follows precisely as in work of Hejhal, who computed the parabolic expan-sions of eigenfunctions under a differential operator closely related to the hyperbolic Laplacian inProposition 4.3 of [14].(2) The stabilizer group Γ z ⊆ Γ ( N ) of z is cyclic and we denote by E one of its generators. In(2a.15) of [22], Petersson showed that X z ( Ez ) = e πiω z X z ( z ) . n particular, r z ( z ) is invariant under Γ z and modularity of g together with uniqueness of expansionsin e inθ implies that, for c n = a n or c n = b n ,(2.4) c n ( z − z ) k − = c n e πinω z j ( E, z ) k − ( Ez − z ) k − . Moreover, using E z = z , we have j ( E, z ) k − ( Ez − z ) k − = j ( E, z ) k − ( z − z ) k − . Then, by (26) of [23], we have j ( M, z ) = e − πiω z , and thus c n = c n e πi ( n +1 − k ) ω z . Hence (2.4) holds if and only if c n = 0 or n ≡ k − ω z ), yielding the first statement in (2).To conclude the second statement in (2), we apply ξ − k to (2.3). Since ξ − k ( F ( z )) = − (4 z ) k − ( z − z ) − k X n ≤ n b n X − n − z ( z )is a cusp form, we require that b n = 0 for n ≥ (cid:3) Weight zero polar harmonic Maass Poincar´e series
In this section, we define a family of Poincar´e series P N,s via the Hecke trick and analyticallycontinue them to s = 0. We follow an argument of Petersson [19], who analytically continuedcertain cuspidal elliptic Poincar´e series. However, technical difficulties arise because the Poincar´eseries P N,s have poles in H . We show in Lemma 4.4 that the analytic continuations z y Ψ ,N ( z , z )of P N,s to s = 0 are elements of H cusp0 ( N ). Applying certain differential operators in the z variable,we construct a family of functions z
7→ Y ,m,N ( z , z ) ∈ H cusp0 ( N ) with principal parts X m z ( z ) for m ∈ − N . In Proposition 5.6 we then prove that these functions, together with constant functionsand the harmonic Maass form Poincar´e series P ̺ ,n,N (with n < H cusp0 ( N ).3.1. Construction of the Poincar´e series and their analytic continuations.
Define P N,s ( z , z ) := X M ∈ Γ ( N ) ϕ s ( M z , z ) j ( M, z ) | j ( M, z ) | s with ϕ s ( z , z ) := y s ( z − z ) − ( z − z ) − | z − z | − s . The goal is to analytically continue P N,s to s = 0 and show that this continuation is Γ ( N )-invariant, as a function of z . Note that the analytic continuation P N, , if it exists, is the weight 2analogue (as a function of z ) of Petersson’s elliptic Poincar´e seriesΨ k,N,ν k ( z , z ) := X M ∈ Γ ( N ) ν k ( M ) − j ( M, z ) − k ( M z − z ) − k ( M z − z ) − , where ν k is any multiplier system on Γ ( N ) which is consistent with weight 2 k modularity. Thisfunction converges absolutely uniformly for k > z , z in compact sets in which M z = z for any M ∈ Γ ( N ) (see Sections 1 and 2 of [20]). In particular, the absolute convergence of P N,s followsfrom Petersson’s work by majorizing by the absolute values in Ψ σ,N,ν σ for σ = Re( s ) > L ∈ Γ ( N ), P N,s ( z , Lz ) = P N,s ( z , z ) . In particular, the analytic continuation to s = 0, if it exists, satisfies P N, ( z , Lz ) = P N, ( z , z ) , nd hence is Γ ( N )-invariant. In order to study further properties of the resulting function, thegoal of this section is to analytically continue P N,s to s ∈ C with σ > − /
4. To explicitly statethe result, we require the Riemann zeta function ζ , the Euler totient function φ , and the standard Kloosterman sums K ( m, n ; c ) := X a,d (mod c ) ad ≡ c ) e πic ( na + md ) . Theorem 3.1.
The function X n ∈ Z ϕ s ( z + n, z ) + 2 X M = (cid:16) a bc d (cid:17) ∈ Γ ( N ) c ≥ ϕ s ( M z , z ) − ϕ s (cid:0) ac , z (cid:1) j ( M, z ) | j ( M, z ) | s + 2 X n ∈ Z Z R ϕ s ( t, z ) e − πint dt X m ∈ Z ( m,n ) =(0 , Z R ( z + w ) − − s ( z + w ) − s e − πimw dw X c ≥ N | c K ( m, n ; c ) c s − √ π s Γ (cid:0) + s (cid:1) Γ (1 + s ) z − − s · s ζ (2 s + 1) ζ (2 s + 2) φ ( N ) N s Y p | N − p − − s Z R ϕ s ( t, z ) dt provides the analytic continuation of P N,s to σ > − / for every z , z ∈ H such that M z = z has nosolutions in Γ ( N ) . We begin the proof of Theorem 3.1 by rewriting P N,s for σ > X n ∈ Z ϕ s ( z + n, z ) + 2 X M = (cid:16) a bc d (cid:17) ∈ Γ ( N ) c ≥ ϕ s ( M z , z ) − ϕ s (cid:0) ac , z (cid:1) j ( M, z ) | j ( M, z ) | s + 2 X M = (cid:16) a bc d (cid:17) ∈ Γ ( N ) c ≥ ϕ s (cid:0) ac , z (cid:1) j ( M, z ) | j ( M, z ) | s =: X + X + X . We break the proof of Theorem 3.1 into Lemmas 3.2, 3.3, and 3.5, in which we obtain the analyticcontinuation of P , P , and P , respectively. We show that P and P converge absolutelylocally uniformly in s if σ > − / z , z for which M z = z has no solution M ∈ Γ ( N ), whichwe assume throughout. Furthermore, we claim that P converges absolutely locally uniformly for σ > s with σ > − /
4. To validatereordering, we note that since the overall expression is absolutely locally uniformly convergent for σ > P is absolutely locally uniformly convergent if both P and P converge absolutely locallyuniformly. We prove this convergence for P in Lemma 3.3 and for P in Lemma 3.5.3.2. Analytic continuation of P . We begin by analytically continuing P in (3.1). Lemma 3.2.
The function (3.2) 2 X n ∈ Z Z R ϕ s ( t, z ) e − πint dt X m ∈ Z ( m,n ) =(0 , Z R ( z + w ) − − s ( z + w ) − s e − πimw dw X c ≥ N | c K ( m, n ; c ) c s − √ π s Γ (cid:0) + s (cid:1) Γ (1 + s ) z − − s · s ζ (2 s + 1) ζ (2 s + 2) φ ( N ) N s Y p | N (cid:0) − p − − s (cid:1) − Z R ϕ s ( t, z ) dt rovides an analytic continuation of P to σ > − / . Proof:
For σ >
0, we have, using Poisson summation twice,(3.3) X = 2 X c ≥ N | c c − − s X n ∈ Z X a,d (mod c ) ad ≡ c ) Z R ϕ s (cid:16) ac + t, z (cid:17) e − πint dt × X m ∈ Z Z R (cid:18) z + dc + w (cid:19) − − s (cid:18) z + dc + w (cid:19) − s e − πimw dw. We next rewrite the right-hand side of (3.3). Shifting t t − a/c , the integral over t becomes(3.4) e πinac Z R ϕ s ( t, z ) e − πint dt = e πinac y s Z R ( t − z ) − − s ( t − z ) − − s e − πint dt. Similarly, letting w w − d/c , the integral over w equals(3.5) e πimdc Z R ( z + w ) − − s ( z + w ) − s e − πimw dw. Thus one formally obtains(3.6) X = 2 X n ∈ Z Z R ϕ s ( t, z ) e − πint dt X m ∈ Z Z R ( z + w ) − − s ( z + w ) − s e − πimw dw X c ≥ N | c K ( m, n ; c ) c s . To validate (3.6), one needs to verify that the triple sum converges absolutely for σ >
0. Forthis, we bound the Kloosterman sums trivially, to estimate the sum over c against X c ≥ c − − σ φ ( c ) ≪ X c ≥ c − − σ < ∞ . It remains to show that the double sum over n and m converges absolutely. Since the integrands in(3.4) and (3.5) are analytic in the integration variable for | Im( t ) | < y and | Im( w ) | < z respectively,we may shift the path of integration to Im( t ) = − sgn( n ) α and Im( w ) = − sgn( m ) β , respectivelyfor any 0 < α < y and 0 < β < z . A straightforward change of variables then shows that for any − / < σ < σ , the absolute value of (3.4) may be bounded against(3.7) e − π | n | α y σ Z R | t − i ( y + sgn( n ) α ) | − − σ | t + i ( y − sgn( n ) α ) | − − σ dt ≤ e − π | n | α Z R | t | − − σ (cid:18) t y + 1 (cid:19) − σ dt ≪ y,σ e − π | n | α , while(3.8) (cid:12)(cid:12)(cid:12)(cid:12)Z R ( z + w ) − − s ( z + w ) − s e − πimw dw (cid:12)(cid:12)(cid:12)(cid:12) ≪ z ,σ e − π | m | β . This validates (3.6).We next split the m = n = 0 term in (3.6) off and show that the remaining terms convergeabsolutely locally uniformly in s for − / < σ < σ . For this we require a well-known result ofWeil, which implies that for, any ε > | K ( m, n ; c ) | ≤ τ ( c ) c ( m, n, c ) ≪ c + ε ( m, n, c ) , here τ ( c ) is the number of divisors of c . Combining (3.9) with (3.7) and (3.8), the terms in (3.6)with ( m, n ) = (0 ,
0) may be bounded against ≪ α,β,σ , z ,y X c ≥ c − + ε − σ X n ∈ Z e − π | n | α X m ∈ Z (cid:16)p | m | + p | n | (cid:17) e − π | m | β . Hence all sums converge absolutely uniformly in s for − / < σ < σ .For m = n = 0, we use (52a) of [18] (see (40a) of [18] for the definition of A ) to evaluate(3.10) Z R ( z + w ) − − s ( z + w ) − s dw = − √ π s Γ (cid:0) + s (cid:1) Γ (1 + s ) s z − − s . Moreover the sum over c equals in this case F ( N, s ), where(3.11) F ( N, s ) := X n ≥ N | n φ ( n ) n s = φ ( N ) N s X n ≥ φ ( N n ) φ ( N ) n − s . Using that φ ( N n ) /φ ( N ) is multiplicative and comparing Euler factors on both sides gives that(3.12) F ( N, s ) = φ ( N ) N s Y p | N (cid:0) − p − s (cid:1) − ζ ( s − ζ ( s ) . Thus(3.13) X c ≥ N | c φ ( c ) c s = φ ( N ) N s +2 Y p | N (cid:0) − p − − s (cid:1) − ζ (2 s + 1) ζ (2 s + 2) . Plugging (3.10) and (3.13) into (3.3) for the m = n = 0 term, we thus obtain that P equals (3.2).It remains to show that the m = n = 0 term is indeed analytic in s for σ > − /
2. Since ζ (2 s + 2) does not vanish for σ > − / ζ (2 s + 1) only has a simple pole at s = 0, the function s ζ (2 s +1) ζ (2 s +2) is analytic for σ > − /
2. The finite factor in (3.13) is clearly analytic away from a pole at s = − σ > − /
2. Similarly, the ratio of the gamma factorsin (3.10) is analytic for s if σ > − / z − − s is analytic for s ∈ C . It thus remains to showthat R R ϕ s ( t, z ) dt is analytic in s . Since s ϕ s ( t, z ) is clearly analytic, it suffices to bound theintegrand locally uniformly. For this, we shift t ty + Re( z ), to obtain Z R | ϕ s ( t, z ) | dt = y σ Z R | t − z | − − σ dt = y − σ Z R (cid:0) t + 1 (cid:1) − − σ dt. Assuming − / < σ < σ < σ this is less than (cid:0) y − σ + y − σ (cid:1) Z R (cid:0) t + 1 (cid:1) − − σ dt. This verifies that the last term in (3.2) is analytic for − / < σ < σ < σ , finishing the proof. (cid:3) Analytic continuation of P . In this subsection we show that P converges absolutelyuniformly inside the rectangle R defined by | Im( s ) | ≤ R and − / < σ ≤ σ ≤ σ . Lemma 3.3. If M z = z has no solution M ∈ Γ ( N ) , then the series P converges absolutelyuniformly in R . Before proving Lemma 3.3, we show a technical lemma which proves useful later. emma 3.4. Let R be the rectangle defined by | Im( s ) | ≤ R and − / < σ ≤ σ ≤ σ . Then, forevery | W | < / , we have (cid:12)(cid:12) | W + 1 | − s − (cid:12)(cid:12) ≪ R | W | , (3.14) (cid:12)(cid:12) ( W + 1) − s (cid:12)(cid:12) ≪ R . (3.15) Proof:
To prove (3.14), we write W = re iθ and let f θ ( r ) = f θ,s ( r ) := (cid:12)(cid:12)(cid:12) re iθ + 1 (cid:12)(cid:12)(cid:12) − s = (cid:0) r cos( θ ) + r (cid:1) − s . Since r < /
2, Taylor’s Theorem yields that f θ ( r ) = 1 + f ′ θ ( c ) r for some 0 < c < r < /
2. But for 0 < c < / σ > −
1, we have (cid:12)(cid:12) f ′ θ ( c ) (cid:12)(cid:12) = | s | (cid:12)(cid:12) c cos( θ ) + c (cid:12)(cid:12) − σ − | θ ) + 2 c | ≤ | s | σ . Since inside the rectangle R , | s | and σ are bounded from above, we can conclude (3.14)To obtain (3.15), we note that the above proof of (3.14) implies that (cid:12)(cid:12) ( W + 1) − s (cid:12)(cid:12) = f θ,σ ( r ) ≪ R . (cid:3) We next prove Lemma 3.3.
Proof of Lemma 3.3:
In order to show absolute locally uniform convergence of P , we rewrite M as T n M with n ∈ Z and M = (cid:0) a bc d (cid:1) ∈ Γ ∞ \ Γ ( N ) such that | ac | ≤ . Abbreviating w := ac + n − z and M ∗ z := M z − ac = − c ( c z + d ) , the terms in the series P equal(3.16) y s j ( M, z ) | j ( M, z ) | s | M ∗ z + w | s (cid:18) M ∗ z + w ) ( M ∗ z + w ) − | w | (cid:19) + y s j ( M, z ) | j ( M, z ) | s | w | (cid:18) | M ∗ z + w | s − | w | s (cid:19) . We next determine the asymptotic growth of (3.16) in | w | and | M ∗ z | , with constants only de-pending on R . For this, we rewrite the first term in (3.16) as(3.17) − y s M ∗ z ( M ∗ z + 2 Re( w )) j ( M, z ) | j ( M, z ) | s | w | s +4 (cid:12)(cid:12)(cid:12) M ∗ z w + 1 (cid:12)(cid:12)(cid:12) s (cid:16) M ∗ z w + 1 (cid:17) (cid:16) M ∗ z w + 1 (cid:17) . Noting that | M ∗ z | = | c |·| c z + d | ≤ z , we estimate(3.18) M ∗ z + 2 Re( w ) ≪ z + | w | . We next rewrite the second term in (3.16) as(3.19) y s j ( M, z ) | j ( M, z ) | s | w | s +2 (cid:12)(cid:12)(cid:12)(cid:12) M ∗ z w + 1 (cid:12)(cid:12)(cid:12)(cid:12) − s − ! . In order to bound (3.19), we split the range on n and apply Lemma 3.4 for all n with | n | sufficiently large. In particular, one can show that if | n | ≥ | z | + 1 / / z , then (3.14) implies that(3.19) can be bounded against ≪ R y σ | j ( M, z ) | σ | M ∗ z | | w | − σ − ≤ y σ | j ( M, z ) | σ | M ∗ z | (cid:18) | n | − | z | − (cid:19) − σ − . oreover, for these n , (3.18) can be bounded against 3 / · | w | . We then use (3.15), once with s and twice with s = 1 /
2, estimating (3.17) against ≪ y σ | j ( M, z ) | σ | M ∗ z | | w | − σ − ≪ y σ | j ( M, z ) | σ | M ∗ z | (cid:18) | n | − | z | − (cid:19) − σ − . Using that | M ∗ z | = | c | − | j ( M, z ) | − , the contribution to P from | n | ≥ | z | + 1 / / z may hencebe bounded by(3.20) ≪ R y σ X M ∈ Γ ∞ \ Γ ( N ) c ≥ | c | − | j ( M, z ) | − − σ X n ≥ n − − σ ≤ (cid:0) y σ + y σ (cid:1) ζ (3 + 2 σ ) X M ∈ Γ ∞ \ Γ ( N ) c ≥ | j ( M, z ) | − − σ . The sum on M is half of the termwise absolute value of the weight 3 + 2 σ Eisenstein series withoutits constant term, and hence converges absolutely uniformly in z and σ > σ > − / P with | n | ≤ | z | + 1 / / z . For these, we have(3.21) y ≤ | w | ≤ | z | + 12 + | n | ≪ | z | + 1 + 1 z . In particular, if(3.22) (cid:12)(cid:12)(cid:12)(cid:12) M ∗ z y (cid:12)(cid:12)(cid:12)(cid:12) ≤ , then Lemma 3.4 can be applied. Thus, by (3.15), (3.18), and (3.21), the absolute value of the termsin (3.17) with | n | ≤ | z | + 1 / / z which satisfy (3.22) may be bounded by ≪ R y σ · | M ∗ z | (cid:16) z + | w | (cid:17) | j ( M, z ) | σ · | w | σ ≪ (cid:18) | z | + 1 + 1 z (cid:19) y − − σ | j ( M, z ) | σ | M ∗ z | . Similarly, (3.14) and (3.21) imply that (3.19) can be estimated against ≪ R y σ · | M ∗ z || j ( M, z ) | σ · | w | σ +3 ≪ y − − σ | j ( M, z ) | σ | M ∗ z | . Hence the sum over the absolute value of those terms in P for which | n | ≤ | z | + 1 / / z and(3.22) is satisfied may be bounded by(3.23) ≪ (cid:18) y (cid:19) y − − σ (cid:18) | z | + 1 + 1 z (cid:19) X M ∈ Γ ∞ \ Γ ( N ) c ≥ | j ( M, z ) | − − σ ≤ (cid:18) y (cid:19) (cid:0) y − − σ + y − − σ (cid:1) (cid:18) | z | + 1 + 1 z (cid:19) X M ∈ Γ ∞ \ Γ ( N ) c ≥ | j ( M, z ) | − − σ . Again, the sum over M is a majorant for the weight 3 + 2 σ Eisenstein series minus its constantterm, and hence uniformly converges for z ∈ H and σ > σ > − / | n | ≪ | z | + 1 + 1 / z and M which does not satisfy (3.22), denoting the set of such ( M, n ) by T . We naively bound the ontributions of T to (3.1), using the original splitting from the definition of P instead of thesplitting from (3.16). If M does not satisfy (3.22), then c z ≤ | c | · | c z + d | < y − and hence(3.24) | c | < √ y z ) − . For each c satisfying (3.24), if (3.22) is not satisfied, then | c z + d | < | c z + d | < yc ) − , and hence(3.25) | d | < | c z | + 2( yc ) − ≤ | c z | + 2 y − < | c z | + 2 y − . We conclude that T is finite, with T bounded by a constant only depending on z and z . Wemoreover bound y s j ( M, z ) | j ( M, z ) | s ≪ y σ z − − σ . Thus the contribution from elements in T to P (recalling that we are using the original splittingin (3.1)) may be estimated against y σ z − − σ max ( M,n ) ∈T (cid:12)(cid:12)(cid:12)(cid:12) M ∗ z + w ) ( M ∗ z + w ) | M ∗ z + w | s − | w | s +2 (cid:12)(cid:12)(cid:12)(cid:12) T≪ z ,z, R max ( M,n ) ∈T (cid:18) | M ∗ z + w | | M ∗ z + w | σ + 1 | w | σ +2 (cid:19) . Since Im ( M ∗ z + w ) ≥ y and | w | ≥ y , by (3.21), | M ∗ z + w | − − σ and | w | − σ − may be boundedagainst ≪ y y − σ + y − σ ≪ R ,y
1. We finally note that since T is finite, max ( M,n ) ∈T | M ∗ z + w | − exists unless M ∗ z + w = 0. However, M ∗ z + w = 0 if and only if T n M z = z , which is not solvableby assumption. This implies absolute locally uniform convergence in R . (cid:3) Analytic continuation of P . We finally consider P . Lemma 3.5. If M z = z has no solution M ∈ Γ ( N ) , then the series P converges absolutelyuniformly in R . Proof:
We have | ϕ s ( z + n, z ) | = y σ | n | − − σ (cid:12)(cid:12)(cid:12)(cid:12) z − zn (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) z − zn (cid:12)(cid:12)(cid:12)(cid:12) − − σ . Hence, by (3.15), the contribution over | n | ≥ | z | + | z | ) to P can be estimated against(3.26) ≪ R (cid:0) y σ + y σ (cid:1) X n ≥ | z | + | z | ) n − − σ ≪ y, R X n ≥ n − − σ = ζ (2 + 2 σ ) ≤ ζ (2 + 2 σ ) . For the terms with | n | ≤ | z | + | z | ), we obtain the estimate ≪ ( | z | + | z | + 1) max | n |≤ | z | + | z | ) | ϕ s ( z + n, z ) | . For each of these (finitely many) n , we use | T n z − z | ≥ Im ( T n z − z ) ≥ y to estimate | ϕ s ( z + n, z ) | = y σ | T n z − z | − | T n z − z | − − σ ≤ | T n z − z | − (cid:0) y − σ + y − σ (cid:1) . Furthermore, since T n z = z for all n ∈ Z by assumption, max | n |≤ | z | + | z | ) | T n z − z | − exists. Wemay hence (uniformly in R ) bound the contribution of these terms by(3.27) ( | z | + | z | + 1) (cid:0) y σ + y − σ (cid:1) max | n |≤ | z | + | z | ) | T n z − z | − , completing the proof. (cid:3) Theorem 3.1 is now a direct consequence of (3.1) and Lemmas 3.2, 3.3, and 3.5. . Properties of y Ψ ,N In this section, we explicitly compute the analytic continuation y Ψ ,N ( z , z ) := P N, ( z , z ) andinvestigate its properties. In particular, we show that it is modular and harmonic in both variables.4.1. The term P . In this section, we evaluate the analytic continuation of P . To state theresult, we let c N := − N − Q p | N (1 + p − ) − = − / [SL ( Z ) : Γ ( N )]. Proposition 4.1.
The analytic continuation of P to s = 0 is explicitly given by (4.1) c N z − π X m ≥ me πim z X n ≤ X c ≥ N | c K ( m, n ; c ) c e − πinz + X n ≥ X c ≥ N | c K ( m, n ; c ) c e − πinz . This function is annihilated by ∆ ,z and ∆ , z .Remark. The fact that (4.1) is annihilated by ∆ ,z alternatively follows by Lemma 4.4. Proof of Proposition 4.1:
We first evaluate the second term in (3.2) for s = 0. Using the factthat lim s → sζ (2 s + 1) = 1 /
2, Γ(1 /
2) = √ π , and ζ (2) = π /
6, it becomes(4.2) c N π z Z R ϕ ( t, z ) dt. To compute the integral in (4.2), we define more generally, for w ∈ H ∪ − H , w ∈ H , and n ∈ Z g n ( w , w ) := Z R ( w + t ) − ( w + t ) − e − πint dt. Shifting the path of integration to − i sgn( n − / ∞ , the Residue Theorem yields(4.3) g n ( w , w ) = n ≤ w ∈ H , πi ( w − w ) − e πinw if n ≤ w ∈ − H , πi ( w − w ) − e πinw if n > w ∈ − H , πi ( w − w ) − (cid:0) e πinw − e πinw (cid:1) if n > w = w ∈ H , − π ne πinw if n > w = w . Thus we in particular obtain Z R ϕ ( t, z ) dt = yg ( − z, − z ) = π, and hence (4.2) equals c N / z , giving the first term in (4.1).Next we turn to the first term in (3.2) with s = 0. To simplify the sums over m and n , we rewrite Z R ϕ ( t, z ) e − πint dt = yg n ( − z, − z ) , Z R z + w ) e − πimw dw = g m ( z , z ) . Plugging in (4.3) yields the sum over m > H × H due to the exponential decay in z and y in the sums over m and n , respectively. Hence the function is harmonic in both z and z because it is termwise. (cid:3) .2. The term P . We next consider P in (3.1). Proposition 4.2.
The series P with s = 0 converges absolutely locally uniformly in z and z if M z = z is not solvable for M ∈ Γ ( N ) and is meromorphic as a function of z and harmonic as afunction of z . Proof:
After rewriting, the series P with s = 0 becomes(4.4) i X M = (cid:16) a bc d (cid:17) ∈ Γ ( N ) c ≥ M z − z ) ( ac − z ) − M z − z ) ( ac − z ) cj ( M, z ) . Each summand is meromorphic as a function of z and harmonic as a function of z . It hence sufficesto prove locally uniform convergence in z and z to show that P has the desired properties. Sincethe argument for z is similar, we only prove the statement for z .We begin by constructing local neighborhoods around each z ∈ H for which M z = z with M ∈ Γ ( N ) does not have a solution. Since for each z ∈ H , { M z | M ∈ Γ ( N ) } is a lattice in H , δ z ( z ) = δ z,N ( z ) := min M ∈ Γ ( N ) | M z − z | exists. For δ, V, R >
0, we then define the set N z ( δ, V, R ) := (cid:26) z ∈ H (cid:12)(cid:12)(cid:12)(cid:12) δ z ( z ) ≥ δ, R ≤ | z | ≤ R, V ≤ z ≤ V (cid:27) . We first claim that for every τ ∈ H with δ z ( τ ) = 0, there exists δ > τ ) such that for R = | τ | and V = v , the set N z ( δ, V, R ) is a neighborhood of τ . In particular, we show that for ε > N z ( δ, v , | τ | ) contains the ball ofradius ε around z = τ . Firstly, if | z − τ | < ε for ε > N z ( δ, V, R ) are satisfied. It hence remains to show that forall M ∈ Γ ( N ) | M z − z | > δ if δ and ε are sufficiently small. To see this, we first note that β z ( z ) := min M ∈ Γ ( N ) | τ − M − z | > β z ( z ) = 0 if and only if δ z ( z ) = 0. If ε < β z ( τ ),then the triangle inequality implies that | M z − z | = (cid:12)(cid:12)(cid:12)(cid:12) j ( M, z ) j ( M, z ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) z − M − z (cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) j ( M, z ) j ( M, z ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:0)(cid:12)(cid:12) τ − M − z (cid:12)(cid:12) − | z − τ | (cid:1) ≥ (cid:12)(cid:12)(cid:12)(cid:12) j ( M, z ) j ( M, z ) (cid:12)(cid:12)(cid:12)(cid:12) ( β z ( τ ) − ε ) . For c = 0 this immediately gives that | M z − z | ≥ β z ( τ ) − ε , and hence, for δ sufficiently small(depending on τ but independent of M ), we have | M z − z | > δ . For c = 0 we rewrite(4.5) (cid:12)(cid:12)(cid:12)(cid:12) j ( M, z ) j ( M, z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) z + dc (cid:12)(cid:12)(cid:12)(cid:12) z + dc (cid:12)(cid:12) . As d/c → ±∞ , (4.5) converges to 1, while for d/c →
0, (4.5) converges to | z /z | . Thus (4.5) attainsa minimum J z ( z ) := min M ∈ Γ ( N ) | j ( M, z ) /j ( M, z ) | > . One sees directly that J z ( z ) is continuousas a function of z , and hence, for | z − τ | < ε satisfying ε < β z ( τ ) and ε < J z ( τ ), we have | M z − z | ≥ ( J z ( τ ) − ε ) ( β z ( τ ) − ε ) > . Choosing ε and δ sufficiently small (again, depending on τ but independent of M ), we concludethat | M z − z | ≥ δ and hence z ∈ N z ( δ, v , | τ | ). But then N z ( δ, v , | τ | ) contains the open ballaround τ of radius ε , and is hence a neighborhood of τ .We next claim that the series P converges uniformly in N z ( δ, V, R ). For this, we require auniform bound for P M ∈ Γ ∞ \ Γ ( N ) | j ( M, z ) | − − σ . Since this series is the termwise absolute valueof the weight 3 + 2 σ Eisenstein series, it is well-known to be smaller than a uniform constanttimes the value with z = i . Thus (3.20) implies that the contribution to P from the termswith | n | > | z | + 1 / / z may be bounded absolutely uniformly on any compact subset of H . imilarly, (3.23) implies a uniform estimate on compact subsets for the contribution of the termswith | n | ≤ | z | + 1 / / z satisfying (3.22).It hence remains to uniformly estimate the sum of the absolute value of those ( M, n ) ∈ T for z ∈ N z ( δ, V, R ). We begin by bounding y X ( M,n ) ∈T (cid:12)(cid:12)(cid:12)(cid:12) M ∗ z + w ) ( M ∗ z + w ) − | w | (cid:12)(cid:12)(cid:12)(cid:12) ≤ y X ( M,n ) ∈T | M ∗ z + w | | M ∗ z + w | + y X ( M,n ) ∈T | w | . Since | w | ≥ y by (3.21), the second sum is bounded by y X ( M,n ) ∈T | w | ≤ y T . For the first sum, we use the inequalities | M ∗ z + w | ≥ Im( M ∗ z + w ) ≥ y and | M ∗ z + w | ≥ min ( M,n ) ∈T | M ∗ z + w | ≥ δ z ( z ) ≥ δ to obtain y X ( M,n ) ∈T | M ∗ z + w | | M ∗ z + w | ≤ δ T . Thus y X ( M,n ) ∈T (cid:12)(cid:12)(cid:12)(cid:12) M ∗ z + w ) ( M ∗ z + w ) − | w | (cid:12)(cid:12)(cid:12)(cid:12) ≤ T (cid:18) δ + 1 y (cid:19) ≪ δ,y T . We next note that
T ≪ (cid:18) | z | + 1 + 1 z (cid:19) n M ∈ Γ ∞ \ Γ ( N ) (cid:12)(cid:12)(cid:12) | M ∗ z | > y o . However, (3.24) and (3.25) imply that(4.6) n M ∈ Γ ∞ \ Γ ( N ) (cid:12)(cid:12)(cid:12) | M ∗ z | > y o ≪ ( z y ) − (cid:16) ( z y ) − | z | + y − (cid:17) . Since the right-hand side of (4.6) is continuous in | z | and z , it may be uniformly bounded on N z ( δ, V, R ). This implies absolute locally uniform convergence of P , as desired. (cid:3) The term P . We next investigate the properties of the analytic continuation of P to s = 0. Proposition 4.3.
The series P with s = 0 converges locally uniformly in z and z for which M z = z is not solvable with M ∈ Γ ( N ) and is meromorphic as a function of z and harmonic as afunction of z . Proof:
For s = 0, the series P becomes(4.7) 2 X n ∈ Z ϕ ( z + n, z ) = − i X n ∈ Z (cid:18) z + n − z − z + n − z (cid:19) . Each term in (4.7) is holomorphic as a function of z and harmonic as a function of z . We againonly prove locally uniform convergence in z and leave the analoguous argument for z to the reader.Noting that, for every z ∈ N z ( δ, V, R ), the inequalitymax | n |≤ | z | + | z | ) | T n z − z | ≤ δ is satisfied, one obtains a uniform bound by (3.26) and (3.27). (cid:3) .4. Image under ξ ,z . As mentioned in the introduction, we obtain the principal part conditionfrom the Riemann–Roch Theorem via a pairing of Bruinier and Funke [7]. This pairing is betweenweight 2 k cusp forms and weight 2 − k polar harmonic Maass forms which map to cusp formsunder the operator ξ k,z in the splitting (2.1). For this reason, it is important to compute the imageof y Ψ ,N ( z , z ) under ξ ,z and prove that it is a cusp form, as we do in this section.For s, s ′ with real parts > τ , z ∈ H , we additionally require the functions P N,s,s ′ ( z , τ , z, z ) := X M ∈ Γ ( N ) j ( M, z ) | j ( M, τ ) | s ( M z − z ) | M τ − z | s ′ . Petersson [19] studied related functions on Γ( N ); one obtains P N,s,s ′ by taking a trace of Peters-son’s functions. Section 4 of [19] then implies that P N,s,s ′ ( z , τ , z, z ) has an analytic continuationto s ′ = s = 0 which is independent of τ and z , denoted here by Φ N ( z , z ). Lemma 4.4.
We have z y Ψ ,N ( z , z ) ∈ H cusp0 ( N ) with (4.8) ξ ,z ( y Ψ ,N ( z , z )) = Φ N ( z, z ) . Proof:
A direct calculation gives ∂∂z (cid:16) y s ( M z − z ) − (1+ s ) (cid:17) = i s ) y s ( M z − z ) − (2+ s ) ( M z − z ) . Thus, using locally uniform convergence in z , which can be shown by an argument similar to theproofs of Propositions 4.1, 4.2, and 4.3,(4.9) ∂∂z P N,s ( z , z ) = i s ) y s P N, s, s ( z , z , z, z ) . Taking the analytic continuation of both sides of (4.9) to s = 0, we conclude that ∂∂z ( y Ψ ,N ( z , z )) = i N ( z , z ) . By conjugating both P N,s,s ′ and s and then using (23) of [23] to switch the role of the variables,we conclude that Φ N ( z , z ) = Φ N ( z, z ) , implying (4.8). Finally, by Satz 2 of [19], Φ N ( z, z ) is a cusp form, giving the final claim. (cid:3) Expansions of y Ψ ,N ( z , z ) in other cusps and limiting behavior towards the cuspsand the proof of Theorem 1.3 In this section, we determine the principal parts of z y Ψ ,N ( z , z ) and then construct a basisof H cusp0 ( N ) by applying differential operators to y Ψ ,N ( z , z ) in the z variable. For this, we showthat y Ψ ,N ( z , z ) is the k = 1 analogue of y k − Ψ k,N ( z , z ) in the sense that their principal parts areas expected if k = 1. Before stating the proposition, we note that the principal parts coming fromthe meromorphic parts of (2 iy ) k − Ψ k,N ( z , z ) were computed as a special case of (50) of [23]; weexpound further upon this analogy in Lemma 5.5. Proposition 5.1. If z is an elliptic fixed point, then z y Ψ ,N ( z , z ) vanishes identically. If z ∈ H is not an elliptic fixed point, then, for every M ∈ Γ ( N ) , y Ψ ,N ( z , z ) has principal part j ( M, z )2 z j ( M, z ) X − M z ( z ) around z = M z and is bounded towards all cusps. o see the statement for an elliptic fixed point z , we rewrite each element in the sum over Γ ( N )in the definition of P N,s as M E r with M ∈ Γ ( N ) / Γ z , with E generating Γ z and r running(mod 2 ω z ). The sum over r then becomes X r (mod 2 ω z ) j ( E r , z ) − | j ( E r , z ) | − s = X r (mod 2 ω z ) e πirω z = ( ω z = 1 , ω z = 1 , where we used (26) of [23] (note that here a = −
1) to evaluate j ( E r , z ) = e − πirω z . Thus P N,s vanishes if z is an elliptic fixed point, and hence its analytic continuation is simply the zero function.If z is not an elliptic fixed point, then the possible poles of y Ψ ,N ( z , z ) come from the terms in P and P for which M z = z , since P converges for all z , z ∈ H . Furthermore, by determiningterms of (4.4) and (4.7) which contribute to the pole, a direct calculation yields that the residue ofthe principal part at z = M z must be i/j ( M, z ) . From this, one concludes that the principal partis j ( M, z )2 z j ( M, z ) X − M z ( z ).In order to prove Proposition 5.1, it hence remains to determine the growth of y Ψ ,N ( z , z ) as z approaches a cusp. This is proven in a series of lemmas.5.1. Cusp expansions.
In this section we rewrite the function P N,s in order to understand thebehavior if z is close to a cusp α/γ , with γ | N , γ = N , and ( α, N ) = 1. Letting L := (cid:16) α βγ δ (cid:17) with αδ ≡ N ), it is easy to see that P N,s ( z , Lz ) = X M ∈ L − Γ ( N ) ϕ s ( M z , z ) j ( M, z ) | j ( M, z ) | s . Hence, for σ > − /
4, the behavior of (the analytic continuation of) P N,s as z approaches the cusp L ( i ∞ ) may be determined by taking z → i ∞ on (the analytic continuation of) the right-hand side.To determine this continuation, we decompose as in (3.1) and denote the corresponding sumsby P ( α, γ ) , P ( α, γ ), and P ( α, γ ) . Note, that since γ = N , P ( α, γ ) cannot occur. Moreover P ( α, γ ) is treated analogously to P , and in particular converges uniformly for − / < σ ≤ σ ≤ σ under the assumption that M z = z is not solvable in L − Γ ( N ). Thus, if σ > − /
2, we maydirectly compute the Fourier expansion of P ( α, γ ) for y sufficiently large to determine the growthtowards the cusp α/γ . We do so in the proof of Lemma 5.4.We are thus left to consider P ( α, γ ). A direct calculation shows that L − Γ ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) c ≡ − aαγ (mod N ) (cid:27) . It is not hard to see that the cusp width of α/γ , the minimal ℓ such that Γ ℓ ∞ acts on L − Γ ( N )from the left, is ℓ = ℓ ̺ := N/γ ( N/γ,γ ) . Moreover Γ ∞ acts on L − Γ ( N ) from the right. Thus we obtain,as in Section 3.2,(5.1) X ( α, γ ) = 2 ℓ − − s X n ∈ Z Z R ϕ s (cid:16) t, zℓ (cid:17) e − πint dt × X m ∈ Z Z R ( z + w ) − − s ( z + w ) − s e − πimw dw X c ≥ c − − s K α,γ ( m, n ; c ) , where K α,γ is defined in (2.2).We next prove that the analogue of (3.9) holds. For this, we write K α,γ in terms of the classicalKloosterman sums. To state the resulting identity, we require the natural splitting ℓ = ℓ ℓ with | γ ∞ and ( ℓ , γ ) = 1, where ℓ | γ ∞ means that there exists n ∈ N such that ℓ | γ n . Astraightforward calculation then shows the following. Lemma 5.2.
We have K α,γ ( m, n ; c ) = 1 N e ℓ (cid:0) − [ ℓ γα ] ℓ n (cid:1) X r (mod N ) e N (cid:18) r [ ℓ α ] N cγ (cid:19) K (cid:18) ℓ [ ℓ ] ℓ c m, n + ℓ cN r ; ℓ c (cid:19) , where N/γ = N ℓ , and [ a ] b denotes the inverse of a (mod b ) . From Lemma 5.2, the analogue of (3.9) then follows easily and we can argue as before for theterms ( m, n ) = (0 , m = n = 0 in a form which yields its analytic contin-uation to s = 0. In order to state the result, we let ℓ γ = A A , with A | N ∞ and ( A , N ) = 1and denote the M¨obius function by µ . Lemma 5.3.
The analytic continuation to σ > − / of the m = n = 0 term in P ( α, γ ) existsand equals (5.2) − √ π s ℓ s Γ (cid:0) + s (cid:1) Γ(1 + s ) s z − − s A N φ ( ℓ A ) ζ (2 s + 1) ζ (2 s + 2) Z R ϕ s (cid:16) t, zℓ (cid:17) dt × X g | Nγ µ ( g ) φ ( gγℓ )( gγℓ ) s Y p | gγℓ − p − − s . Proof:
A direct calculation shows that K α,γ (0 , c ) vanishes unless ( c, N ) = γ , in which case itequals K α,γ (0 , c ) = A N φ (cid:18) A cγ (cid:19) . Using (3.10) and letting c cγ , we then easily obtain that the m = n = 0 term in P ( α, γ ) equals(5.3) − ℓ − − s √ π s Γ (cid:0) + s (cid:1) Γ(1 + s ) s z − − s A N X c ≥ (cid:16) c, Nγ (cid:17) =1 φ ( A c )( cγ ) s Z R ϕ s (cid:16) t, zℓ (cid:17) dt. We next rewrite the sum over c as X c ≥ (cid:16) c, Nγ (cid:17) =1 φ ( A c )( cγ ) s = ℓ s φ ( A ℓ ) X c ≥ c, Nγ )=1 φ ( A c ) φ ( A ℓ )( cγℓ ) s . One easily checks that ( A c, A ℓ ) = 1 and then uses the multiplicativity of φ together with A A ℓ = γℓ to rewrite the right-hand side as(5.4) ℓ s φ ( A ℓ ) X c ≥ (cid:16) c, Nγ (cid:17) =1 φ ( cγℓ )( cγℓ ) s = ℓ s φ ( A ℓ ) X c ≥ γℓ | c (cid:16) cγℓ , Nγ (cid:17) =1 φ ( c ) c s = ℓ s φ ( A ℓ ) X c ≥ γℓ | c X g | (cid:16) cγℓ , Nγ (cid:17) µ ( g ) φ ( c ) c s . Here the last equality holds by the well-known identity X g | n µ ( g ) = ( n = 1 , n > . eversing the order of summation then yields that the right-hand side of (5.4) equals ℓ s φ ( A ℓ ) X g | Nγ µ ( g ) F ( gγℓ, s ) , where F ( N, s ) is defined in (3.11). Using (3.12), (5.3) hence becomes (5.2), completing the proof. (cid:3)
Behavior towards the cusps.
We are now ready to determine the growth as z approachesa cusp. For this we compute the Fourier expansion of z y Ψ ,N ( z , z ). Lemma 5.4.
For L ∈ SL ( Z ) , consider the cusp L ( i ∞ ) = α/γ with γ | N and ( α, N ) = 1 , and let v > . Then, for x L + iy L = z L := Lz satisfying v < z < y − /v , we have y L Ψ ,N ( z , z L ) =2 πδ αγ , ∞ X n ≥ e − πin z e πinz + X n ≥ e πin z e − πinz + 4 π X c ≥ c − X n ≥ n − e πinzℓ X m ≥ m e πim z K α,γ ( m, − n ; c ) I (cid:18) π √ mncℓ (cid:19) + 4 π X c ≥ c − X n ≥ n − e − πinzℓ X m ≥ m e πim z K α,γ ( m, n ; c ) J (cid:18) π √ mncℓ (cid:19) + c N z − π ℓ X m ≥ me πim z X c ≥ K α,γ ( m, c ) c . (5.5) In particular, lim z → αγ y Ψ ,N ( z , z ) = c N z + 2 πδ αγ , ∞ − π ℓ X m ≥ me πim z X c ≥ K α,γ ( m, c ) c . Proof:
One determines (5.5) by computing the Fourier expansion of the analytic continuation to s = 0 of each of the sums P ( α, γ ), P ( α, γ ), and P ( α, γ ) in the splitting analogous to (3.1) forthe coset L − Γ ( N ). The behavior towards the cusp α/γ then follows by taking the limit z → i ∞ termwise. The sum P ( α, γ ) vanishes unless L ∈ Γ ( N ). For P ( α, γ ) we plug s = 0 into the( m, n ) = (0 ,
0) terms of (5.1) and note that by Lemma 5.3 the contribution from m = n = 0 is c N ( γ ) / z , where c N ( γ ) is some constant. However, observing that, by Propositions 4.1, 4.2, and4.3, z y Ψ ,N ( z , z ) − c N / z is meromorphic for every z ∈ H , we conclude that c N ( γ ) = c N .Since the proofs of the Fourier expansions for P ( α, γ ) and for the limit of P ( α, γ ) at differentcusps are similar, we only consider the cusp i ∞ . In order to compute the expansions for P and P , we note that the assumption on y and z implies that y > Im( M z ) for all M ∈ Γ ( N ). Thisfollows because, for M ∈ Γ ∞ , we have Im( M z ) = z < y and for M = (cid:0) a bc d (cid:1) with c = 0, we haveIm( M z ) = z | c z + d | ≤ c z ≤ z < v < y − z < y. Hence, for P , we apply Poisson summation to obtain, using (4.3), X = 2 y X n ∈ Z z − z + n )( z − z + n ) = 2 y X n ∈ Z g n ( z − z, z − z )(5.6) = 2 π X n ≥ e − πin z e πinz + 2 π X n ≥ e πin z e − πinz . e finally consider P , using the representation directly from the definition (3.1) with s = 0plugged in, namely X = 2 y X M = (cid:16) a bc d (cid:17) ∈ Γ ( N ) c ≥ M z − z )( M z − z ) − ( ac − z )( ac − z ) j ( M, z ) . To determine the Fourier expansion of the right-hand side we obtain, using Poisson summation,2 y X c ≥ N | c c − X a,d (mod c ) ad ≡ c ) X n ∈ Z X m ∈ Z × Z R e − πimw (cid:0) z + dc + w (cid:1) Z R e − πint (cid:18) − c ( z + dc + w ) + ac − z + t (cid:19) (cid:18) − c ( z + dc + w ) + ac − z + t (cid:19) dtdw − g n (cid:16) ac − z, ac − z (cid:17) g m (cid:18) z + dc , z + dc (cid:19)(cid:19) . Using (4.3), we evaluate g n (cid:16) ac − z, ac − z (cid:17) = ( πy e πin ( ac − z ) if n ≤ , πy e πin ( ac − z ) if n > , and g m (cid:18) z + dc , z + dc (cid:19) = ( m ≤ , − π me πim ( z + dc ) if m > . To compute the remaining double integral, we shift w w − z − d/c and t t − a/c , so thatthe double integral becomes(5.7) e πi ( na + md ) c e πim z Z R + i z e − πimw w Z R e − πint (cid:0) − c w − z + t (cid:1) (cid:0) − c w − z + t (cid:1) dtdw. From the restrictions on y and z , one concludes that − c w − z ∈ − H and hence (4.3) implies thatthe integral over t equals g n (cid:18) − c w − z, − c w − z (cid:19) = πy e − πin (cid:16) c w + z (cid:17) if n ≤ , πy e − πin (cid:16) c w + z (cid:17) if n > . Therefore (5.7) equals πy e πi ( na + md ) c e πim z e − πin ( x − i sgn( n ) y ) I m,n ( z ) , where I m,n ( z ) := Z R + i z e − πi (cid:16) mw + nc w (cid:17) w dw. Since the integrand in I m,n ( z ) is meromorphic, the path of integration may be shifted to Im( w ) = α for any α >
0, implying that I m,n ( z ) = I m,n ( α ) is independent of z . Taking the limit Im( w ) →∞ yields that I m,n ( z ) vanishes for m ∈ − N . Moreover, for n = 0 and m ∈ N , (4.3) implies that I m,n ( z ) = e πm z g m ( i z , i z ) = − π m. We conclude that the n = 0 term precisely cancels the product of g n and g m computed above.Finally, for n = 0 and m ∈ N , we make the change of variables w i | n | / m − / c − w and then hift to Re( w ) = α >
0, to obtain I m,n ( z ) = − ic r m | n | Z α + i R e πc √ m | n | ( w − sgn( n ) w ) w dw. Using the fact that, for fixed µ, κ >
0, the functions t ( t/κ ) ( µ − / J µ − (2 √ κt ) and s s − µ e − κ/s (resp. t ( t/κ ) ( µ − / I µ − (2 √ κt ) and s s − µ e κ/s ) are inverse to each other withrespect to the Laplace transform, by (29.3.80) and (29.3.81) of [1], then yields I m,n ( z ) = 2 πc r m | n | × J (cid:16) π √ mnc (cid:17) if n > ,I (cid:18) π √ m | n | c (cid:19) if n < . Hence we obtain for P (5.8) X = 4 π X c ≥ N | c c − X n ≥ n − e πinz X m ≥ m e πim z K ( m, − n ; c ) (cid:18) I (cid:18) π √ mnc (cid:19) + 2 π √ mnc (cid:19) + 4 π X c ≥ N | c c − X n ≥ n − e − πinz X m ≥ m e πim z K ( m, n ; c ) (cid:18) J (cid:18) π √ mnc (cid:19) + 2 π √ mnc (cid:19) . To finish the computation of the Fourier expansion of yψ ,N ( z , z ), we note that the expansion of P is given in Proposition 4.1 and the terms n = 0 precisely cancel the terms appearing after the I -Bessel and J -Bessel functions. Combining (5.8) with (4.1) and (5.6) yields the claimed expansion.We finally compute the limit of y Ψ ,N ( z , z ) as z → α/γ , or equivalently, the behavior of y L Ψ ,N ( z , z L ) as z → i ∞ . For this we take the limit termwise, but to do so we first verify absoluteuniform convergence for y sufficiently large. The Fourier expansion in Proposition 4.1 convergesuniformly in y > y for any fixed y >
0, so the contribution to the limit from P equals c N z − π X m ≥ m X c ≥ N | c K ( m, c ) c e πim z , which is a constant with respect to z . The sums P and P converge absolutely uniformly under theassumptions given in the lemma, so we may also take the limits z → i ∞ termwise; the contributioncoming from P is 2 π and the limit of (5.8) vanishes, completing the proof. (cid:3) A basis of polar Maass forms.
In this section, we use Proposition 5.1 to construct a familyof polar Maass forms with arbitrary principal parts. For m ∈ − N and τ ∈ H , we let(5.9) Y − k,m,N ( τ , z ) := i (2 i ) k − v ω τ ( − m − × ∂ − m − ∂X − m − τ ( z ) h ( z − τ ) k (cid:16) y k − Ψ k,N ( z , z ) + π c N δ k =1 b E ( z ) (cid:17)i z = τ . Remark.
For k >
1, Petersson applied his differential operator ∂ − m − ∂X − m − τ ( z ) to the meromorphic partof y k − Ψ k,N ( z , z ) (see (49a) of [23]). He investigated this function and used the Residue Theoremto compute its principal part in (50) of [23].The following lemma extends Lemma 4.4 of [5] to include k = 1 and level N . emma 5.5. For τ ∈ H , n ∈ − N , and k ∈ N , there exists F ∈ H cusp2 − k ( N ) with principal part ( z − τ ) k − X nτ ( z ) + O (cid:16) ( z − τ ) n +1 (cid:17) around z = τ and no other singularities modulo Γ ( N ) if and only if n ≡ k − ω τ ) . Inparticular, if n ≡ k − ω τ ) , then the principal part of z
7→ Y − k,n,N ( τ , z ) ∈ H cusp2 − k ( N ) equals ( z − τ ) k − X nτ ( z ) . Proof:
The necessary condition follows as in the proof of Lemma 4.4 of [5]. It hence suffices toshow that z
7→ Y − k,n,N ( z , z ) have prescribed principal parts and are indeed elements of H cusp2 − k ( N ).The principal parts for k > H k,N ( z , z ). However,in Proposition 3.1 of [5] it was shown that H k,N is the meromorphic part of (2 iy ) k − Ψ k,N ( z , z ).Furthermore, for k >
1, we also have y k − Ψ k,N ( z , z ) ∈ H cusp2 − k ( N ) by Proposition 3.1 of [5].Since acting by a differential operator in the independent variable z preserves both modularity andharmonicity in z , one easily concludes that Y − k,n,N ∈ H cusp2 − k ( N ). Since the non-meromorphicpart of z (2 iy ) k − Ψ k,N ( z , z ) is real analytic, its image under Petersson’s differential operatoris also real analytic, and hence does not contribute to the principal part. To conclude the claimfor k > z H k,N ( z , z ) is not modular, it was necessary for him to computethe principal part around z = Lτ separately for each L ∈ Γ ( N ). Due to the modularity of y Ψ ,N ( z , z ), in order to determine the principal parts at all such points, it suffices to compute theprincipal part at z = τ . Translating (50) of [23] into the notation in this paper and specializingto the case we are looking at, we have l = ω τ , ω = ω = τ , t = X τ ( z ), and η = q = 1.If the appropriate congruence condition is satisfied, then by (50) of [23] the principal part of Y − k,n,N ( τ , z ) around z = τ equals i v ω τ (cid:16) − ω τ (2 iv )( z − τ ) k − X nτ ( z ) (cid:17) = ( z − τ ) k − X nτ ( z ) . We next turn to the case k = 1. By Lemma 4.4, we have y Ψ ,N ( z , z ) ∈ H cusp0 . Furthermore, theprincipal parts of y Ψ ,N ( z , z ) are the same as for k > Y ,n,N ( τ , z ) = 1 v ω τ − n − ∂ − n − ∂X − n − τ ( z ) (cid:20) − ω z z ( z − τ ) Y , − ,N ( z , z ) (cid:21) z = τ . Since Y , − ,N is meromorphic as a function of z , the computation of the principal parts of Y ,n,N follows the proof of (50) in [23], except that one must be careful to verify that, for each n ∈ − N , ∂ − n − ∂X − n − τ ( z ) [[ P ] s =0 ] z = τ does not contribute to the principal part, where [ P ] s =0 denotesthe analytic continuation to s = 0 of P from (3.1). Acting by Petersson’s differential operatortermwise and noting absolute locally uniform convergence due to exponential decay in y and z ,one easily determines that the resulting Fourier expansion converge for every z , z ∈ H and hencethese terms do not contribute to the principal parts. This completes the proof. (cid:3) Lemma 5.5 then yields the following proposition. roposition 5.6. For each choice of τ , . . . , τ r ∈ H and k ≥ , there exists F ∈ H cusp2 − k ( N ) withprincipal parts in H given by r X d =1 ( z − τ d ) k − X n< n ≡ k − ω τd ) b τ d ( n ) X nτ d ( z ) and principal part at each cusp ̺ given by P n< a ̺ ( n ) e πinzℓ̺ . Furthermore, F is unique up toaddition by a constant (for k = 1 ) and is explicitly given by X ̺ ∈S N X n< a ̺ ( n ) P ̺ − k,n,N ( z ) + r X d =1 X n< n ≡ k − ω τd ) b τ d ( n ) Y − k,n,N ( τ d , z ) . Proof:
The existence of F follows directly by Lemma 5.5, and the uniqueness comes from the factthat the only harmonic Maass forms with trivial principal parts are holomorphic modular forms,which follows by Proposition 3.5 of [7]. (cid:3) Differentials.
In this section, we consider the properties of Y , − ,N . Using the connectionbetween weight 2 forms and differentials, we prove Theorem 1.3. Proof of Theorem 1.3: (1) We have to show that y Ψ ,N ( z , z ) has precisely a simple pole at z = z .As in the proof of Proposition 5.1 for non-elliptic fixed points, these correspond to the terms from P and P for which M z = z , since P converges absolutely in H . One sees directly that thepoles are at most simple and the only matrices contributing to the pole at z = z are M = ± I ,yielding the principal part − i/ ( z − z ). For elliptic fixed points, the argument is similar, except thatwe must make sure that the congruence conditions for the coefficients are satisfied. The congruencecondition is − ≡ − k (mod ω z ), which in this case ( k = 1) is always satisfied.(2) By Lemma 4.4 and the fact that b E ( z ) in annihilated by ξ ,z , we obtain the cusp form ξ ,z ( y Ψ ,N ( z , z )) = Φ N ( z, z ) . (3) We may differentiate (4.1), (4.4), and (4.7) directly. There is no contribution to the residue at z = z from (4.1) because it is real analytic. Since z M z − z ) (cid:0) ac − z (cid:1) and z z − n − z are both meromorphic, they have Laurent expansions around z = z , and hence their derivativeshave trivial residue, as asserted here. (cid:3) Bruinier–Funke pairing and the proofs of Theorem 1.1 and Corollary 1.2
In this section, we use a slight variant of the Bruinier–Funke pairing [7] to finish the proof ofTheorem 1.1. For g ∈ S k ( N ) and F ∈ H cusp2 − k ( N ) , define the pairing { g, F } := ( g, ξ − k ( F )) , where, for g, h ∈ S k ( N ) and dµ := dxdyy ,( g, h ) := 1 µ N Z Γ ( N ) \ H g ( z ) h ( z ) y k dµ is the standard Petersson inner-product with µ N := [SL ( Z ) : Γ ( N )]. The pairing { g, F } wascomputed for F ∈ H cusp2 − k ( N ) by Bruinier and Funke in Proposition 3.5 of [7]. The followingproposition extends their evaluation of { g, F } to the entire space H cusp2 − k ( N ). roposition 6.1. Assume that g ∈ S k ( N ) and write its elliptic expansion around each z ∈ H as g ( z ) = ( z − z ) − k P n ≥ a g, z ( n ) X n z ( z ) and its expansion at each cusp ̺ as g ̺ ( z ) = P n ≥ a g,̺ ( n ) e πinzℓ̺ .Suppose that F ∈ H cusp2 − k ( N ) and write its expansion around z ∈ H as F ( z ) = ( z − z ) k − X n ≫−∞ b F, z ( n ) X n z ( z ) + ( z − z ) k − X n ≤− c F, z ( n ) β (cid:0) r z ( z ); − n, k − (cid:1) X n z ( z ) . Moreover, write the Fourier expansion of F at the cusp ̺ as F ̺ ( z ) = X n ≫−∞ c + F,̺ ( n ) e πinzℓ̺ + X n< c − F,̺ ( n )Γ (cid:18) k − , π | n | yℓ ̺ (cid:19) e πinzℓ̺ . Then we have (6.1) { g, F } = πµ N X z ∈ Γ ( N ) \ H z ω z X n ≥ b F, z ( − n ) a g, z ( n −
1) + 1 µ N X ̺ ∈S N X n ≥ c + F,̺ ( − n ) a g,̺ ( n ) . Proof:
The proof closely follows the proof of Proposition 3.5 in [7]. By Proposition 5.6 and linearity, H cusp2 − k ( N ) decomposes into the direct sum of H cusp2 − k ( N ) and H cusp2 − k ( N ). Each of these subspacesthen further splits into direct sums of subspaces consisting of forms with poles at precisely one cusp ̺ or precisely one point z ∈ Γ ( N ) \ H . Hence it suffices to assume that F has a pole at exactly one z ∈ Γ ( N ) \ H .We may choose a fundamental domain F ( N ) of Γ ( N ) \ H for which z is in the interior ofΓ z F ( N ) = { M z | M ∈ Γ z , z ∈ F ( N ) } . Since Γ z fixes z , there is precisely one copy of z moduloΓ ( N ) inside Γ z F ( N ). Denoting the ball around z with radius ε > B ε ( z ) and noting that ξ − k ( F ) = y − k L − k ( F ), we may rewrite { g, F } = lim ε → µ N ω z Z Γ z F ( N ) \ B ε ( z ) g ( z ) L − k ( F ( z )) dµ. Using d ( g ( z ) F ( z ) dz ) = − g ( z ) L − k ( F ( z )) dµ, and applying Stokes’ Theorem, the integral becomes(6.2) 1 µ N ω z Z ∂B ε ( z ) g ( z ) F ( z ) dz, where the integral is taken counter-clockwise. We next insert the expansions of F and g around z = z . Since the non-meromorphic part of F is real analytic (and hence in particular has no poles)in H , we may directly plug in ε = 0 to see that the contribution from non-meromorphic part to(6.2) vanishes. Therefore, the limit ε → ε → µ N ω z X n ≫−∞ m ≥ b F, z ( n ) a g, z ( m ) Z ∂B ε ( z ) ( z − z ) − − ( n + m ) ( z − z ) n + m dz. Setting n + m = − ℓ , we obtain, by the Residue Theorem,(6.3) Z ∂B ε ( z ) ( z − z ) ℓ − ( z − z ) − ℓ dz = 2 πi Res z = z (cid:16) ( z − z ) ℓ − ( z − z ) − ℓ (cid:17) . From this, one sees immediately that there is no contribution from ℓ ≤
0. If ℓ ≥ z − z = z − z + 2 i z , ( z − z ) ℓ − = ℓ − X j =0 (cid:18) ℓ − j (cid:19) (2 i z ) j ( z − z ) ℓ − − j , hus the residue in (6.3) vanishes. If ℓ = 1, then we obtain( z − z ) − ( z − z ) − = 12 i z ( z − z ) (cid:16) z − z i z (cid:17) = 12 i z ∞ X j =0 ( − i z ) − j ( z − z ) − j . So only the term with j = 0 contributes to the residue, giving the claim. (cid:3) We are finally ready to extend Satz 3 of [22] to include the case k = 1. Proof of Theorem 1.1:
By Proposition 5.6, the linear combination (1.2) is a (unique, up toaddition by a constant if k = 1) weight 2 − k polar harmonic Maass form F ∈ H cusp2 − k ( N ) with theprincipal parts as in the theorem. Since F ∈ M − k ( N ) if and only if ξ − k ( F ) = 0 and ξ − k ( F )is a cusp form, we conclude that F ∈ M − k ( N ) if and only if ξ − k ( F ) is orthogonal to every cuspform g ∈ S k ( N ). However, by Proposition 6.1, we see that this occurs if and only if (6.1) holdsfor every g ∈ S k ( N ). This is the statement of the theorem. (cid:3) We are now ready to give an explicit version of Corollary 1.2, which is an easy consequenceof Theorem 1.1 together with Lemma 5.4. To state the theorem, we require the sum-of-divisorsfunction σ ( m ) := P d | m d . Theorem 6.2.
Suppose that τ , . . . , τ r ∈ H are given such that F ( z ) := X ̺ ∈S N X n< a ̺ ( n ) P ̺ ,n,N ( z ) + r X d =1 X n ≡ ω τd ) n< b τ d ( n ) Y ,n,N ( τ d , z ) satisfies (1.1) (with k = 1 ). For y sufficiently large (depending on v , . . . , v d ), we have the expansion F ( z ) = X ̺ = αγ ∈S N X n< a ̺ ( n ) δ ̺, ∞ e πinz + 4 π ℓ ̺ | n | X c ≥ K α,γ ( n, c ) c + 2 π (cid:18) | n | ℓ ̺ (cid:19) X j ≥ j − X c ≥ K α,γ ( n, j ; c ) c I πc s | n | jℓ ̺ ! e πijz − r X d =1 X n ≡ ω τd ) n< b τ d ( n ) πv d ω τ d − n − ∂ − n − ∂X − n − z ( z ) ( z − τ d ) X j ≥ e πijz e − πij z + 2 π ( z − τ d ) X j ≥ j − e πijz X m ≥ m e πim z X c ≥ N | c c − K ( m, − j ; c ) I (cid:18) π √ mjc (cid:19) − π ( z − τ d ) X m ≥ me πim z X c ≥ N | c K ( m, c ) c − z − τ d ) N Q p | N (1 + p − ) − X m ≥ σ ( m ) e πim z z = τ d . Proof:
The claim follows by computing the Fourier expansions of P ̺ ,n,N and Y ,n,N for each n ∈− N . However, since F is meromorphic by Theorem 1.1, we only need to compute the meromorphicparts of each Fourier expansion. We begin by plugging in the meromorphic parts of the the Fourierexpansions of P ̺ ,n,N given in Theorem 2.1.In order to compute the Fourier expansions of the Y ,n,N , we assume that y is sufficiently largeso that in particular there exists v > v < v d < y − /v for every d ∈ { , . . . , r } . y (5.10), to determine the expansions of Y +0 ,n,N , we only need to apply the differential operatorsin the definition (5.9) to the expansion of Y +0 , − ,N . Furthermore, the expansion of Y +0 , − ,N may bedirectly obtained by taking the meromorphic part of the expansion given in Lemma 5.4 plus π c N b E ( z ) = − c N z + 4 πN Y p | N (cid:0) p − (cid:1) − − X m ≥ σ ( m ) e πim z . This yields the statement of the theorem. (cid:3)
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E-mail address : [email protected]@maths.hku.hk