Heegner divisors, L -functions and harmonic weak Maass forms
aa r X i v : . [ m a t h . N T ] D ec HEEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAKMAASS FORMS JAN H. BRUINIER AND KEN ONO
Abstract.
Recent works, mostly related to Ramanujan’s mock theta functions, make useof the fact that harmonic weak Maass forms can be combinatorial generating functions.Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serveas “generating functions” for central values and derivatives of quadratic twists of weight 2modular L -functions. To obtain these results, we construct differentials of the third kindwith twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weakMaass forms. The connection with periods, Fourier coefficients, derivatives of L -functions,and points in the Jacobian of modular curves is obtained by analyzing the properties ofthese differentials using works of Scholl, Waldschmidt, and Gross and Zagier. Introduction and Statement of Results
Half-integral weight modular forms play important roles in arithmetic geometry andnumber theory. Thanks to the theory of theta functions, such forms include importantgenerating functions for the representation numbers of integers by quadratic forms. Amongweight 3/2 modular forms, one finds Gauss’ function ( q := e πiτ throughout) X x,y,z ∈ Z q x + y + z = 1 + 6 q + 12 q + 8 q + 6 q + 24 q + · · · , which is essentially the generating function for class numbers of imaginary quadratic fields,as well as Gross’s theta functions [Gro2] which enumerate the supersingular reductions ofCM elliptic curves.In the 1980s, Waldspurger [Wa], and Kohnen and Zagier [KZ, K] established that half-integral weight modular forms also serve as generating functions of a different type. Us-ing the Shimura correspondence [Sh], they proved that certain coefficients of half-integralweight cusp forms essentially are square-roots of central values of quadratic twists of mod-ular L -functions. When the weight is 3/2, these results appear prominently in works onthe ancient “congruent number problem” [T], as well as the deep works of Gross, Zagierand Kohnen [GZ, GKZ] on the Birch and Swinnerton-Dyer Conjecture.In analogy with these works, Katok and Sarnak [KS] employed a Shimura correspondenceto relate coefficients of weight 1/2 Maass forms to sums of values and sums of line integrals Date : October 1, 2007.2000
Mathematics Subject Classification. of Maass cusp forms. We investigate the arithmetic properties of the coefficients of adifferent class of Maass forms, the weight 1/2 harmonic weak Maass forms.A harmonic weak Maass form of weight k ∈ Z on Γ ( N ) (with 4 | N if k ∈ Z \ Z ) is asmooth function on H , the upper half of the complex plane, which satisfies:(i) f | k γ = f for all γ ∈ Γ ( N );(ii) ∆ k f = 0, where ∆ k is the weight k hyperbolic Laplacian on H (see (2.4));(iii) There is a polynomial P f = P n ≤ c + ( n ) q n ∈ C [ q − ] such that f ( τ ) − P f ( τ ) = O ( e − εv ) as v → ∞ for some ε >
0. Analogous conditions are required at all cusps.Throughout, for τ ∈ H , we let τ = u + iv , where u, v ∈ R , and we let q := e πiτ . Remark . The polynomial P f , the principal part of f at ∞ , is uniquely determined. If P f is non-constant, then f has exponential growth at the cusp ∞ . Similar remarks apply atall of the cusps. Remark . The results in the body of the paper are phrased in terms of vector valuedharmonic weak Maass forms. These forms are defined in Section 2.2.Spaces of harmonic weak Maass forms include weakly holomorphic modular forms , thosemeromorphic modular forms whose poles (if any) are supported at cusps. We are interestedin those harmonic weak Maass forms which do not arise in this way. Such forms have beena source of recent interest due to their connection to Ramanujan’s mock theta functions(see [BO1, BO2, BOR, O2, Za3, Zw1, Zw2]). For example, it turns out that(1.1) M f ( τ ) := q − f ( q ) + 2 i √ · N f ( τ )is a weight 1/2 harmonic weak Maass form, where N f ( τ ) := Z i ∞− τ P ∞ n = −∞ (cid:0) n + (cid:1) e πi ( n + ) z p − i ( z + 24 τ ) dz = i √ π X n ∈ Z Γ(1 / , π (6 n + 1) v ) q − (6 n +1) is a period integral of a theta function, Γ( a, x ) is the incomplete Gamma function, and f ( q )is Ramanujan’s mock theta function f ( q ) := 1 + ∞ X n =1 q n (1 + q ) (1 + q ) · · · (1 + q n ) . This example reveals two important features common to all harmonic weak Maass formson Γ ( N ). Firstly, all such f have Fourier expansions of the form(1.2) f ( τ ) = X n ≫−∞ c + ( n ) q n + X n< c − ( n ) W (2 πnv ) q n , where W ( x ) = W k ( x ) := Γ(1 − k, | x | ). We call P n ≫−∞ c + ( n ) q n the holomorphic part of f ,and we call its complement its non-holomorphic part . Secondly, the non-holomorphic partsare period integrals of weight 2 − k modular forms. Equivalently, ξ k ( f ) is a weight 2 − k modular form on Γ ( N ), where ξ k is a differential operator (see (2.6)) which is essentiallythe Maass lowering operator.Every weight 2 − k cusp form is the image under ξ k of a weight k harmonic weak Maassform. The mock theta functions correspond to those forms whose images under ξ are EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 3 weight 3/2 theta functions. We turn our attention to those weight 1/2 harmonic weak Maassforms whose images under ξ are orthogonal to the elementary theta series. Unlike the mocktheta functions, whose holomorphic parts are often generating functions in the theory ofpartitions (for example, see [BO1, BO2, BOR]), we show that these other harmonic weakMaass forms can be “generating functions” simultaneously for both the values and centralderivatives of quadratic twists of weight 2 modular L -functions.Although we treat the general case in this paper, to simplify exposition, in the intro-duction we assume that p is prime and that G ( τ ) = P ∞ n =1 B G ( n ) q n ∈ S new (Γ ( p )) is anormalized Hecke eigenform with the property that the sign of the functional equation of L ( G, s ) = ∞ X n =1 B G ( n ) n s is ǫ ( G ) = −
1. Therefore, we have that L ( G,
1) = 0.By Kohnen’s theory of plus-spaces [K], there is a half-integral weight newform(1.3) g ( τ ) = ∞ X n =1 b g ( n ) q n ∈ S + (Γ (4 p )) , unique up to a multiplicative constant, which lifts to G under the Shimura correspondence.For convenience, we choose g so that its coefficients are in F G , the totally real number fieldobtained by adjoining the Fourier coefficients of G to Q . We shall prove that there is aweight 1/2 harmonic weak Maass form on Γ (4 p ) in the plus space, say(1.4) f g ( τ ) = X n ≫−∞ c + g ( n ) q n + X n< c − g ( n ) W (2 πnv ) q n , whose principal part P f g has coefficients in F G , which also enjoys the property that ξ ( f g ) = k g k − g , where k g k denotes the usual Petersson norm.A calculation shows that if n > b g ( n ) = − √ πn k g k · c − g ( − n ) . The coefficients c + g ( n ) are more mysterious. We show that both types of coefficients arerelated to L -functions. To make this precise, for fundamental discriminants D let χ D bethe Kronecker character for Q ( √ D ), and let L ( G, χ D , s ) be the quadratic twist of L ( G, s )by χ D . These coefficients are related to these L -functions in the following way. Theorem 1.1.
Assume that p is prime, and that G ∈ S new (Γ ( p )) is a newform. If thesign of the functional equation of L ( G, s ) is ǫ ( G ) = − , then the following are true: (1) If ∆ < is a fundamental discriminant for which (cid:16) ∆ p (cid:17) = 1 , then L ( G, χ ∆ ,
1) = 32 k G k k g k π p | ∆ | · c − g (∆) . (2) If ∆ > is a fundamental discriminant for which (cid:16) ∆ p (cid:17) = 1 , then L ′ ( G, χ ∆ ,
1) = 0 if and only if c + g (∆) is algebraic. JAN H. BRUINIER AND KEN ONO
Remark . In Theorem 1.1 (2), we have that L ( G, χ ∆ ,
1) = 0 since the sign of the functionalequation of L ( G, χ ∆ , s ) is −
1. Therefore it is natural to consider derivatives in these cases.
Remark . The f g are uniquely determined up to the addition of a weight 1/2 weaklyholomorphic modular form with coefficients in F G . Furthermore, absolute values of thenonvanishing coefficients c + g ( n ) are typically asymptotic to subexponential functions in n .For these reasons, Theorem 1.1 (2) cannot be simply modified to obtain a formula for L ′ ( G, χ ∆ , c + g (∆). Remark . We give some numerical examples illustrating the theorem in Section 8.3.
Remark . Here we comment on the construction of the weak harmonic Maass forms f g .Due to the general results in this paper, we discuss the problem in the context of vectorvalued forms. It is not difficult to see that this problem boils down to the question ofproducing inverse images of classical Poincar´e series under ξ . A simple observation estab-lishes that these preimages should be weight 1/2 Maass-Poincar´e series which are explicitlydescribed in Chapter 1 of [Br]. Since standard Weil-type bounds fall short of establishingconvergence of these series, we briefly discuss a method for establishing convergence. Onemay employ a generalization of work of Goldfeld and Sarnak [GS] (for example, see [P]) onKloosterman-Selberg zeta functions. This theory proves that the relevant zeta functionsare holomorphic at s = 3 /
4, the crucial point for the task at hand. One then deducesconvergence using standard methods relating the series expansions of Kloosterman-Selbergzeta functions with their integral representations (for example, using Perron-type formulas).The reader may see [FO] where an argument of this type is carried out in detail.Theorem 1.1 (1) follows from Kohnen’s theory (see Corollary 1 on page 242 of [K]) ofhalf-integral newforms, the existence of f g , and (1.5). The proof of Theorem 1.1 (2) ismore difficult, and it involves a detailed study of Heegner divisors. We establish that thealgebraicity of the coefficients c + g (∆) is dictated by the vanishing of certain twisted Heegnerdivisors in the Jacobian of X ( p ). This result, when combined with the work of Gross andZagier [GZ], will imply Theorem 1.1 (2).To make this precise, we first recall some definitions. Let d < > p . Let Q d,p be the set of discriminant d = b − ac integral binary quadratic forms aX + bXY + cY with the property that p | a .For these pairs of discriminants, we define the twisted Heegner divisor Z ∆ ( d ) by(1.6) Z ∆ ( d ) := X Q ∈Q ∆ d,p / Γ ( p ) χ ∆ ( Q ) · α Q w Q , where χ ∆ denotes the generalized genus character corresponding to the decomposition ∆ · d as in [GKZ], α Q is the unique root of Q ( x,
1) in H , and w Q denotes the order of the stabilizerof Q in Γ ( p ). Then Z ∆ ( d ) is a divisor on X ( p ) defined over Q ( √ ∆) (see Lemma 5.1). Weuse these twisted Heegner divisors to define the degree 0 divisor(1.7) y ∆ ( d ) := Z ∆ ( d ) − deg( Z ∆ ( d )) · ∞ . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 5 Finally, we associate a divisor to f g by letting(1.8) y ∆ ( f g ) := X n< c + g ( n ) y ∆ ( n ) ∈ Div ( X ( p )) ⊗ F G . Recall that we have selected f g so that the coefficients of P f g are in F G .To state our results, let J be the Jacobian of X ( p ), and let J ( F ) denote the points of J over a number field F . The Hecke algebra acts on J ( F ) ⊗ C , which by the Mordell-Weil Theorem is a finite dimensional vector space. The main results of Section 7 (seeTheorems 7.5 and 7.6) imply the following theorem. Theorem 1.2.
Assuming the notation and hypotheses above, the point corresponding to y ∆ ( f g ) in J ( Q ( √ ∆)) ⊗ C is in its G -isotypical component. Moreover, the following areequivalent: (i) The Heegner divisor y ∆ ( f g ) vanishes in J ( Q ( √ ∆)) ⊗ C . (ii) The coefficient c + g (∆) is algebraic. (iii) The coefficient c + g (∆) is contained in F G . We then obtain the following generalization of the Gross-Kohnen-Zagier theorem [GKZ](see Corollary 6.7 and Theorem 7.7).
Theorem 1.3.
Assuming the notation and hypotheses above, we have that X n> y G ∆ ( − n ) q n = g ( τ ) ⊗ y ∆ ( f g ) ∈ S + (Γ (4 p )) ⊗ J ( Q ( √ ∆)) , where y G ∆ ( − n ) denotes the projection of y ∆ ( − n ) onto its G -isotypical component. This result, when combined with the Gross-Zagier theorem [GZ], gives the conclusion (seeTheorem 7.8) that the Heegner divisor y ∆ ( f g ) vanishes in J ( Q ( √ ∆)) ⊗ C if and only if L ′ ( G, χ ∆ ,
1) = 0, thereby proving Theorem 1.1 (2).These results arise from the interplay between Heegner divisors, harmonic weak Maassforms and Borcherds products, relations which are of independent interest. We extend thetheory of regularized theta lifts of harmonic weak Maass forms, and we apply these resultsto obtain generalized Borcherds products. In that way harmonic weak Maass forms areplaced in the central position which allows us to obtain the main results in this paper.In view of Theorem 1.1, it is natural to investigate the algebraicity and the nonvanishingof the coefficients of harmonic weak Maass forms, questions which are of particular interestin the context of elliptic curves. As a companion to Goldfeld’s famous conjecture forquadratic twists of elliptic curves [G], which asserts that “half” of the quadratic twists ofa fixed elliptic curve have rank zero (resp. one), we make the following conjecture.
Conjecture.
Suppose that f ( τ ) = X n ≫−∞ c + ( n ) q n + X n< c − ( n ) W (2 πnv ) q n is a weight / harmonic weak Maass form on Γ ( N ) whose principal parts at cusps aredefined over a number field. If ξ / ( f ) is non-zero and is not a linear combination of JAN H. BRUINIER AND KEN ONO elementary theta series, then { < n < X : c − ( − n ) = 0 } ≫ f X, { < n < X : c + ( n ) is transcendental } ≫ f X. Remark . Suppose that G is as in Theorem 1.1. If G corresponds to a modular ellipticcurve E , then the truth of a sufficiently precise form of the conjecture for f g , combined withKolyvagin’s Theorem on the Birch and Swinnerton-Dyer Conjecture, would prove that a“proportion” of quadratic twists of E have Mordell-Weil rank zero (resp. one). Remark . In a recent paper, Sarnak [Sa] characterized those Maass cusp forms whoseFourier coefficients are all integers. He proved that such a Maass cusp form must corre-spond to even irreducible 2-dimensional Galois representations which are either dihedralor tetrahedral. More generally, a number of authors such as Langlands [L], and Booker,Str¨ombergsson, and Venkatesh [BSV] have considered the algebraicity of coefficients ofMaass cusp forms. It is generally believed that the coefficients of generic Maass cusp formsare transcendental. The conjecture above suggests that a similar phenomenon should alsohold for harmonic weak Maass forms. In this setting, we believe that the exceptional har-monic weak Maass forms are those which arise as preimages of elementary theta functionssuch as those forms associated to the mock theta functions.In the direction of this conjecture, we combine Theorem 1.1 with works by the secondauthor and Skinner [OSk] and Perelli and Pomykala [PP] to obtain the following result.
Corollary 1.4.
Assuming the notation and hypotheses above, as X → + ∞ we have {− X < ∆ < fundamental : c − g (∆) = 0 } ≫ f g X log X , { < ∆ < X fundamental : c + g (∆) is transcendental } ≫ f g ,ǫ X − ǫ . Remark . One can typically obtain better estimates for c − g (∆) using properties of 2-adicGalois representations. For example, if L ( G, s ) is the Hasse-Weil L -function of an ellipticcurve where p is not of the form x + 64, where x is an integer, then using Theorem 1 of [O]and the theory of Setzer curves [Se], one can find a rational number 0 < α < {− X < ∆ < c − g (∆) = 0 } ≫ f g X (log X ) − α . Now we briefly provide an overview of the ideas behind the proofs of our main theorems.They depend on the construction of canonical differentials of the third kind for twistedHeegner divisors. In Section 5 we produce such differentials of the form η ∆ ,r ( z, f ) = − ∂ Φ ∆ ,r ( z, f ), where Φ ∆ ,r ( z, f ) are automorphic Green functions on X ( N ) which areobtained as liftings of weight 1/2 harmonic weak Maass forms f . To define these liftings, inSection 5 we generalize the regularized theta lift due to Borcherds, Harvey, and Moore (forexample, see [Bo1], [Br]). We then employ transcendence results of Waldschmidt and Scholl(see [W], [Sch]), for the periods of differentials, to relate the vanishing of twisted Heegnerdivisors in the Jacobian to the algebraicity of the corresponding canonical differentials ofthe third kind. By means of the q -expansion principle, we obtain the connection to thecoefficients of harmonic weak Maass forms. EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 7 In Section 6 we construct generalized Borcherds products for twisted Heegner divisors,and we study their properties and multiplier systems (Theorem 6.1). In particular, we givea necessary and sufficient condition that the character of such a Borcherds product hasfinite order (Theorem 6.2).In Section 7, we consider the implications of these results when restricting to Hecke stablecomponents. We obtain the general versions of Theorems 1.1, 1.2, and 1.3, and we proveCorollary 1.4. In particular, Theorem 1.3, the Gross-Kohnen-Zagier theorem for twistedHeegner divisors, is proved by adapting an argument of Borcherds [Bo2], combined with ananalysis of the Hecke action on cusp forms, harmonic weak Maass forms, and the Jacobian.In the body of the paper we consider weight 2 newforms G of arbitrary level and functionalequation. For the regularized theta lift, it is convenient to identify SL with Spin( V ) for acertain rational quadratic space V of signature (2 , V ).We define the basic setup in Section 2, and in Section 3 we recall some results of Schollon canonical differentials of the third kind. The relevant theta kernels are then studiedin Section 4. They can be viewed as vector valued weight 1 / / not given by Borcherds lifts of weakly holomorphicmodular forms. They are obtained as lifts of harmonic weak Maass forms. One of theseexamples is related to a famous example of Gross [Za1], and the other is related to Ra-manujan’s mock theta function f ( q ). We also derive the infinite product expansions ofZagier’s twisted modular polynomials [Za2]. Acknowledgements
We thank Fredrik Str¨omberg for his numerical computations of harmonic weak Maassforms, and we thank Tonghai Yang for many fruitful conversations. We thank the referee(s)for their comments on a preliminary version of this paper.2.
Preliminaries
To ease exposition, the results in the introduction were stated using the classical languageof half-integral weight modular forms. To treat the case of general levels and functionalequations, it will be more convenient to work with vector valued forms and certain Weilrepresentations. Here we recall this framework, and we discuss important theta functionswhich will be used to study differentials of the third kind.We begin by fixing notation. Let (
V, Q ) be a non-degenerate rational quadratic spaceof signature ( b + , b − ). Let L ⊂ V be an even lattice with dual L ′ . The discriminant group L ′ /L , together with the Q / Z -valued quadratic form induced by Q , is called the discriminantform of the lattice L . JAN H. BRUINIER AND KEN ONO
The Weil representation.
Let H = { τ ∈ C ; ℑ ( τ ) > } be the complex upper halfplane. We write Mp ( R ) for the metaplectic two-fold cover of SL ( R ). The elements of thisgroup are pairs ( M, φ ( τ )), where M = ( a bc d ) ∈ SL ( R ) and φ : H → C is a holomorphicfunction with φ ( τ ) = cτ + d . The multiplication is defined by( M, φ ( τ ))( M ′ , φ ′ ( τ )) = ( M M ′ , φ ( M ′ τ ) φ ′ ( τ )) . We denote the integral metaplectic group, the inverse image of Γ := SL ( Z ) under thecovering map, by ˜Γ := Mp ( Z ). It is well known that ˜Γ is generated by T := (( ) , S := (( −
11 0 ) , √ τ ). One has the relations S = ( ST ) = Z , where Z := (cid:0)(cid:0) − − (cid:1) , i (cid:1) isthe standard generator of the center of ˜Γ. We let ˜Γ ∞ := h T i ⊂ ˜Γ.We now recall the Weil representation associated with the discriminant form L ′ /L (forexample, see [Bo1], [Br]). It is a representation of ˜Γ on the group algebra C [ L ′ /L ]. Wedenote the standard basis elements of C [ L ′ /L ] by e h , h ∈ L ′ /L , and write h· , ·i for thestandard scalar product (antilinear in the second entry) such that h e h , e h ′ i = δ h,h ′ . The Weilrepresentation ρ L associated with the discriminant form L ′ /L is the unitary representationof ˜Γ on C [ L ′ /L ] defined by ρ L ( T )( e h ) := e ( h / e h , (2.1) ρ L ( S )( e h ) := e (( b − − b + ) / p | L ′ /L | X h ′ ∈ L ′ /L e ( − ( h, h ′ )) e h ′ . (2.2)Note that ρ L ( Z )( e h ) = e (( b − − b + ) / e − h . (2.3)2.2. Vector valued modular forms. If f : H → C [ L ′ /L ] is a function, we write f = P λ ∈ L ′ /L f h e h for its decomposition in components with respect to the standard basis of C [ L ′ /L ]. Let k ∈ Z , and let M ! k,ρ L denote the space of C [ L ′ /L ]-valued weakly holomorphicmodular forms of weight k and type ρ L for the group ˜Γ. The subspaces of holomorphicmodular forms (resp. cusp forms) are denoted by M k,ρ L (resp. S k,ρ L ).Now assume that k ≤
1. A twice continuously differentiable function f : H → C [ L ′ /L ]is called a harmonic weak Maass form (of weight k with respect to ˜Γ and ρ L ) if it satisfies:(i) f ( M τ ) = φ ( τ ) k ρ L ( M, φ ) f ( τ ) for all ( M, φ ) ∈ ˜Γ;(ii) there is a C > f ( τ ) = O ( e Cv ) as v → ∞ ;(iii) ∆ k f = 0.Here we have that(2.4) ∆ k := − v (cid:18) ∂ ∂u + ∂ ∂v (cid:19) + ikv (cid:18) ∂∂u + i ∂∂v (cid:19) is the usual weight k hyperbolic Laplace operator, where τ = u + iv (see [BF]). We denotethe vector space of these harmonic weak Maass forms by H k,ρ L . Moreover, we write H k,ρ L for the subspace of f ∈ H k,ρ L whose singularity at ∞ is locally given by the pole of ameromorphic function. In particular, this means that f satisfies f ( τ ) = P f ( τ ) + O ( e − εv ) , v → ∞ , EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 9 for some Fourier polynomial P f ( τ ) = X h ∈ L ′ /L X n ∈ Z + Q ( h ) −∞≪ n ≤ c + ( n, h ) e ( nτ ) e h and some ε >
0. In this situation, P f is uniquely determined by f . It is called the principalpart of f . (The space H k,ρ L was called H + k,L in [BF].) We have M ! k,ρ L ⊂ H k,ρ L ⊂ H k,ρ L . TheFourier expansion of any f ∈ H k,ρ L gives a unique decomposition f = f + + f − , where f + ( τ ) = X h ∈ L ′ /L X n ∈ Q n ≫−∞ c + ( n, h ) e ( nτ ) e h , (2.5a) f − ( τ ) = X h ∈ L ′ /L X n ∈ Q n< c − ( n, h ) W (2 πnv ) e ( nτ ) e h , (2.5b)and W ( x ) = W k ( x ) := R ∞− x e − t t − k dt = Γ(1 − k, | x | ) for x < ξ k : H k,ρ L −→ S − k, ¯ ρ L , f ( τ ) ξ k ( f )( τ ) := v k − L k f ( τ ) . Here L k := − iv ∂∂ ¯ τ is the usual Maass lowering operator. Note that ξ − k ξ k = ∆ k . TheFourier expansion of ξ k ( f ) is given by ξ k ( f ) = − X h ∈ L ′ /L X n ∈ Q n> (4 πn ) − k c − ( − n, h ) e ( nτ ) e h . (2.7)The kernel of ξ k is equal to M ! k,ρ L . By Corollary 3.8 of [BF], the following sequence is exact0 / / M ! k,ρ L / / H k,ρ L ξ k / / S − k, ¯ ρ L / / . (2.8)Moreover, by Proposition 3.11 of [BF], for any given Fourier polynomial of the form Q ( τ ) = X h ∈ L ′ /L X n ∈ Z + Q ( h ) n< a ( n, h ) e ( nτ ) e h with a ( n, h ) = ( − k − sig( L ) / a ( n, − h ), there is an f ∈ H k,ρ L with principal part P f = Q + c for some T -invariant constant c ∈ C [ L ′ /L ].Using the Petersson scalar product, we obtain a bilinear pairing between M − k, ¯ ρ L and H k,ρ L defined by(2.9) { g, f } = (cid:0) g, ξ k ( f ) (cid:1) − k := Z Γ \ H h g, ξ k ( f ) i v − k du dvv , where g ∈ M − k, ¯ ρ L and f ∈ H k,ρ L . If g has the Fourier expansion g = P h,n b ( n, h ) e ( nτ ) e h ,and we denote the expansion of f as in (2.5), then by Proposition 3.5 of [BF] we have(2.10) { g, f } = X h ∈ L ′ /L X n ≤ c + ( n, h ) b ( − n, h ) . Hence { g, f } only depends on the principal part of f . The exactness of (2.8) implies thatthe induced pairing between S − k, ¯ ρ L and H k,ρ L /M ! k,ρ L is non-degenerate. Moreover, thepairing is compatible with the natural Q -structures on M − k, ¯ ρ L and H k,ρ L /M ! k,ρ L given bymodular forms with rational coefficients and harmonic weak Maass forms with rationalprincipal part, respectively.We conclude this subsection with a notion which will be used later in the paper. Aharmonic weak Maass form f ∈ H k,ρ L is said to be orthogonal to cusp forms of weight k iffor all s ∈ S k,ρ L we have ( f, s ) reg := Z reg F h f ( τ ) , s ( τ ) i v k du dvv = 0 . Here F denotes the standard fundamental domain for the action of SL ( Z ) on H , and theintegral has been regularized as in [Bo1].2.3. Siegel theta functions.
Now we recall some basic properties of theta functions as-sociated to indefinite quadratic forms. LetGr( V ) := { z ⊂ V ( R ) : z is a b + -dimensional subspace with Q | z > } be the Grassmannian of 2-dimensional positive definite subspaces of V ( R ). If λ ∈ V ( R ) and z ∈ Gr( V ), we write λ z and λ z ⊥ for the orthogonal projection of λ to z and z ⊥ , respectively.Let α, β ∈ V ( R ). For τ = u + iv ∈ H and z ∈ Gr( V ), we define a theta function by ϑ L ( τ, z, α, β ) := v b − / X λ ∈ L e (cid:18) ( λ + β ) z τ + ( λ + β ) z ⊥ τ − ( λ + β/ , α ) (cid:19) . (2.11) Proposition 2.1.
We have the transformation formula ϑ L ( − /τ, z, − β, α ) = (cid:16) τi (cid:17) b + − b − | L ′ /L | − / ϑ L ′ ( τ, z, α, β ) . Proof.
This follows by Poisson summation (for example, see Theorem 4.1 of [Bo1]). (cid:3)
The proposition can be used to define vector valued Siegel theta functions of weight k = ( b + − b − ) / ρ L (see Section 4 of [Bo1]).2.4. A lattice related to Γ ( N ) . Let N be a positive integer. We consider the rationalquadratic space V := { X ∈ Mat ( Q ) : tr( X ) = 0 } (2.12)with the quadratic form Q ( X ) := − N det( X ). The corresponding bilinear form is given by( X, Y ) = N tr( XY ) for X, Y ∈ V . The signature of V is (2 , C ( V ) of V can be identified with Mat ( Q ). The Clifford norm on C ( V ) is identified withthe determinant. The group GL ( Q ) acts on V by γ.X = γXγ − , γ ∈ GL ( Q ) , leaving the quadratic form invariant, inducing isomorphisms of algebraic groups over Q GL ∼ = GSpin( V ) , SL ∼ = Spin( V ) . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 11 There is an isometry from (
V, Q ) to the trace zero part of ( C ( V ) , − N det). We let L bethe lattice L := (cid:26)(cid:18) b − a/Nc − b (cid:19) : a, b, c ∈ Z (cid:27) . (2.13)Then the dual lattice is given by L ′ := (cid:26)(cid:18) b/ N − a/Nc − b/ N (cid:19) : a, b, c ∈ Z (cid:27) . (2.14)We identify L ′ /L with Z / N Z . Here the quadratic form on L ′ /L is identified with thequadratic form x x on Z / N Z . The level of L is 4 N .If D ∈ Z , let L D be the set of vectors X ∈ L ′ with Q ( X ) = D/ N . Notice that L D isempty unless D is a square modulo 4 N . For r ∈ L ′ /L with r ≡ D (mod 4 N ) we define L D,r := { X ∈ L ′ : Q ( X ) = D/ N and X ≡ r (mod L ) } . We write L D for the subset of primitive vectors in L D , and L D,r for the primitive vectorsin L D,r , respectively. If X = (cid:16) b/ N − a/Nc − b/ N (cid:17) ∈ L D,r , then the matrix(2.15) ψ ( X ) := (cid:18) a b/ b/ N c (cid:19) = X (cid:18) N − N (cid:19) defines an integral binary quadratic form of discriminant D = b − N ac = 4
N Q ( X ) with b ≡ r (mod 2 N ).It is easily seen that the natural homomorphism SO( L ) → Aut( L ′ /L ) is surjective. Itskernel is called the discriminant kernel subgroup, which we denote by Γ( L ). We writeSO + ( L ) for the intersection of SO( L ) and the connected component of the identity ofSO( V )( R ). The group Γ ( N ) ⊂ Spin( V ) takes L to itself and acts trivially on L ′ /L . Proposition 2.2.
The image of Γ ( N ) in SO( L ) is equal to Γ( L ) ∩ SO + ( L ) . The imagein SO( L ) of the extension of Γ ( N ) by all Atkin-Lehner involutions is equal to SO + ( L ) . In particular, Γ ( N ) acts on L D,r and L D,r . By reduction theory, the number of orbitsof L D,r is finite. The number of orbits of L D,r is finite if D = 0.3. Differentials of the third kind
We shall construct differentials of the third kind associated to twisted Heegner divisorsusing regularized Borcherds products. We begin by recalling some general facts concerningsuch differentials [Sch] and [Gri]. Let X be a non-singular projective curve over C of genus g . A differential of the first kind on X is a holomorphic 1-form. A differential of the secondkind is a meromorphic 1-form on X whose residues all vanish. A differential of the thirdkind on X is a meromorphic 1-form on X whose poles are all of first order with residues in Z . In Section 5.1 and the subsequent sections we will relax the condition on the integralityof the residues. Let ψ be a differential of the third kind on X that has poles at the points P j , with residues c j , and is holomorphic elsewhere. Then the residue divisor of ψ isres( ψ ) := X j c j P j . By the residue theorem, the restriction of this divisor to any component of X has degree 0.Conversely, if D = P j c j P j is any divisor on X whose restriction to any component of X has degree 0, then the Riemann-Roch theorem and Serre duality imply that there is adifferential ψ D of the third kind with residue divisor D (see e.g. [Gri], p. 233). Moreover, ψ D is determined by this condition up to addition of a differential of the first kind. Let U = X \ { P j } . The canonical homomorphism H ( U, Z ) → H ( X, Z ) is surjective and itskernel is spanned by the classes of small circles δ j around the points P j . In particular, wehave R δ j ψ D = 2 πic j .Using the Riemann period relations, it can be shown that there is a unique differentialof the third kind η D on X with residue divisor D such that ℜ (cid:18)Z γ η D (cid:19) = 0for all γ ∈ H ( U, Z ). It is called the canonical differential of the third kind associatedwith D . For instance, if f is a meromorphic function on X with divisor D , then df /f is a canonical differential of the third kind on X with residue divisor div( f ). A differentcharacterization of η D is given in Proposition 1 of [Sch]. Proposition 3.1.
The differential η D is the unique differential of the third kind with residuedivisor D which can be written as η D = ∂h , where h is a harmonic function on U . Let ¯ Q ⊂ C be a fixed algebraic closure of Q . We now assume that the curve X andthe divisor D are defined over a number field F ⊂ ¯ Q . The following theorem by Scholl onthe transcendence of canonical differentials of the third kind will be important for us (seeTheorem 1 of [Sch]). Its proof is based on results by Waldschmidt on the transcendence ofperiods of differentials of the third kind (see Section 5.2 of [W], and Theorem 2 of [Sch]). Theorem 3.2 (Scholl) . If some non-zero multiple of D is a principal divisor, then η D isdefined over F . Otherwise, η D is not defined over ¯ Q . Differentials of the third kind on modular curves.
We consider the modularcurve Y ( N ) := Γ ( N ) \ H . By adding cusps in the usual way, we obtain the compactmodular curve X ( N ). It is well known that X ( N ) is defined over Q . The cusps aredefined over Q ( ζ N ), where ζ N denotes a primitive N -th root of unity. The action of theGalois group on them can be described explicitly (for example, see [Ogg]). In particular,it turns out that the cusps are defined over Q when N or N/ J be the Jacobian of X ( N ), and let J ( F ) denote its points overany number field F . They correspond to divisor classes of degree zero on X ( N ) which arerational over F . By the Mordell-Weil theorem, J ( F ) is a finitely generated abelian group.Let ψ be a differential of the third kind on X ( N ). We may write ψ = 2 πif dz , where f is a meromorphic modular form of weight 2 for the group Γ ( N ). All poles of f lie on Y ( N ) and are of first order, and they have residues in Z . In a neighborhood of the cusp ∞ , the modular form f has a Fourier expansion f ( z ) = ∞ X n ≥ a ( n ) q n . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 13 The constant coefficient a (0) is the residue of ψ at ∞ . We have analogous expansions atthe other cusps. According to the q -expansion principle, ψ is defined over a number field F , if and only if all Fourier coefficients a ( n ) are contained in F . Combining these factswith Theorem 3.2, we obtain the following criterion. Theorem 3.3.
Let D be a divisor of degree on X ( N ) defined over F . Let η D be thecanonical differential of the third kind associated to D and write η D = 2 πif dz . If somenon-zero multiple of D is a principal divisor, then all the coefficients a ( n ) of f at the cusp ∞ are contained in F . Otherwise, there exists an n such that a ( n ) is transcendental. Twisted Siegel theta functions
To define a generalized theta lift in the next section, we first must consider twisted Siegeltheta functions. We begin with some notation. Let N be a positive integer, and let L bethe lattice defined in Section 2.4. Let ∆ ∈ Z be a fundamental discriminant and r ∈ Z such that ∆ ≡ r (mod 4 N ). Following [GKZ], we define a generalized genus character for λ = (cid:16) b/ N − a/Nc − b/ N (cid:17) ∈ L ′ by putting χ ∆ ( λ ) = χ ∆ ([ a, b, N c ]) := (cid:0) ∆ n (cid:1) , if ∆ | b − N ac and ( b − N ac ) / ∆ is asquare modulo 4 N and gcd( a, b, c, ∆) = 1,0 , otherwise . Here [ a, b, N c ] is the integral binary quadratic form corresponding to λ , and n is any integerprime to ∆ represented by one of the quadratic forms [ N a, b, N c ] with N N = N and N , N > χ ∆ is invariant under the action of Γ ( N ) and under the action of allAtkin-Lehner involutions. Hence it is invariant under SO + ( L ). It can be computed bythe following explicit formula (see Section I.2, Proposition 1 of [GKZ]): If ∆ = ∆ ∆ is afactorization of ∆ into discriminants, and N = N N is a factorization of N into positivefactors such that (∆ , N a ) = (∆ , N c ) = 1, then χ ∆ ([ a, b, N c ]) = (cid:18) ∆ N a (cid:19) (cid:18) ∆ N c (cid:19) . (4.1)If no such factorizations of ∆ and N exist, then we have χ ∆ ([ a, b, N c ]) = 0.We define a twisted variant of the Siegel theta function for L as follows. For a coset h ∈ L ′ /L , and variables τ = u + iv ∈ H , z ∈ Gr( V ), we put θ ∆ ,r,h ( τ, z ) := v / X λ ∈ L + rhQ ( λ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ ) e (cid:18) | ∆ | λ z τ + 1 | ∆ | λ z ⊥ τ (cid:19) (4.2) = v / X λ ∈ L + rhQ ( λ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ ) e (cid:18) | ∆ | λ u + 1 | ∆ | (cid:18) λ z − λ z ⊥ (cid:19) iv (cid:19) . Moreover, we define a C [ L ′ /L ]-valued theta function by puttingΘ ∆ ,r ( τ, z ) := X h ∈ L ′ /L θ ∆ ,r,h ( τ, z ) e h . (4.3)We will often omit the dependency on ∆ , r from the notation if it is clear from the context.In the variable z , the function Θ ∆ ,r ( τ, z ) is invariant under Γ ( N ). In the next theorem weconsider the transformation behavior in the variable τ . Theorem 4.1.
The theta function Θ ∆ ,r ( τ, z ) is a non-holomorphic C [ L ′ /L ] -valued modularform for Mp ( Z ) of weight / . It transforms with the representation ρ L if ∆ > , andwith ¯ ρ L if ∆ < . Theorem 4.1 of [Bo1] gives the ∆ = 1 case. For general ∆, a similar result for Jacobiforms is contained in [Sk2] (see §
2, pp.507). The following is crucial for its proof.
Proposition 4.2.
For h ∈ L ′ /L and λ ∈ L ′ / ∆ L , the exponential sum G h ( λ, ∆ , r ) = X λ ′ ∈ L ′ / ∆ Lλ ′ ≡ rh ( L ) Q ( λ ′ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ ′ ) e (cid:18) − | ∆ | ( λ, λ ′ ) (cid:19) is equal to ε | ∆ | / χ ∆ ( λ ) X h ′ ∈ L ′ /Lλ ≡ rh ′ ( L ) Q ( λ ) ≡ ∆ Q ( h ′ ) (∆) e (cid:0) − sgn(∆)( h, h ′ ) (cid:1) . Here ε = 1 if ∆ > , and ε = i if ∆ < .Proof. By applying a finite Fourier transform in r ′ modulo 2 N , the claim follows fromidentity (3) on page 517 of [Sk2]. (cid:3) Proof of Theorem 4.1.
We only have to check the transformation behavior under the gen-erators T and S of ˜Γ. The transformation law under T follows directly from the definitionin (4.2). For the transformation law under S we notice that we may write θ h ( τ, z ) = X α ∈ L ′ / ∆ Lα ≡ rh ( L ) Q ( α ) ≡ ∆ Q ( h ) (∆) χ ∆ ( α ) | ∆ | − / ϑ L ( | ∆ | τ, z, , α/ | ∆ | ) , where ϑ L is the theta function for the lattice L defined in (2.11). Here we have used that χ ∆ ( λ ) only depends on λ ∈ L ′ modulo ∆ L . By Proposition 2.1, we find that θ h ( − /τ, z ) = r τi | L ′ /L | − / | ∆ | − X α ∈ L ′ / ∆ Lα ≡ rh ( L ) Q ( α ) ≡ ∆ Q ( h ) (∆) χ ∆ ( α ) ϑ L ′ ( τ / | ∆ | , z, α/ | ∆ | , r τi (2 N ) − / | ∆ | − / v / X λ ∈ L ′ G h ( λ, ∆ , r ) e (cid:18) | ∆ | λ z τ + 1 | ∆ | λ z ⊥ τ (cid:19) . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 15 By Proposition 4.2, we obtain θ h ( − /τ, z ) = r τi ε (2 N ) − / X h ′ ∈ L ′ /L e (cid:0) sgn(∆)( h, h ′ ) (cid:1) θ h ′ ( τ, z ) . This completes the proof of the theorem. (cid:3)
Partial Poisson summation.
We now consider the Fourier expansion of θ h ( τ, z )in the variable z . Following [Bo1] and [Br], it is obtained by applying a partial Poissonsummation to the theta kernel.Recall that the cusps of Γ ( N ) correspond to Γ ( N )-classes of primitive isotropic vectorsin L . Let ℓ ∈ L be a primitive isotropic vector. Let ℓ ′ ∈ L ′ with ( ℓ, ℓ ′ ) = 1. The1-dimensional lattice K = L ∩ ℓ ′⊥ ∩ ℓ ⊥ is positive definite. For simplicity we assume that ( ℓ, L ) = Z . In this case we may chose ℓ ′ ∈ L . Then L splits into L = K ⊕ Z ℓ ′ ⊕ Z ℓ, (4.4)and K ′ /K ∼ = L ′ /L . (If N is squarefree, then any primitive isotropic vector ℓ ∈ L satisfies( ℓ, L ) = Z and our assumption is not a restriction. For general N , the results of this sectionstill hold with the appropriate modifications, but the formulas get considerably longer.)We denote by w the orthogonal complement of ℓ z in z . Hence V ( R ) = z ⊕ z ⊥ = w ⊕ R ℓ z ⊕ R ℓ z ⊥ . If λ ∈ V ( R ), let λ w be the orthogonal projection of λ to w . We denote by µ the vector µ = µ ( z ) := − ℓ ′ + ℓ z ℓ z + ℓ z ⊥ ℓ z ⊥ in V ( R ) ∩ ℓ ⊥ . The Grassmannian of K consists of a single point. Therefore we omit thevariable z in the corresponding theta function ϑ K defined in (2.11).Let α, β ∈ Z , and let µ ∈ K ⊗ Z R . For h ∈ K ′ /K and τ ∈ H , we let ξ h ( τ, µ, α, β ) := X λ ∈ K + rh X t (∆) Q ( λ − βℓ ′ + tℓ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ − βℓ ′ + tℓ ) e ( − αt/ | ∆ | )(4.5) × e (cid:18) ( λ + βµ ) τ | ∆ | − | ∆ | ( λ + βµ/ , αµ ) (cid:19) . Moreover, we define a C [ K ′ /K ]-valued theta function by puttingΞ( τ, µ, α, β ) := X h ∈ K ′ /K ξ h ( τ, µ, α, β ) e h . (4.6)Later we will use the following slightly more explicit formula for Ξ( τ, µ, n, Proposition 4.3. If n is an integer, then we have ξ h ( τ, µ, n,
0) = (cid:18) ∆ n (cid:19) ¯ ε | ∆ | / X λ ∈ K + rhQ ( λ ) ≡ ∆ Q ( h ) (∆) e (cid:18) λ τ | ∆ | − n | ∆ | ( λ, µ ) (cid:19) . Here (cid:0) ∆0 (cid:1) = 1 if ∆ = 1 , and (cid:0) ∆0 (cid:1) = 0 otherwise.Proof. By definition we have ξ h ( τ, µ, n,
0) = X λ ∈ K + rhQ ( λ ) ≡ ∆ Q ( h ) (∆) X t (∆) χ ∆ ( λ + tℓ ) e ( − nt/ | ∆ | ) e (cid:18) λ τ | ∆ | − n | ∆ | ( λ, µ ) (cid:19) . Using the SO + ( L )-invariance of χ ∆ , we find that χ ∆ ( λ + tℓ ) = χ ∆ ([ t, ∗ , (cid:0) ∆ t (cid:1) for λ ∈ K + rh with Q ( λ ) ≡ ∆ Q ( h ) (mod ∆). Inserting the value of the Gauss sum X t (∆) (cid:18) ∆ t (cid:19) e ( nt/ | ∆ | ) = (cid:18) ∆ n (cid:19) ε | ∆ | / , (4.7)we obtain the assertion. (cid:3) Theorem 4.4. If ( M, φ ) ∈ Mp ( Z ) with M = ( a bc d ) , then we have that Ξ( M τ, µ, aα + bβ, cα + dβ ) = φ ( τ ) ˜ ρ K ( M, φ ) · Ξ( τ, µ, α, β ) . Here ˜ ρ K is the representation ρ K when ∆ > , and the representation ¯ ρ K when ∆ < . The proof is based on the following proposition.
Proposition 4.5.
Let h ∈ K ′ /K . For κ ∈ K ′ / ∆ K , a ∈ Z / ∆ Z , and s ∈ Z / ∆ Z , theexponential sum g h ( κ, a, s ) = X κ ′ ∈ K ′ / ∆ Kκ ′ ≡ rh ( L ) b ′ (∆) Q ( κ ′ + sℓ ′ + b ′ ℓ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( κ ′ + sℓ ′ + b ′ ℓ ) e (cid:18) − | ∆ | (( κ, κ ′ ) + ab ′ ) (cid:19) is equal to ε | ∆ | / X h ′ ∈ K ′ /Kκ ≡ rh ′ ( K ) e (cid:0) − sgn(∆)( h, h ′ ) (cid:1) X b (∆) Q ( κ + aℓ ′ + bℓ ) ≡ ∆ Q ( h ′ ) (∆) χ ∆ ( κ + aℓ ′ + bℓ ) e ( bs/ | ∆ | ) . Here ε = 1 if ∆ > , and ε = i if ∆ < .Proof. This follows from Proposition 4.2 for λ = κ + aℓ ′ + bℓ , by applying a finite Fouriertransform in b modulo ∆. (cid:3) Proof of Theorem 4.4.
We only have to check the transformation behavior under the gen-erators T and S of ˜Γ. The transformation law under T follows directly from (4.5). For the EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 17 transformation law under S we notice that we may write ξ h ( τ, µ, α, β ) = X λ ∈ K ′ / ∆ Kλ ≡ rh ( K ) t (∆) Q ( λ − βℓ ′ + tℓ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ − βℓ ′ + tℓ ) e (cid:18) − αt | ∆ | − ( λ , αµ )2 | ∆ | (cid:19) × ϑ K (cid:18) | ∆ | τ, αµ, λ | ∆ | + βµ | ∆ | (cid:19) , where ϑ K is the theta function for the lattice K in (2.11). By Proposition 2.1, we have ξ h ( − /τ, µ, − β, α ) = r τi | K ′ /K | − / | ∆ | − / X λ ∈ K ′ g h ( λ, − β, − α ) × e (cid:18) ( λ + βµ ) τ | ∆ | − | ∆ | ( λ + βµ/ , αµ ) (cid:19) . By Proposition 4.5, we find that ξ h ( − /τ, µ, − β, α ) = r τi ε | K ′ /K | − / X h ′ ∈ K ′ /K e (cid:0) sgn(∆)( h, h ′ ) (cid:1) ξ h ( τ, µ, α, β ) . This concludes the proof of the theorem. (cid:3)
Lemma 4.6.
We have that θ h ( τ, z ) = 1 p | ∆ | ℓ z X λ ∈ rh + L/ Z ℓ X d ∈ Z X t (∆) Q ( λ + tℓ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ + tℓ ) e ( − dt/ | ∆ | ) × e (cid:18) λ w τ | ∆ | − d | ∆ | ( λ, ℓ z − ℓ z ⊥ )2 ℓ z − | d + ( λ, ℓ ) τ | i | ∆ | vℓ z (cid:19) . Proof.
The proof follows the argument of Lemma 5.1 in [Bo1] (see also Lemma 2.3 in [Br]).In the definition of θ h ( τ, z ), we rewrite the sum over λ ∈ rh + L as a sum over λ ′ + d | ∆ | ℓ ,where λ ′ runs through rh + L/ Z ∆ ℓ and d runs through Z . We obtain θ h ( τ, z ) = v / X λ ∈ rh + L/ Z ∆ ℓQ ( λ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ ) X d ∈ Z g ( | ∆ | τ, z, λ/ | ∆ | ; d ) , where the function g ( τ, z, λ ; d ) is defined by g ( τ, z, λ ; d ) = e (cid:18) ( λ + dℓ ) z τ + ( λ + dℓ ) z ⊥ τ (cid:19) for τ ∈ H , z ∈ Gr( V ), λ ∈ V ( R ), and d ∈ R . We apply Poisson summation to the sumover d . The Fourier transform of g as a function in d isˆ g ( τ, z, λ, d ) = 1 p vℓ z e (cid:18) λ w τ − d ( λ, ℓ z − ℓ z ⊥ )2 ℓ z − | d + ( λ, ℓ ) τ | ivℓ z (cid:19) (see [Br], p. 43). We obtain θ h ( τ, z ) = v / X λ ∈ rh + L/ Z ∆ ℓQ ( λ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ ) X d ∈ Z ˆ g ( | ∆ | τ, z, λ/ | ∆ | ; d )= 1 p | ∆ | ℓ z X λ ∈ rh + L/ Z ∆ ℓQ ( λ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ ) X d ∈ Z e (cid:18) λ w τ | ∆ | − d ( λ, ℓ z − ℓ z ⊥ )2 | ∆ | ℓ z − | d + ( λ, ℓ ) τ | iv | ∆ | ℓ z (cid:19) . The claim follows by rewriting the sum over λ ∈ rh + L/ Z ∆ ℓ as a sum over λ ′ + tℓ , where λ ′ runs through rh + L/ Z ℓ and t runs through Z / ∆ Z , and by using the facts that ℓ w = 0and ( ℓ, ℓ z − ℓ z ⊥ ) / ℓ z = 1. (cid:3) Lemma 4.7.
We have that θ h ( τ, z ) = 1 p | ∆ | ℓ z X c,d ∈ Z exp (cid:18) − π | cτ + d | | ∆ | vℓ z (cid:19) ξ h ( τ, µ, d, − c ) . Proof.
Using rh + L/ Z ℓ = rh + K + Z ℓ ′ and the identities ℓ ′ w = − µ w , − µ ℓ ′ , ℓ z − ℓ z ⊥ )2 ℓ z , ( λ, µ ) = ( λ, ℓ z − ℓ z ⊥ )2 ℓ z for λ ∈ K ⊗ R , the formula of Lemma 4.6 can rewritten as θ h ( τ, z ) = 1 p | ∆ | ℓ z X λ ∈ rh + K X c,d ∈ Z X t (∆) Q ( λ + cℓ ′ + tℓ ) ≡ ∆ Q ( h ) (∆) χ ∆ ( λ + cℓ ′ + tℓ ) e ( − dt/ | ∆ | ) × e (cid:18) ( λ − cµ ) w τ | ∆ | − | ∆ | ( λ − cµ/ , dµ ) − | cτ + d | i | ∆ | vℓ z (cid:19) . Inserting the definition (4.5) of ξ h ( τ, µ, α, β ), we obtain the assertion. (cid:3) Theorem 4.8.
We have that Θ ∆ ,r ( τ, z ) = 1 p | ∆ | ℓ z Ξ( τ, , , p | ∆ | ℓ z X n ≥ X γ ∈ ˜Γ ∞ \ ˜Γ (cid:20) exp (cid:18) − πn | ∆ |ℑ ( τ ) ℓ z (cid:19) Ξ( τ, µ, n, (cid:21) | / , ˜ ρ K γ. EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 19 Proof.
According to Lemma 4.7, we haveΘ ∆ ,r ( τ, z ) = 1 p | ∆ | ℓ z X c,d ∈ Z exp (cid:18) − π | cτ + d | | ∆ | vℓ z (cid:19) Ξ( τ, µ, d, − c )= 1 p | ∆ | ℓ z Ξ( τ, µ, , p | ∆ | ℓ z X n ≥ X c,d ∈ Z ( c,d )=1 exp (cid:18) − πn | cτ + d | | ∆ | vℓ z (cid:19) Ξ( τ, µ, nd, − nc ) . Writing the sum over coprime integers c, d as a sum over ˜Γ ∞ \ ˜Γ and using the transformationlaw for Ξ( τ, µ, α, β ) of Theorem 4.4, we obtain the assertion. (cid:3) According to Proposition 4.3, the function Ξ( τ, , ,
0) in Theorem 4.8 vanishes when∆ = 1. When ∆ = 1 we have Ξ( τ, , ,
0) = X λ ∈ K ′ e ( Q ( λ ) τ ) e λ , and so Ξ( τ, , ,
0) is the usual vector valued holomorphic theta function of the one-dimensional positive definite lattice K .Let k be a basis vector for K . If y ∈ K ⊗ R , we write y > y is a positive multiple of k . Let f ∈ H / , ˜ ρ L . We define the Weyl vector corresponding to f and ℓ to be the unique ρ f,ℓ ∈ K ′ ⊗ R such that( ρ f,ℓ , y ) = p ( y, y )8 π p | ∆ | Z reg F h f ( τ ) , Ξ( τ, , , i v / du dvv (4.8)for all y ∈ K ⊗ R with y >
0. Here F denotes the standard fundamental domain forthe action of SL ( Z ) on H , and the integral has to be regularized as in [Bo1]. We have, ρ f,ℓ = 0 when ∆ = 1. (This is also true for cusps given by primitive isotropic vectors ℓ with ( ℓ, L ) = Z .) One can show that ρ f,ℓ does not depend on the choice of the vector ℓ ′ .The sign of ρ f,ℓ depends on the choice of k .We conclude this section with an important fact on the rationality of Weyl vectors ρ f,ℓ . Proposition 4.9.
Let f ∈ H / , ˜ ρ L be a harmonic weak Maass form with coefficients c ± ( m, h ) as in (2.5) . If c + ( m, h ) ∈ Q for all m ≤ and f is orthogonal to weight / cusp forms, then ρ f,ℓ ∈ Q .Proof. The idea of the proof is similar to § N is a prime. Therefore we need some additional care. Since ρ f,ℓ = 0when ∆ = 1, we only need to consider the case ∆ = 1. Let E / ( τ ) be the weight 3 / ρ L normalized to have constant term e . It turnsout that ξ / ( E / )( τ ) = C √ N π Ξ( τ, , ,
0) + s ( τ ) , where C is a non-zero rational constant and s ∈ S / ,ρ L is a cusp form. Hence the integralin (4.8) can be computed by means of (2.10) in terms of the coefficients of the holomorphic part of the Eisenstein series E / . These coefficients are known to be generalized classnumbers and thereby rational. Since f is orthogonal to cusp forms, the cusp form s doesnot give any contribution to the integral. This concludes the proof of the proposition. (cid:3) Remark . Proposition 4.9 does not hold without the hypothesis that f is orthogonal toweight 1 / Regularized theta lifts of weak Maass forms
In this section we generalize the regularized theta lift of Borcherds, Harvey, and Moorein two ways to construct automorphic forms on modular curves. First, we work with thetwisted Siegel theta functions of the previous section as kernel functions, and secondly, weconsider the lift for harmonic weak Maass forms.Such generalizations have been studied previously in other settings. In [Ka], Kawaiconstructed twisted theta lifts of weakly holomorphic modular forms in a different way.However, his automorphic products are of higher level, and only twists by even real Dirichletcharacters are considered. In [Br] and [BF], the (untwisted) regularized theta lift wasstudied on harmonic weak Maass forms and was used to construct automorphic Greenfunctions and harmonic square integrable representatives for the Chern classes of Heegnerdivisors. However, the Chern class construction only leads to non-trivial information aboutHeegner divisors if the modular variety under consideration has dimension ≥ , n ) with n ≥ , r ∈ Z such that ∆ ≡ r (mod 4 N ). We let ℓ, ℓ ′ ∈ L be the isotropic vectors ℓ = (cid:18) /N (cid:19) , ℓ ′ = (cid:18) (cid:19) . Then we have K = Z ( − ). For λ ∈ K ⊗ R , we write λ > λ is a positive multipleof ( − ). Following Section 13 of [Bo1], and Section 3.2 of [Br], we identify the complexupper half plane H with an open subset of K ⊗ C by mapping t ∈ H to ( − ) ⊗ t . Moreover,we identify H with the Grassmannian Gr( V ) by mapping t ∈ H to the positive definitesubspace z ( t ) = R ℜ (cid:18) t − t − t (cid:19) + R ℑ (cid:18) t − t − t (cid:19) EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 21 of V ( R ). Under this identification, the action of Spin( V ) on H by fractional linear trans-formations corresponds to the linear action on Gr( V ) through SO( V ). We have that ℓ z = 12 N ℑ ( t ) , ( λ, µ ) = ( λ, ℜ ( t )) ,λ /ℓ z = ( λ, ℑ ( t )) , for λ ∈ K ⊗ R . In the following we will frequently identify t and z ( t ) and simply write z for this variable. We let z = x + iy be the decomposition into real and imaginary part.We now define twisted Heegner divisors on the modular curve X ( N ). For any vector λ ∈ L ′ of negative norm, the orthogonal complement λ ⊥ ⊂ V ( R ) defines a point Z ( λ )in Gr( V ) ∼ = H . For h ∈ L ′ /L and a negative rational number m ∈ Z + sgn(∆) Q ( h ), weconsider the twisted Heegner divisor Z ∆ ,r ( m, h ) := X λ ∈ L d ∆ ,hr / Γ ( N ) χ ∆ ( λ ) w ( λ ) Z ( λ ) ∈ Div( X ( N )) Q , (5.1)where d := 4 N m sgn(∆) ∈ Z . Note that d is a discriminant which is congruent to a squaremodulo 4 N and which has the opposite sign as ∆. Here w ( λ ) is the order of the stabilizerof λ in Γ ( N ). (So w ( λ ) ∈ { , , } , and w ( λ ) = 2 when d ∆ < − y ∆ ,r ( m, h ) := Z ∆ ,r ( m, h ) − deg( Z ∆ ,r ( m, h )) · ∞ . (5.2)We have y ∆ ,r ( f ) = Z ∆ ,r ( f ) when ∆ = 1. By the theory of complex multiplication, the divi-sor Z ∆ ,r ( h, m ) is defined over Q ( √ D, √ ∆) (for example, see §
12 of [Gro1]). The followinglemma shows that it is defined over Q ( √ ∆) and summarizes some further properties. Lemma 5.1.
Let w N be the Fricke involution on X ( N ) , and let τ denote complex con-jugation, and let σ be the non-trivial automorphism of Q ( √ D, √ ∆) / Q ( √ D ) . Then thefollowing are true: (i) w N ( Z ∆ ,r ( m, h )) = Z ∆ ,r ( m, − h ) , (ii) τ ( Z ∆ ,r ( m, h )) = Z ∆ ,r ( m, − h ) , (iii) σ ( Z ∆ ,r ( m, h )) = − Z ∆ ,r ( m, h ) , (iv) Z ∆ ,r ( m, − h ) = sgn(∆) Z ∆ ,r ( m, h ) , (v) Z ∆ ,r ( m, h ) is defined over Q ( √ ∆) .Proof. Properties (i) and (ii) are verified by a straightforward computation, and (iv) im-mediately follows from the definition of the genus character χ ∆ . Moreover, (iii) followsfrom the theory of complex multiplication (see p. 15 of [BS], and [Gro1]). Finally, (v) is aconsequence of (ii), (iii), (iv). (cid:3) Remark . Our definition of Heegner divisors differs slightly from [GKZ]. They considerthe orthogonal complements of vectors λ = (cid:16) b/ N − a/Nc − b/ N (cid:17) ∈ L ′ of negative norm with a > positive definite binary quadratic forms. Recall that ˜ ρ L = ρ L for ∆ >
0, and ˜ ρ L = ¯ ρ L for ∆ <
0. Let f ∈ H / , ˜ ρ L be a harmonicweak Maass form of weight 1 / ρ L . We denote the coefficients of f = f + + f − by c ± ( m, h ) as in (2.5). Note that c ± ( m, h ) = 0 unless m ∈ Z + sgn(∆) Q ( h ).Moreover, by means of (2.3) we see that c ± ( m, h ) = c ± ( m, − h ) if ∆ >
0, and c ± ( m, h ) = − c ± ( m, − h ) if ∆ <
0. Throughout we assume that c + ( m, h ) ∈ R for all m and h .Using the Fourier coefficients of the principal part of f , we define the twisted Heegnerdivisor associated to f by Z ∆ ,r ( f ) := X h ∈ L ′ /L X m< c + ( m, h ) Z ∆ ,r ( m, h ) ∈ Div( X ( N )) R , (5.3) y ∆ ,r ( f ) := X h ∈ L ′ /L X m< c + ( m, h ) y ∆ ,r ( m, h ) ∈ Div( X ( N )) R . (5.4)Notice that y ∆ ,r ( f ) = Z ∆ ,r ( f ) when ∆ = 1. The divisors lie in Div( X ( N )) Q if the coeffi-cients of the principal part of f are rational.We define a regularized theta integral of f byΦ ∆ ,r ( z, f ) = Z regτ ∈F h f ( τ ) , Θ ∆ ,r ( τ, z ) i v / du dvv . (5.5)Here F denotes the standard fundamental domain for the action of SL ( Z ) on H , and theintegral has to be regularized as in [Bo1]. Proposition 5.2.
The theta integral Φ ∆ ,r ( z, f ) defines a Γ ( N ) -invariant function on H \ Z ∆ ,r ( f ) with a logarithmic singularity on the divisor − Z ∆ ,r ( f ) . If Ω denotes the in-variant Laplace operator on H , we have ΩΦ ∆ ,r ( z, f ) = (cid:18) ∆0 (cid:19) c + (0 , . Proof.
Using the argument of Section 6 of [Bo1], one can show that Φ ∆ ,r ( z, f ) defines aΓ ( N )-invariant function on H \ Z ∆ ,r ( f ) with a logarithmic singularity on − Z ∆ ,r ( f ). Toprove the claim concerning the Laplacian, one may argue as in Theorem 4.6 of [Br]. (cid:3) Remark . The fact that the function Φ ∆ ,r ( z, f ) is subharmonic implies that it is realanalytic on H \ Z ∆ ,r ( f ) by a standard regularity theorem for elliptic differential operators.We now describe the Fourier expansion of Φ ∆ ,r ( z, f ). Recall the definition of the Weylvector ρ f,ℓ corresponding to f and ℓ , see (4.8). If X is a normal complex space, D ⊂ X a Cartier divisor, and f a smooth function on X \ supp( D ),then f has a logarithmic singularity along D , if for any local equation g for D on an open subset U ⊂ X ,the function f − log | g | is smooth on U . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 23 Theorem 5.3.
For z ∈ H with y ≫ , we have Φ ∆ ,r ( z, f ) = − X λ ∈ K ′ λ> X b (∆) (cid:18) ∆ b (cid:19) c + ( | ∆ | λ / , rλ ) log | − e (( λ, z ) + b/ ∆) | (5.6) + ( √ ∆ c + (0 , L (1 , χ ∆ ) , if ∆ = 1 , π ( ρ f,ℓ , y ) − c + (0 , πN y ) + Γ ′ (1)) , if ∆ = 1 .Proof. Here we carry out the proof only in the case ∆ = 1, for which the regularization isslightly easier and there is no Weyl vector term. We note that when ∆ = 1, the proof issimilar, and when f is weakly holomorphic it is contained in Theorem 13.3 of [Bo1]. In ourproof we essentially follow the argument of Theorem 7.1 of [Bo1], and Theorem 2.15 of [Br].In particular, all questions regarding convergence can be treated analogously. Inserting theformula of Theorem 4.8 in definition (5.5) and unfolding, we obtainΦ ∆ ,r ( z, f ) = √ p | ∆ | ℓ z X n ≥ Z ∞ v =0 Z u =0 exp (cid:18) − πn | ∆ | vℓ z (cid:19) h f ( τ ) , Ξ( τ, µ, n, i du dvv / . (5.7)Here we have also used the fact that ρ f,ℓ = 0 when ∆ = 1. We temporarily denote theFourier expansion of f by f ( τ ) = X h ∈ L ′ /L X n ∈ Q c ( n, h, v ) e ( nτ ) . Inserting the formula for Ξ( τ, µ, n,
0) of Proposition 4.3 in (5.7), and carrying out theintegration over u , we obtainΦ ∆ ,r ( z, f ) = √ ε | ℓ z | X h ∈ K ′ /K X λ ∈ K + rhQ ( λ ) ≡ ∆ Q ( h ) (∆) X n ≥ (cid:18) ∆ n (cid:19) e (cid:18) n | ∆ | ( λ, µ ) (cid:19) (5.8) × Z ∞ v =0 c ( Q ( λ ) / | ∆ | , h, v ) exp (cid:18) − πn | ∆ | vℓ z − πλ v | ∆ | (cid:19) dvv / . Since ∆ is fundamental, the conditions Q ( λ ) ≡ ∆ Q ( h ) (mod ∆) and λ ≡ rh (mod K ) areequivalent to λ = ∆ λ ′ and rλ ′ ≡ h (mod K ) for some λ ′ ∈ K ′ . Consequently, we haveΦ ∆ ,r ( z, f ) = √ ε | ℓ z | X λ ∈ K ′ X n ≥ (cid:18) ∆ n (cid:19) e (sgn(∆) n ( λ, µ ))(5.9) × Z ∞ v =0 c ( | ∆ | λ / , rλ, v ) exp (cid:18) − πn | ∆ | vℓ z − πλ | ∆ | v (cid:19) dvv / . Notice that only the coefficients c ( | ∆ | λ / , rλ, v ) where λ ∈ K ′ occur in the latter formula.Since K is positive definite, the quantity | ∆ | λ / c ( | ∆ | λ / , rλ, v ) = c + ( | ∆ | λ / , rλ ) , that is, only the coefficients of the “holomorphic part” f + of f give a contribution. Wenow compute the integral over v (for example, using page 77 of [Br]). We obtain Z ∞ v =0 exp (cid:18) − πn | ∆ | vℓ z − πλ | ∆ | v (cid:19) dvv / = p | ∆ | ℓ z n exp( − πn | λ | / | ℓ z | ) . Inserting this and separating the contribution of λ = 0, we getΦ ∆ ,r ( z, f ) = 2 √ ∆ c + (0 , X n ≥ (cid:18) ∆ n (cid:19) n + 4 X λ ∈ K ′ λ> c + ( | ∆ | λ / , rλ ) ℜ √ ∆ X n ≥ n (cid:18) ∆ n (cid:19) e (cid:0) sgn(∆) n ( λ, µ ) + in | λ | / | ℓ z | (cid:1)! . Using the value of the Gauss sum (4.7), we see that this is equal toΦ ∆ ,r ( z, f ) = 2 √ ∆ c + (0 , L (1 , χ ∆ ) − X λ ∈ K ′ λ> X b (∆) (cid:18) ∆ b (cid:19) c + ( | ∆ | λ / , rλ ) log (cid:12)(cid:12)(cid:12)(cid:12) − e (cid:18) b ∆ + ( λ, µ ) + i | λ || ℓ z | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . We finally put in the identities ( λ, µ ) = ( λ, x ) and | λ | / | ℓ z | = | ( λ, y ) | , to derive the theorem. (cid:3) Remark . i) Note that for lattices of signature (2 , n ) with n ≥
2, the lattice K isLorentzian, and one gets a non-trivial contribution from f − to the theta integral, which isinvestigated in [Br]. So the above situation is very special.ii) At the other cusps of X ( N ), the function Φ ∆ ,r ( z, f ) has similar Fourier expansionsas in (5.6).iii) The function Φ ∆ ,r ( z, f ) is a Green function for the divisor Z ∆ ,r ( f ) + C ∆ ,r ( f ) in thesense of [BKK], [BBK]. Here C ∆ ,r ( f ) is a divisor on X ( N ) supported at the cusps, seealso (5.12)5.1. Canonical differentials of the third kind for Heegner divisors.
For the rest ofthis section, we assume that f ∈ H / , ˜ ρ L and that the coefficients c + ( m, h ) are rational forall m ≤ h ∈ L ′ /L . Moreover, we assume that the constant term c + (0 ,
0) of f vanisheswhen ∆ = 1, so that Φ ∆ ,r ( z, f ) is harmonic. We identify Z with K ′ by mapping n ∈ Z to n N ( − ). Then the Fourier expansion of Φ ∆ ,r ( z, f ) given in Theorem 5.3 becomesΦ ∆ ,r ( z, f ) = 2 √ ∆ c + (0 , L (1 , χ ∆ ) + 8 πρ f,ℓ y (5.10) − X n ≥ X b (∆) (cid:18) ∆ b (cid:19) c + ( | ∆ | n N , rn N ) log | − e ( nz + b/ ∆) | . It follows from Proposition 5.2 that η ∆ ,r ( z, f ) = − ∂ Φ ∆ ,r ( z, f )(5.11) EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 25 is a differential of the third kind on X ( N ). It has the residue divisorres( η ∆ ,r ( z, f )) = Z ∆ ,r ( f ) + C ∆ ,r ( f ) , (5.12)where Z ∆ ,r ( f ) ∈ Div( X ( N )) Q , and C ∆ ,r ( f ) ∈ Div( X ( N )) R is a divisor on X ( N ) which issupported at the cusps. (Here we have relaxed the condition that the residues be integral,and only require them to be real.) The multiplicity of any cusp ℓ in the divisor C ∆ ,r ( f ) isgiven by the Weyl vector ρ f,ℓ . According to Proposition 4.9, if f is orthogonal to the cuspforms in S / , ˜ ρ L then ρ f,ℓ is rational. When ∆ = 1, all Weyl vectors vanish and consequently C ∆ ,r ( f ) = 0. Theorem 5.4.
The differential η ∆ ,r ( z, f ) is the canonical differential of the third kindcorresponding to Z ∆ ,r ( f ) + C ∆ ,r ( f ) . It has the Fourier expansion η ∆ ,r ( z, f ) = (cid:18) ρ f,ℓ − sgn(∆) √ ∆ X n ≥ X d | n nd (cid:18) ∆ d (cid:19) c + ( | ∆ | n Nd , rn Nd ) e ( nz ) (cid:19) · πi dz. Proof.
Since Φ ∆ ,r ( z, f ) is harmonic on H \ Z ∆ ,r ( f ), Proposition 3.1 implies that η ∆ ,r ( z, f ) isthe canonical differential of the third kind associated with Z ∆ ,r ( f )+ C ∆ ,r ( f ). Differentiating(5.10), we obtain η ∆ ,r ( z, f ) = (cid:18) ρ f,ℓ − X n ≥ X d ≥ X b (∆) (cid:18) ∆ b (cid:19) c + ( | ∆ | n N , rn N ) ne ( ndz + bd/ ∆) (cid:19) · πi dz. Inserting the value of the Gauss sum (4.7) and reordering the summation, we get theclaimed Fourier expansion. (cid:3)
Theorem 5.5.
Assume that ∆ = 1 . The following are equivalent. (i) A non-zero multiple of y ∆ ,r ( f ) is the divisor of a rational function on X ( N ) . (ii) The coefficients c + ( | ∆ | n N , rn N ) of f are algebraic for all positive integers n . (iii) The coefficients c + ( | ∆ | n N , rn N ) of f are rational for all positive integers n .Proof. Statement (iii) trivially implies (ii). If (ii) holds, then, in view of Theorem 5.4, thecanonical differential η ∆ ,r ( z, f ) of the divisor y ∆ ,r ( f ) ∈ Div( X ( N )) Q is defined over ¯ Q .Consequently, Theorem 3.2 implies that a non-zero multiple of y ∆ ,r ( f ) is the divisor of arational function on X ( N ). Hence (i) holds.It remains to prove that (i) implies (iii). If y ∆ ,r ( z, f ) is a non-zero multiple of the divisorof a rational function on X ( N ), then Lemma 5.1 and Theorem 3.2 imply that η ∆ ,r ( z, f ) isdefined over F = Q ( √ ∆), the field of definition of y ∆ ,r ( f ). Using the q -expansion principleand M¨obius inversion, we deduce from Theorem 5.4, for every positive integer n , that √ ∆ nc + ( | ∆ | n N , rn N ) ∈ F. (5.13)Denote by σ the non-trivial automorphism of F/ Q . It follows from Lemma 5.1 that σ ( y ∆ ,r ( f )) = − y ∆ ,r ( f ). Hence σ ( η ∆ ,r ( z, f )) = − η ∆ ,r ( z, f ). Using the action of σ on the q -expansion of η ∆ ,r ( z, f ), we find that σ fixes the coefficients c + ( | ∆ | n N , rn N ). Consequently,these coefficients are rational. (cid:3) Remark . Theorem 5.5 also holds for ∆ = 1 when S / , ˜ ρ L = 0, or more generally when f is orthogonal to the cusp forms in S / , ˜ ρ L . The latter conditions ensure that the Weylvectors corresponding to f are rational and thereby C ∆ ,r ( f ) ∈ Div(( X ( N )) Q . Observethat Z ∆ ,r ( f ) + C ∆ ,r ( f ) differs from y ∆ ,r ( f ) only by a divisor of degree 0 supported at thecusps. In particular, by the Manin-Drinfeld theorem, the divisors Z ∆ ,r ( f ) + C ∆ ,r ( f ) and y ∆ ,r ( f ) define the same point in J ( Q ) ⊗ R . Remark . Assume that the equivalent conditions of Theorem 5.5 hold.i) It is interesting to consider whether the rational coefficients c + ( | ∆ | n N , rn N ) in (iii) havebounded denominators. This is true if f is weakly holomorphic, since M !1 / , ˜ ρ L has a basis ofmodular forms with integral coefficients. However, if f is an honest harmonic weak Maassform, it is not clear at all.ii) The rational function in Theorem 5.5 (i) has an automorphic product expansion as inTheorem 6.1. It is given by a non-zero power of Ψ ∆ ,r ( z, f ).6. Generalized Borcherds products
In this section we consider certain automorphic products which arise as liftings of har-monic weak Maass forms and which can be viewed as generalizations of the automorphicproducts in Theorem 13.3 of [Bo1]. In particular, for any Heegner divisor Z ∆ ,r ( m, h ),we obtain a meromorphic automorphic product Ψ whose divisor on X ( N ) is the sum of Z ∆ ,r ( m, h ) and a divisor supported at the cusps. But unlike the results in [Bo1], the func-tion Ψ will in general transform with a multiplier system of infinite order under Γ ( N ). Wethen give a criterion when the multiplier system has finite order.As usual, for complex numbers a and b , we let a b = exp( b Log( a )), where Log denotesthe principal branch of the complex logarithm. In particular, if | a | < − a ) b =exp( − b P n ≥ a n n ). Theorem 6.1.
Let f ∈ H / , ˜ ρ L be a harmonic weak Maass form with real coefficients c + ( m, h ) for all m ∈ Q and h ∈ L ′ /L . Moreover, assume that c + ( n, h ) ∈ Z for all n ≤ .The infinite product Ψ ∆ ,r ( z, f ) = e (( ρ f,ℓ , z )) Y λ ∈ K ′ λ> Y b (∆) [1 − e (( λ, z ) + b/ ∆)]( ∆ b ) c + ( | ∆ | λ / ,rλ ) converges for y sufficiently large and has a meromorphic continuation to all of H with thefollowing properties. (i) It is a meromorphic modular form for Γ ( N ) with a unitary character σ which mayhave infinite order. (ii) The weight of Ψ ∆ ,r ( z, f ) is c + (0 , when ∆ = 1 , and is when ∆ = 1 . (iii) The divisor of Ψ ∆ ,r ( z, f ) on X ( N ) is given by Z ∆ ,r ( f ) + C ∆ ,r ( f ) . (iv) We have Φ ∆ ,r ( z, f ) = ( − c + (0 , πN ) + Γ ′ (1)) − | Ψ ∆ ,r ( z, f ) y c + (0 , / | , if ∆ = 1 , √ ∆ c (0 , L (1 , χ ∆ ) − | Ψ ∆ ,r ( z, f ) | , if ∆ = 1 . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 27 Proof.
By means of the same argument as in Section 13 of [Bo1], or Chapter 3 of [Br], itcan be deduced from Proposition 5.2, Remark 12, and Theorem 5.3 that Ψ ∆ ,r ( z, f ) has acontinuation to a meromorphic function on H satisfying (iii) and (iv). Moreover, using theΓ ( N )-invariance of Φ ∆ ,r ( z, f ) one finds that it satisfies the transformation lawΨ ∆ ,r ( γz, f ) = σ ( γ )( cz + d ) c + (0 , Ψ ∆ ,r ( z, f ) , for γ = ( a bc d ) ∈ Γ ( N ), where σ : Γ ( N ) → C × is a unitary character of Γ ( N ). (cid:3) Theorem 6.2.
Suppose that ∆ = 1 . Let f ∈ H / , ˜ ρ L be a harmonic weak Maass form withreal coefficients c + ( m, h ) for all m ∈ Q and h ∈ L ′ /L . Moreover, assume that c + ( n, h ) ∈ Z for all n ≤ . The following are equivalent. (i) The character σ of the function Ψ ∆ ,r ( z, f ) defined in Theorem 6.1 is of finite order. (ii) The coefficients c + ( | ∆ | λ / , rλ ) are rational for all λ ∈ K ′ .Proof. If (i) holds, then there is a positive integer M such that Ψ ∆ ,r ( z, f ) M is a rationalfunction on X ( N ) with divisor M · Z ∆ ,r ( f ). By means of Theorem 5.5 we find that c + ( | ∆ | λ / , rλ ) ∈ Q for all λ ∈ K ′ . Conversely, if (ii) holds, using Theorem 5.5 we mayconclude that M · Z ∆ ,r ( f ) is the divisor of a rational function R on X ( N ) for some positiveinteger M . But this implies thatlog | R | − M log | Ψ ∆ ,r ( z, f ) | is a harmonic function on X ( N ) (without any singularities). By the maximum principle, itis constant. Hence R/ Ψ ∆ ,r ( z, f ) M is a holomorphic function on H with constant modulus,which must be constant. Consequently, σ M is the trivial character. (cid:3) Remark . Theorem 6.2 also holds for ∆ = 1 when S / , ˜ ρ L = 0, or more generally when f is orthogonal to the cusp forms in S / , ˜ ρ L . The latter conditions ensure that Theorem 5.5still applies (see Remark 14). Notice that we may reduce to the case that the constantterm c + (0 ,
0) of f vanishes by adding a suitable rational linear combination of O( K ′ /K )-translates of the vector valued weight 1 / K .The rationality of the coefficients c + ( m, h ) of a harmonic weak Maass form is usually noteasy to verify. In view of Theorem 6.2, it is related to the vanishing of twisted Heegnerdivisors in the Jacobian, which is a deep question (see [GZ]). However for special harmonicweak Maass forms, such as the mock theta functions, one can read off the rationality directlyfrom the construction. This leads to explicit relations among certain Heegner divisors on X ( N ) comparable to the relations among cuspidal divisors coming from modular units.For weakly holomorphic modular forms f ∈ M !1 / , ˜ ρ L the rationality of the Fourier coeffi-cients is essentially dictated by the principal part (with minor complications caused by thepresence of cusp forms of weight 1 / Lemma 6.3.
Suppose that f ∈ M !1 / , ˜ ρ L . If c + ( m, h ) ∈ Q for all m ≤ , then there exists acusp form f ′ ∈ S / , ˜ ρ L such that f + f ′ has rational coefficients.Proof. This follows from the fact that the spaces M !1 / , ˜ ρ L and S / , ˜ ρ L have bases of modularforms with rational coefficients (see [McG]). (cid:3) Remark . Observe that M / , ¯ ρ L is always trivial, by a result of Skoruppa (see Theorem 5.7of [EZ]). However, M / ,ρ L may contain non-zero elements (which are linear combinationsof theta series of weight 1 / Lemma 6.4. If f ∈ S / ,ρ L , then the coefficient c + ( m, h ) of f vanishes unless m = λ / for some λ ∈ K ′ .Proof. This could be proved using the Serre-Stark basis theorem. Here we give a moreindirect proof using the twisted theta lifts of the previous section.Let ∆ = 1 be a fundamental discriminant and let r be an integer satisfying ∆ ≡ r (mod 4 N ). We consider the canonical differential of the third kind η ∆ ,r ( z, f ). Since f isa cusp form and ∆ = 1, the divisor Z ∆ ,r ( f ) vanishes. Consequently, η ∆ ,r ( z, f ) ≡
0. ByTheorem 5.4, we find that c + ( | ∆ | λ / , rλ ) = 0 for all λ ∈ K ′ . This proves the lemma. (cid:3) Lemma 6.5. If f ∈ M !1 / , ˜ ρ L , then the following are true: (i) If c + ( m, h ) = 0 for all m < , then f vanishes identically. (ii) If c + ( m, h ) ∈ Q for all m < , then all coefficients of f are contained in Q .Proof. (i) The hypothesis implies that f / ∆ is a holomorphic modular form of weight − / ρ L . Hence it has to vanish identically. (ii) The assertion follows from(i) using the Galois action on M !1 / , ˜ ρ L . (cid:3) Remark . i) If f ∈ M !1 / , ˜ ρ L is a weakly holomorphic modular form with rational coeffi-cients c + ( m, h ) and c + ( m, h ) ∈ Z for m ≤
0, then Theorem 6.2 and Remark 16 show thatthe automorphic product Ψ ∆ ,r ( z, f ) of Theorem 6.1 is a meromorphic modular form forΓ ( N ) with a character of finite order. When ∆ = 1, this result is contained in Theorem13.3 of [Bo1]. Borcherds proved the finiteness of the multiplier system in [Bo3] using theembedding trick. (However, the embedding trick argument does not work for harmonicweak Maass forms.) In the special case that N = 1 and ∆ >
0, twisted Borcherds productswere first constructed by Zagier in a different way (see § / ρ L is isomorphic to the space of weakly skew holomorphic Jacobi forms in the sense of[Sk1]. For these forms, Theorem 6.1 gives the automorphic products Ψ ∆ ,r for any positive fundamental discriminant ∆. The space of weakly holomorphic modular forms of weight1 / ρ L is isomorphic to the space of “classical” weakly holomorphicJacobi forms in the sense of [EZ]. For these forms, the theorem gives the automorphicproducts Ψ ∆ ,r for any negative fundamental discriminant ∆. Corollary 6.6.
Let F be a number field. If f ∈ M !1 / , ˜ ρ L has the property that all of itscoefficients lie in F , then the divisor y ∆ ,r ( f ) vanishes in the Jacobian J ( Q ( √ D )) ⊗ Z F .Proof. The assertion follows from Theorems 6.1 and 6.2, using the Galois action on M !1 / , ˜ ρ L . (cid:3) As another corollary, we obtain the following generalization of the Gross-Kohnen-Zagiertheorem [GKZ]. It can derived from Corollary 6.6 using Serre duality as in [Bo2].
EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 29 Corollary 6.7.
Let ρ = ρ L when ∆ > , and let ρ = ¯ ρ L when ∆ < . The generatingseries A ∆ ,r ( τ ) = X h ∈ L ′ /L X n> y ∆ ,r ( − n, h ) q n e h is a cusp form of weight / for ˜Γ of type ¯ ρ with values in J ( Q ( √ ∆)) ( i.e. A ∆ ,r ∈ S / , ¯ ρ ⊗ Z J ( Q ( √ ∆))) . (cid:3) Hecke eigenforms and isotypical components of the Jacobian
Now we consider the implications of the results of the previous section when the actionof the Hecke algebra is introduced. We begin with some notation. Let L be the latticeof discriminant 2 N defined in Section 2.4. Let k ∈ Z \ Z . The space of vector valuedholomorphic modular forms M k,ρ L is isomorphic to the space of skew holomorphic Jacobiforms J skewk +1 / ,N of weight k + 1 / N . Moreover, M k, ¯ ρ L is isomorphic to the spaceof holomorphic Jacobi forms J k +1 / ,N . There is an extensive Hecke theory for Jacobi forms(see [EZ], [Sk1], [SZ]), which gives rise to a Hecke theory on M k,ρ L and M k, ¯ ρ L , and whichis compatible with the Hecke theory on vector valued modular forms considered in [BrSt].In particular, there is an Atkin-Lehner theory for these spaces.The subspace S newk,ρ L of newforms of S k,ρ L is isomorphic as a module over the Hecke alge-bra to the space of newforms S new, +2 k − ( N ) of weight 2 k − ( N ) on which the Frickeinvolution acts by multiplication with ( − k − / . The isomorphism is given by the Shimuracorrespondence. Similarly, the subspace S newk, ¯ ρ L of newforms of S k, ¯ ρ L is isomorphic as a mod-ule over the Hecke algebra to the space of newforms S new, − k − ( N ) of weight 2 k − ( N ) onwhich the Fricke involution acts by multiplication with ( − k +1 / (see [SZ], [GKZ], [Sk1]).Observe that the Hecke L -series of any G ∈ S new, ± k − ( N ) satisfies a functional equation under s k − − s with root number ±
1. If G ∈ S new, ± k − ( N ) is a normalized newform (inparticular a common eigenform of all Hecke operators), we denote by F G the number fieldgenerated by the Hecke eigenvalues of G . It is well known that we may normalize thepreimage of G under the Shimura correspondence such that all its Fourier coefficients arecontained in F G .Let ρ be one of the representations ρ L or ¯ ρ L . For every positive integer l there is a Heckeoperator T ( l ) on M k,ρ which is self adjoint with respect to the Petersson scalar product.The action on the Fourier expansion g ( τ ) = P h,n b ( n, h ) e ( mτ ) e h of any g ∈ M k,ρ can bedescribed explicitly (for example, see § § p is aprime coprime to N and we write g | k T ( p ) = P h,n b ∗ ( n, h ) e ( nτ ) e h , we have b ∗ ( n, h ) = b ( p n, ph ) + p k − / (cid:18) N σnp (cid:19) b ( n, h ) + p k − b ( n/p , h/p ) , (7.1)where σ = 1 if ρ = ρ L , and σ = − ρ = ¯ ρ L . There are similar formulas for general l .The Hecke operators act on harmonic weak Maass forms and on weakly holomorphicmodular forms in an analogous way. In particular, the formula for the action on Fouriercoefficients is the same. In the following we assume that k ≤ /
2. Recall from Section 2.2 that there is a bilinear pairing {· , ·} between the spaces S − k, ¯ ρ and H k,ρ , which induces anon-degenerate pairing of S − k, ¯ ρ and H k,ρ /M ! k,ρ . Proposition 7.1.
The Hecke operator T ( l ) is (up to a scalar factor) self adjoint withrespect to the pairing {· , ·} . More precisely we have { g, f | k T ( l ) } = l k − { g | − k T ( l ) , f } for any g ∈ S − k, ¯ ρ and f ∈ H k,ρ .Proof. From the definition of the Hecke operator or from its action on the Fourier expansionof f one sees that ξ k ( f | k T ( l )) = l k − ( ξ k f ) | − k T ( l ) . (7.2)Hence the assertion follows from the the self-adjointness of T ( l ) with respect to the Peters-son scalar product on S − k, ¯ ρ . (cid:3) Recall that (see [McG]) the space S − k, ¯ ρ has a basis of cusp forms with rational coeffi-cients. Let F be any subfield of C . We denote by S − k, ¯ ρ ( F ) the F -vector space of cuspforms in S − k, ¯ ρ with Fourier coefficients in F . Moreover, we write H k,ρ ( F ) for the F -vectorspace of harmonic weak Maass forms whose principal part has coefficients in F . We write M ! k,ρ ( F ) for the subspace of weakly holomorphic modular forms whose principal part hascoefficients in F . Using the pairing {· , ·} , we identify H k,ρ ( F ) /M ! k,ρ ( F ) with the F -dual of S − k, ¯ ρ ( F ). Lemma 7.2.
Let g ∈ S − k, ¯ ρ , and suppose that f ∈ H k,ρ has the property that { g, f } = 1 ,and also satisfies { g ′ , f } = 0 for all g ′ ∈ S − k, ¯ ρ orthogonal to g . Then ξ k ( f ) = k g k − g ,where k g k denotes the Petersson norm of g .Proof. This follows directly from the definition of the pairing. (cid:3)
Lemma 7.3.
Let F be a subfield of C , and let g ∈ S − k, ¯ ρ ( F ) be a newform. There is a f ∈ H k,ρ ( F ) such that ξ k ( f ) = k g k − g. Proof.
Since g ∈ S − k, ¯ ρ ( F ) is a newform, the orthogonal complement of g with respect tothe Petersson scalar product has a basis consisting of cusp forms with coefficients in F .Let g , . . . , g d ∈ S − k, ¯ ρ ( F ) be a basis of the orthogonal complement of g . Let f , . . . , f d ∈ H k,ρ ( F ) be the dual basis of the basis g, g , . . . , g d with respect to {· , ·} . In particular { g, f } = 1, and { g, f j } = 0 for all j = 2 , . . . , d . According to Lemma 7.2 we have that ξ k ( f ) = k g k − g . This completes the proof of the lemma. (cid:3) Lemma 7.4.
Let f ∈ H k,ρ ( F ) and assume that ξ k ( f ) | − k T ( l ) = λ l ξ k ( f ) with λ l ∈ F .Then f | k T ( l ) − l k − λ l f ∈ M ! k,ρ ( F ) . Proof.
The formula for the action of T ( l ) on the Fourier expansion implies that f | k T ( l ) ∈ H k,ρ ( F ). Moreover, it follows from (7.2) that ξ k (cid:0) f | k T ( l ) − l k − λ l f (cid:1) = 0 . This proves the lemma. (cid:3)
EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 31 We now come to the main results of this section. Let G ∈ S new ( N ) be a normalizednewform of weight 2 and write F G for the number field generated by the eigenvalues of G .If G ∈ S new, − ( N ), we put ρ = ρ L and assume that ∆ is a positive fundamental discriminant.If G ∈ S new, +2 ( N ), we put ρ = ¯ ρ L and assume that ∆ is a negative fundamental discriminant.There is a newform g ∈ S new / , ¯ ρ mapping to G under the Shimura correspondence. We maynormalize g such that all its coefficients are contained in F G . Therefore by Lemma 7.3,there is a harmonic weak Maass form f ∈ H / ,ρ ( F G ) such that ξ / ( f ) = k g k − g. This form is unique up to addition of a weakly holomorphic form in M !1 / ,ρ ( F G ). Theorem 7.5.
The divisor y ∆ ,r ( f ) ∈ Div( X ( N )) ⊗ F G defines a point in the G -isotypicalcomponent of the Jacobian J ( Q ( √ ∆)) ⊗ C .Proof. We write λ l for eigenvalue of the Hecke operator T ( l ) corresponding to G (where l ∈ Z > ). Let p be any prime coprime to N . It suffices to show that under the action ofthe Hecke algebra on the Jacobian we have T ( p ) y ∆ ,r ( f ) = λ p y ∆ ,r ( f ) . It is easily seen that T ( p ) y ∆ ,r ( m, h ) = y ∆ ,r ( p m, ph ) + (cid:18) N σmp (cid:19) y ∆ ,r ( m, h ) + py ∆ ,r ( m/p , h/p ) , (7.3)where m ∈ Q is negative, h ∈ L ′ /L , and σ = sgn(∆). (For example, see p. 507 and p. 542of [GKZ] for the case ∆ = 1. The general case is analogous.) Combining this with (7.1),we see that T ( p ) y ∆ ,r ( f ) = py ∆ ,r ( f | / T ( p )) . (7.4)In view of Lemma 7.4, there is a f ′ ∈ M !1 / ,ρ ( F G ) such that f | / T ( p ) = p − λ p f + f ′ . Combining this with (7.4), we find that T ( p ) y ∆ ,r ( f ) = λ p y ∆ ,r ( f ) + py ∆ ,r ( f ′ ) . But Lemma 6.3 and Corollary 6.6 imply that y ∆ ,r ( f ′ ) vanishes in J ( Q ( √ ∆)) ⊗ C . (cid:3) Theorem 7.6.
Assume that ∆ = 1 . Let f ∈ H / ,ρ ( F G ) be a weak Maass form suchthat ξ / ( f ) is a newform which maps to G ∈ S new ( N ) under the Shimura correspondence.Denote the Fourier coefficients of f + by c + ( m, h ) . Then the following are equivalent: (i) The Heegner divisor y ∆ ,r ( f ) vanishes in J ( Q ( √ ∆)) ⊗ C . (ii) The coefficient c + ( | ∆ | N , r N ) is algebraic. (iii) The coefficient c + ( | ∆ | N , r N ) is contained in F G . Proof.
If (i) holds, then Theorem 5.5 implies that the coefficients c + ( | ∆ | n N , rn N ) are in F G for all positive integers n . Hence (iii) holds. Moreover, it is clear that (iii) implies (ii).Now we show that (ii) implies (i). If y ∆ ,r ( f ) = 0 is in J ( Q ( √ ∆)) ⊗ C , then Theorem 5.5implies that there are positive integers n for which c + ( | ∆ | n N , rn N ) is transcendental. Let n be the smallest of these integers. We need to show that n = 1.Assume that n = 1, and that p | n is prime. Let λ p be the eigenvalue of the Heckeoperator T ( p ) corresponding to G . By Lemma 7.4, there is a f ′ ∈ M !1 / ,ρ ( F G ) such that f | / T ( p ) = p − λ p f + f ′ . Using the formula for the action of T ( p ) on the Fourier expansion of f , we see that c + ( | ∆ | n N , rn N ) is an algebraic linear combination of Fourier coefficients c + ( | ∆ | n N , r n N ) of f with n ≤ n /p and coefficients of f ′ . In view of Lemma 6.3 and Lemma 6.4, this impliesthat c + ( | ∆ | n N , rn N ) is algebraic, contradicting our assumption. (cid:3) Remark . Theorem 7.6 also holds for ∆ = 1 when S / ,ρ = 0. More generally, it shouldalso hold for ∆ = 1 when f is chosen to be orthogonal to the cusp forms in S / ,ρ . Thelatter condition ensures that Theorem 5.5 still applies, see Remark 14. However, it remainsto show that a weakly holomorphic form f ′ ∈ M !1 / ,ρ ( F G ) which is orthogonal to cusp forms,automatically has all coefficients in F G .Using the action of the Hecke algebra on the Jacobian, we may derive a more precise ver-sion of Corollary 6.7. Let y G ∆ ,r ( m, h ) denote the projection of the Heegner divisor y ∆ ,r ( m, h )onto its G -isotypical component. We consider the generating series A G ∆ ,r ( τ ) = X h ∈ L ′ /L X n> y G ∆ ,r ( − n, h ) q n e h ∈ S / , ¯ ρ ⊗ Z J ( Q ( √ ∆)) . Theorem 7.7.
We have the identity A G ∆ ,r ( τ ) = g ( τ ) ⊗ y ∆ ,r ( f ) . In particular, the space in ( J ( Q ( √ ∆)) ⊗ C ) G spanned by the y G ∆ ,r ( m, h ) is at most one-dimensional and is generated by y ∆ ,r ( f ) .Proof. We write λ l for eigenvalue of the Hecke operator T ( l ) corresponding to G (where l ∈ Z > ). Let p be any prime coprime to N . By means of (7.1) and (7.3), we see that T ( p ) A ∆ ,r = A ∆ ,r | / T ( p ) , where on the left hand side the Hecke operator acts through the Jacobian, while on theright hand side it acts through S / , ¯ ρ . Consequently, for the G -isotypical part we find A G ∆ ,r | / T ( p ) = λ p A G ∆ ,r . Hence A G ∆ ,r is an eigenform of all the T ( p ) for p coprime to N with the same eigenvaluesas g . By “multiplicity one” for S new / , ¯ ρ , we find that A G ∆ ,r = Cg for some constant C ∈ EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 33 ( J ( Q ( √ ∆)) ⊗ C ) G . To compute the constant, we determine the pairing with f . We have { A G ∆ ,r , f } = y ∆ ,r ( f ) , { Cg, f } = C ( g, ξ / ( f )) = C. This concludes the proof of the theorem. (cid:3)
Note that in the case ∆ = 1, the theorem (in a different formulation) was proved in adifferent way in [GKZ]. We may use the Gross-Zagier theorem to relate the vanishing of y ∆ ,r ( f ) to the vanishing of a twisted L -series associated with G . Theorem 7.8.
Let the hypotheses be as in Theorem 7.6. The following are equivalent. (i)
The Heegner divisor y ∆ ,r ( f ) vanishes in J ( Q ( √ ∆)) ⊗ C . (ii) We have L ′ ( G, χ ∆ ,
1) = 0 .Proof.
We denote the Fourier coefficients of g by b ( n, h ) for n ∈ Z − sgn(∆) Q ( h ) and h ∈ L ′ /L . Since g is a newform, by Lemma 3.2 of [SZ], there is a fundamental discriminant d coprime to N such that sgn(∆) d < b ( n, h ) = 0 for n = − sgn(∆) d N andsome h ∈ L ′ /L . According to Corollary 1 of Chapter II in [GKZ], and [Sk2], we have theWaldspurger type formula | b ( n, h ) | = k g k π k G k p | d | L ( G, χ d , . In particular, the non-vanishing of b ( n, h ) implies the non-vanishing of L ( G, χ d , J ( H ) of y G ∆ ,r ( − n, h ) is given by h y G ∆ ,r ( − n, h ) , y G ∆ ,r ( − n, h ) i = h K u π k G k p | d ∆ | L ′ ( G, χ ∆ , L ( G, χ d , . Here H is the Hilbert class field of K = Q ( √ d ∆), and 2 u is the number of roots of unityin K , and h K denotes the class number of K .Consequently, the Heegner divisor y G ∆ ,r ( − n, h ) vanishes if and only if L ′ ( G, χ ∆ ,
1) van-ishes. But by Theorem 7.7 we know that y G ∆ ,r ( − n, h ) = y ∆ ,r ( f ) b ( n, h ) . This concludes the proof of the theorem. (cid:3)
As described in the introduction, the results in this section imply Theorem 1.1. Weconclude this section with the proof of Corollary 1.4.
Proof of Corollary 1.4.
By Theorem 1.1, it suffices to show that(7.5) { ≤ ∆ < X fundamental : L ( G, χ ∆ , = 0 } ≫ G X log X , and(7.6) {− X < ∆ < L ′ ( G, χ ∆ , = 0 } ≫ G,ǫ X − ǫ . Corollary 3 of [OSk] implies (7.5), and the proof of Theorem 1 of [PP] implies (7.6). (cid:3) Examples
Here we give some examples related to the main results in this paper.8.1.
Twisted modular polynomials.
Here we use Theorems 6.1 and 6.2 to deduce theinfinite product expansion of twisted modular polynomials found by Zagier (see § N = 1. Then we have L ′ /L ∼ = Z / Z . Moreover, H / ,ρ L = M !1 / ,ρ L and H / , ¯ ρ L = 0.Therefore we consider the case where ∆ > r ∈ Z such that ∆ ≡ r (mod 4). By § M !1 / ,ρ L can be identified withthe space M !1 / of scalar valued weakly holomorphic modular forms of weight 1 / (4)satisfying the Kohnen plus space condition. For every negative discriminant d , there is aunique f d ∈ M !1 / , whose Fourier expansion at the cusp ∞ has the form f d = q d + X n ≥ n ≡ , c d ( n ) q n . The expansions of the first few f d are given in [Za2], and one sees that the coefficients arerational. Theorems 6.1 and 6.2 gives a meromorphic modular form Ψ ∆ ( z, f d ) := Ψ ∆ ,r ( z, f d )of weight 0 for the group Γ = SL ( Z ) whose divisor on X (1) is given by Z ∆ ( d ) := Z ∆ ,r ( d/ , d/
2) = X λ ∈ L ∆ d / Γ χ ∆ ( λ ) w ( λ ) · Z ( λ ) . By (2.15), L ∆ d / Γ corresponds to the Γ-classes of integral binary quadratic forms of dis-criminant ∆ d . Moreover, for sufficiently large ℑ ( z ), we have the product expansionΨ ∆ ( z, f d ) = ∞ Y n =1 Y b (∆) [1 − e ( nz + b/ ∆)]( ∆ b ) c d (∆ n ) . (8.1)From these properties it follows thatΨ ∆ ( z, f d ) = Y λ ∈ L ∆ d / Γ (cid:0) j ( z ) − j ( Z ( λ )) (cid:1) χ ∆ ( λ ) . (8.2)As an example, let ∆ := 5 and d := −
3. There are two classes of binary quadratic formsof discriminant −
15, represented by [1 , ,
4] and [2 , , − √− and − √− . It is well known that the singular moduli of j ( τ ) of these pointsare − − √
5, and − + √
5. The function f − has the Fourier expansion f − = q − − q + 26752 q − q + 1707264 q − q + . . . . Multiplying out the product over b in (8.1), we obtain the infinite product expansionΨ ( z, f − ) = j ( z ) + + √ j ( z ) + − √ ∞ Y n =1 −√ q n + q n √ q n + q n ! c − (5 n ) . EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 35 Ramanujan’s mock theta functions f ( q ) and ω ( q ) . Here we give an example ofa Borcherds product arising from Ramanujan’s mock theta functions. We first recall themodular transformation properties of f ( q ), defined in (1.1), and ω ( q ) := ∞ X n =0 q n +2 n ( q ; q ) n +1 = 1(1 − q ) + q (1 − q ) (1 − q ) + q (1 − q ) (1 − q ) (1 − q ) + . . . . (8.3)Using these functions, define the vector valued function F ( τ ) by(8.4) F ( τ ) = ( F ( τ ) , F ( τ ) , F ( τ )) T := ( q − f ( q ) , q ω ( q ) , q ω ( − q )) T . Similarly, let G ( τ ) be the vector valued non-holomorphic function defined by(8.5) G ( τ ) = ( G ( τ ) , G ( τ ) , G ( τ )) T := 2 i √ Z i ∞− τ ( g ( z ) , g ( z ) , − g ( z )) T p − i ( τ + z ) dz, where the g i ( τ ) are the cuspidal weight 3 / g ( τ ) := ∞ X n = −∞ ( − n (cid:18) n + 13 (cid:19) e πi ( n + ) τ ,g ( τ ) := − ∞ X n = −∞ (cid:18) n + 16 (cid:19) e πi ( n + ) τ ,g ( τ ) := ∞ X n = −∞ (cid:18) n + 13 (cid:19) e πi ( n + ) τ . (8.6)Using these vector valued functions, Zwegers [Zw1] let H ( τ ) := F ( τ ) − G ( τ ), and he proved[Zw1] that it is a vector valued weight 1 / H ( τ + 1) = ζ − ζ ζ H ( τ ) , (8.7) H ( − /τ ) = √− iτ · − H ( τ ) . (8.8)Now let N := 6. One can check the following lemma which asserts that this representationof ˜Γ is an irreducible piece of the Weil representation ¯ ρ L . Lemma 8.1.
Assume that H = ( h , h , h ) T : H → C is a vector valued modular form ofweight k for ˜Γ transforming with the representation defined by (8.7) and (8.8) . Then thefunction ˜ H = (0 , h , h − h , , − h − h , − h , , h , h + h , , h − h , − h ) T (8.9) is a vector valued modular form of weight k for ˜Γ with representation ¯ ρ L . Here we haveidentified C [ L ′ /L ] with C by mapping the standard basis vector of C [ L ′ /L ] correspondingto the coset j/
12 + Z ∈ L ′ /L to the standard basis vector e j of C (where j = 0 , . . . , ). This lemma shows that H gives rise to an element ˜ H ∈ H / , ¯ ρ L . Let c ± ( m, h ) be thecoefficients of ˜ H . For any fundamental discriminant ∆ < r such that ∆ ≡ r (mod 24), we obtain a twisted generalized Borcherds lift Ψ ∆ ,r ( z, ˜ H ). By Theorems 6.1and 6.2, it is a weight 0 meromorphic modular function on X (6) with divisor2 Z ∆ ,r ( − , ) − Z ∆ ,r ( − , ) . Moreover, it has the infinite product expansionΨ ∆ ,r ( z, ˜ H ) = ∞ Y n =1 P ∆ ( q n ) c + ( | ∆ | n / , rn/ , (8.10)where P ∆ ( X ) := Y b (∆) [1 − e ( b/ ∆) X ]( ∆ b ) . (8.11)For instance, let ∆ := − r := 4. The set L − , / Γ (6) is represented by the binaryquadratic forms Q = [6 , ,
1] and Q = [ − , , − L − , − / Γ (6) is represented by − Q and − Q . The Heegner points in H corresponding to Q and Q respectively are α = − √− , α = 2 + √− . Consequently, the divisor of Ψ − , ( z, ˜ H ) on X (6) is given by 2( α ) − α ). In this casethe infinite product expansion (8.10) only involves the coefficients of the components of ˜ H of the form ± ( h + h ). To simplify the notation, we put − q / (cid:0) ω ( q / ) + ω ( − q / ) (cid:1) =: X n ∈ Z +1 / a ( n ) q n = − q / − q / − q / − q / − . . . . We have P − ( X ) := 1 + √− X − X − √− X − X , and the infinite product expansion (8.10) can be rewritten asΨ − , ( z, ˜ H ) = ∞ Y n =1 P − ( q n )( n ) a ( n / . (8.12)It is amusing to work out an expression for Ψ − , ( z, ˜ H ). We use the Hauptmodul forΓ ∗ (6), the extension of Γ (6) by all Atkin-Lehner involutions, which is j ∗ ( z ) = (cid:18) η ( z ) η (2 z ) η (3 z ) η (6 z ) (cid:19) + 4 + 3 (cid:18) η (3 z ) η (6 z ) η ( z ) η (2 z ) (cid:19) = q − + 79 q + 352 q + 1431 q + . . . . Here η ( z ) = q / Q ∞ n =1 (1 − q n ) denotes the Dedekind eta function. We have j ∗ ( α ) = j ∗ ( α ) = −
10. Hence j ∗ ( z ) + 10 is a rational function on X (6) whose divisor consists of EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 37 the 4 cusps with multiplicity − α , α with multiplicity 2. The uniquenormalized cusp form of weight 4 for Γ ∗ (6) is δ ( z ) := η ( z ) η (2 z ) η (3 z ) η (6 z ) = q − q − q + 4 q + 6 q + 6 q − q − q + . . . . Using these functions, we find that φ ( z ) := Ψ − , ( z, ˜ H ) · ( j ∗ ( z ) + 10) δ ( z )is a holomorphic modular form of weight 4 for Γ (6) with divisor 4( α ). Using the classicalEisenstein series, it turns out that450 φ ( z ) = (3360 − √− δ ( z ) + (1 − √− E ( z ) + (4 − √− E (2 z )+ (89 + 7 √− E (3 z ) + (356 + 28 √− E (6 z ) . Putting this all together, (8.12) becomes ∞ Y n =1 (cid:18) √− q n − q n − √− q n − q n (cid:19) ( n ) a ( n / = φ ( z )( j ∗ ( z ) + 10) δ ( z )= 1 − √− q − (64 − √− q + (384 + 168 √− q + (64 − √− q + . . . . Relations among Heegner points and vanishing derivatives of L -functions. Let G ∈ S new (Γ ( N )) be a newform of weight 2, and let ∆ be a fundamental discriminantsuch that L ( G, χ ∆ , s ) has an odd functional equation. By the Gross-Zagier formula, thevanishing of L ′ ( G, χ ∆ ,
1) is equivalent to the vanishing of a certain Heegner divisor in theJacobian. More precisely, let f be a harmonic weak Maass form of weight 1 / G as in Section 7. Then L ′ ( G, χ ∆ ,
1) vanishes if and only if the divisor y ∆ ,r ( f ) vanishes.If G is defined over Q , we may consider the generalized regularized theta lift of f as inSection 6. It gives rise to a rational function Ψ ∆ ,r ( z, f ) on X ( N ) with divisor y ∆ ,r ( f ). (If G is not defined over Q , one also has to consider the Galois conjugates.) Such relationsamong Heegner divisors cannot be obtained as the Borcherds lift of a weakly holomorphicform. They are given by the generalized regularized theta lift of a harmonic weak Maassform.As an example, we consider the the relation for Heegner points of discriminant −
139 on X (37) found by Gross (see § N := 37, ∆ := − r := 3. In ournotation, L − , / Γ (37) can be represented by the quadratic forms Q = [37 , , , Q = [185 , , , Q = [185 , − , ,Q ′ = [ − , , − , Q ′ = [ − , , − , Q ′ = [ − , − , − . Denote the corresponding points on X (37) by α , α , α and α ′ , α ′ , α ′ . Hence we have Z , ( − · , · ) = α + α + α + α ′ + α ′ + α ′ ,Z − , ( − · , · ) = α + α + α − α ′ − α ′ − α ′ . Gross proved that the function r ( z ) = η ( z ) η (37 z ) − √− X (37) has the divisor ( α ) + ( α ) + ( α ) − ∞ ). This easily implies that r ′ ( z ), theimage of r ( z ) under complex conjugation, has the divisor ( α ′ ) + ( α ′ ) + ( α ′ ) − ∞ ).We show how the function r ( z ) can be obtained as a regularized theta lift. Let f ∈ H / ,ρ L be the unique harmonic weak Maass form whose Fourier expansion is of the form f = e ( − · τ ) e + e ( − · τ ) e − + O ( e − εv ) , v → ∞ . It is known that the dual space S / , ¯ ρ L is one-dimensional. Moreover, any element has theproperty that the coefficients with index · vanish (see [EZ], p.145). It follows from (2.10)that ξ / ( f ) = 0, and so f is weakly holomorphic. Its Borcherds lift is equal toΨ , ( z, f ) = r ( z ) · r ′ ( z ) · η (37 z ) η ( z ) . (8.13)On the other hand, we consider the unique harmonic weak Maass form f ∈ H / , ¯ ρ L whose Fourier expansion is of the form f = e ( − · τ ) e − e ( − · τ ) e − + O ( e − εv ) , v → ∞ . It is known that the dual space S / ,ρ L is two-dimensional. For a fixed λ in the positivedefinite one-dimensional sublattice K ⊂ L , the theta series g ( τ ) = X λ ∈ K ′ ( λ, λ ) · q Q ( λ ) e λ is a non-zero element. Under the Shimura correspondence it is mapped to the Eisensteinseries in M +2 (Γ (37)). Let g be a generator of the orthogonal complement of g in S / ,ρ L .Then g is a Hecke eigenform and we may normalize it such that it has rational coefficients.The Shimura lift of g is the newform G ∈ S +2 (Γ (37)), which corresponds to the conductor37 elliptic curve E : y = x + 10 x − x + 8 . Its L -function has an even functional equation, and it is known that L ( G , = 0. By theWaldspurger type formula for skew holomorphic Jacobi forms, see [Sk2], this implies thatthe coefficients of g with index · do not vanish. In view of (2.10), we find that ξ / ( f ) = c g + c g with non-zero constants c and c . So f is not weakly holomorphic. Nevertheless, we maylook at the twisted generalized Borcherds lift of f . We obtain thatΨ − , ( z, f ) = r ( z ) /r ′ ( z ) = η ( z ) − √− η (37 z ) η ( z ) − −√− η (37 z ) . (8.14)F. Str¨omberg computed a large number of coefficients of f numerically. The first fewcoefficients of the holomorphic part of f (indexed by the corresponding discriminants)are listed in Table 1. Details on the computations and some further results will be givenin [BrStr]. The rationality of the coefficients c + (139) and c + (823) corresponds to the EEGNER DIVISORS, L -FUNCTIONS AND HARMONIC WEAK MAASS FORMS 39 Table 1.
Coefficients of f ∆ c + ( − ∆) − − . . . . − − . . . . − − . . . . − − . . . . ... ... −
136 0 . . . . −
139 0 − − . . . . ... ... − − . . . . − − − − . . . . vanishing of the Heegner divisors Z − , ( − · , · ) and Z − , ( − · , · ) in the Jacobianof X (37).To obtain an element of H / , ¯ ρ L corresponding to g as in Lemma 7.3 we consider theunique harmonic weak Maass form f ∈ H / , ¯ ρ L whose Fourier expansion is of the form f = e ( − · τ ) e − e ( − · τ ) e − + O ( e − εv ) , v → ∞ . Arguing as above we see that ξ / ( f ) is a non-zero multiple of g . Table 2 includes some ofthe coefficients of f and the corresponding values of L ′ ( G , χ ∆ ,
1) as numerically computedby F. Str¨omberg. We have that L ′ ( G , χ − ,
1) = L ′ ( G , χ − ,
1) = 0 by the Gross-Zagierformula. We see that the values are in accordance with Theorem 7.6 and Theorem 7.8.
Table 2.
Coefficients of f ∆ c + ( − ∆) L ′ ( G , χ ∆ , − . . . . . . . . − . . . . . . . . − . . . . . . . . −
11 0 . . . . . . . . ... ... ... − − . . . . . . . . − − − − . . . . . . . . ... ... ... −
815 121 . . . . . . . . −
823 312 0 − − . . . . . . . . Observe that the numerics suggest that the holomorphic part of 3 f − f has integralcoefficients. This harmonic weak Maass form is mapped to a non-zero multiple of the thetafunction g under ξ / . So it should be viewed as an analogue of the function ˜ H in theexample of Section 8.2. References [BS]
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Fachbereich Mathematik, Technische Universit¨at Darmstadt, Schlossgartenstrasse 7,D–64289 Darmstadt, Germany
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