Harmonic Maass forms associated to real quadratic fields
aa r X i v : . [ m a t h . N T ] J a n HARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATICFIELDS
PIERRE CHAROLLOIS AND YINGKUN LI
Abstract.
In this paper, we explicitly construct harmonic Maass forms that map to theholomorphic weight one theta series associated by Hecke to odd ray class group charactersof real quadratic fields. From this construction, we give precise arithmetic informationcontained in the Fourier coefficients of the holomorphic part of the harmonic Maass form,establishing the main part of a conjecture of the second author.
Contents
1. Introduction. 2Acknowledgment 52. Theta Lift from O(1 ,
1) to SL . 52.1. Indefinite, anisotropic Z -lattice of rank 2. 52.2. Weil representation and automorphic forms. 62.3. Vector-valued theta functions. 72.4. Theta integral. 93. A Special Function. 103.1. Fourier transform and distribution. 113.2. A special function in L p ( R ). 133.3. Harmonic Maass form of weight . 154. Construction of ˜Θ( τ, L ). 194.1. Special case. 194.2. General case. 205. Deformation of Theta Integral. 235.1. Fourier expansion of I ′ ( τ, − L ). 245.2. Proof of Theorem 5.1. 266. Scalar-Valued Result and Numerical Examples. 276.1. Reducing the level. 27 Date : January 24, 2018.The first author is partially supported by the ANR grant ANR-12-BS01-0002.The second author is partially supported by the DFG grant BR-2163/4 and an NSF postdoctoralfellowship.
Introduction.
In number theory, modular forms of weight one play an important role because of theirspecial relationship to number fields. To each weight one eigenform f ∈ S ,χ (Γ ( N )),Deligne and Serre functorially attached a 2-dimensional, odd, irreducible representation ̺ f of Gal( Q / Q ) [11]. This representation gives rise to a finite Galois extension M/ Q and it isnatural to expect f to encode interesting arithmetic information about M . In fact, Stark’sconjecture [30] predicts that a certain subfield of M can be constructed from the specialvalues of the L -function attached to the modular form f .In [18], Hecke gave several systematic constructions of weight one modular forms. One ofthem attached weight one cusp forms, whose Galois representations have dihedral projectiveimages, to real quadratic fields. Specifically, let F ⊂ R be a real quadratic field withdiscriminant D > ε F >
1. Let m be an integral ideal in F withnorm M and ϕ an odd ray class group character with conductor m · ∞ . Then one canassociate an eigenform f ϕ ( τ ) = X n ≥ c ϕ ( n ) q n := X a ⊂O F ϕ ( a ) q Nm( a ) ∈ S ,χ (Γ ( DM )) , where τ is in the upper half complex plane H , q := e πiτ , and χ = χ D · ϕ | Q (see Section 6.2for details). The Galois representation associated to f ϕ by Deligne and Serre is the inductionof ϕ from Gal( Q /F ) to Gal( Q / Q ). Hecke’s construction was originally in terms of vector-valued modular forms and one of the first instances of producing holomorphic modular formsusing a theta lift for indefinite quadratic forms. This work had been vastly generalized byKudla and Millson [21, 22].It turns out the method of theta liftings has far reaching consequences in number theoryand arithmetic geometry. In [2], Borcherds constructed automorphic forms with singularitieson orthogonal Shimura varieties via a regularized theta lift. The singularities of the outputare controlled by the input, which is a holomorphic modular form with poles at the cusps.Then in [3] and [4], Bruinier replaced this input with certain non-holomorphic Poincar´e series,which are eigenfunctions of the hyperbolic Laplacian. He used the outputs to produce Chernclasses for the Heegner divisors. Motivated by these works, Bruinier and Funke introducedin [5] the notion of harmonic Maass forms generalizing classical modular forms. They areannihilated by the weight- k hyperbolic Laplacian(1.0.1) ∆ k := ξ − k ◦ ξ k , ξ k := 2 iv k ∂∂τ , τ = u + iv ∈ H ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 3 and have polar type singularities at the cusps (see Section 2.2). Using harmonic Maassforms as the input to Borcherds’ regularized theta lift, Bruinier and Funke constructed anadjoint of the Kudla-Millson theta lift for orthogonal groups of arbitrary signature [5]. Thistheta lift then produces automorphic Green’s function for special divisors on orthogonal typeShimura varieties, which enables one to calculate arithmetic intersection numbers and leadsto generalizations of the famous Gross-Zagier formula [7] and the recent proof of an averagedversion of Colmez’s conjecture [1].Besides as input to the theta lift, harmonic Maass forms also have interesting Fouriercoefficients. Because of the annihilation by ∆ k , a harmonic Maass form naturally has a holo-morphic part and a non-holomorphic part in its Fourier expansion. In his ground breakingthesis [32], Zwegers completed Ramanujan’s holomorphic mock theta functions and producedreal-analytic modular forms of weight . They turned out to be harmonic Maass forms thatmap to weight unary theta series under ξ / . In weight , there are many other importantworks that reveal the arithmetic nature of these Fourier coefficients (see e.g. [6], [13]).In the self-dual case of weight k = 1, Kudla, Rapoport and Yang [23] constructed “inco-herent Eisenstein series”, which turned out to be harmonic Maass forms that map under ξ to Eisenstein series associated to an imaginary quadratic field K . The Fourier coefficients ofthe holomorphic part are logarithms of integers, and can be interpreted as arithmetic degreesof special divisors on arithmetic curves. Later Duke and the second author studied in [14]harmonic Maass forms that map to weight one cusp forms associated to non-trivial classgroup characters of K . The Fourier coefficients were shown to be logarithms of algebraicnumbers in the Hilbert class field of K . In his thesis [16], Ehlen gave an arithmetic inter-pretation of the valuation of these algebraic numbers along the lines of [23]. In contrast tothe incoherent Eisenstein series in [23], the harmonic Maass forms in [14] and [16] were notconstructed explicitly. Also, numerical evidence suggests that given any weight one eigen-form f with associated Galois representation ̺ , there exists a harmonic Maass form ˜ f suchthat ξ ˜ f = f and the Fourier coefficients of its holomorphic part are Q -linear combinationsof logarithms of algebraic numbers in the number field cut out by ad ̺ (see [14], [24]).In this paper, we will explicitly construct a harmonic Maass form ˜ f ϕ that maps under ξ to Hecke’s weight one cusp form f ϕ , and study the arithmetic information contained in itsFourier coefficients. One special case of our main result (Theorems 5.1 and 6.5) is as follows. Theorem 1.1.
Let ϕ be an odd ray class group character of conductor m · ∞ with Nm( m ) = M and f ϕ the holomorphic weight one eigenform associated to ϕ as above. Suppose F hasclass number 1. Then there exists an integer κ m dividing M φ (2 M ) and a harmonicMaass form ˜ f ϕ ∈ H ,χ (Γ ( DM )) with holomorphic part ˜ f + ϕ ( τ ) = P n ≫−∞ c + ϕ ( n ) q n such that ξ ˜ f ϕ = f ϕ and (1.0.2) c + ϕ ( n ) − X ( λ ) ⊂O F , Nm(( λ ))= n (cid:0) ϕ − ( λ ) − ϕ − ( λ ′ ) (cid:1) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) ∈ κ m Z [ ϕ ] · log ε F , PIERRE CHAROLLOIS AND YINGKUN LI where φ ( N ) := [SL ( Z ) : Γ ( N )] and Z [ ϕ ] ⊂ C is the subring generated by the values of ϕ .Remark . The class number assumption is only to ease exposition. The more generalresult is in Theorem 6.5. Since Hecke’s construction is naturally stated in the setting ofvector-valued modular forms transforming with respect to the Weil representation, we willfirst obtain the main result in this setting (Theorem 5.1) and deduce Theorem 6.5 from it.
Remark . For each n ≥
1, we can define an analogue of c ϕ ( n ) by(1.0.3) c ϕ ( n ) := 12 X ( λ ) ⊂O F , Nm(( λ ))= n (cid:0) ϕ − ( λ ) − ϕ − ( λ ′ ) (cid:1) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) ∈ C /R ϕ with R ϕ := Z [ ϕ ] · log ε F and form the formal power series(1.0.4) f ϕ := X n ≥ c ϕ ( n ) q n ∈ C /R ϕ J q K . Then the second author conjectured in [24] that f ϕ can be lifted to the holomorphic partof a harmonic Maass form mapping to f ϕ under ξ . Now Theorem 1.1 implies a slightlyweaker result that κ m f ϕ can be lifted, i.e. κ m f ϕ agrees with κ m ˜ f + ϕ as formal power series in C /R ϕ J q K . In specific cases, it is possible to reduce the bound 48 M φ (2 M ) above with carefulanalysis. Note that if n = ℓ or ℓ with ℓ an inert prime in F/ Q , then Theorem 1.1 impliesthat κ m c + ϕ ( n ) ∈ R ϕ . Remark . In [10], Darmon, Lauder and Rotger studied a non-classical, overconvergentgeneralized eigenform associated to f ϕ . In their p -adic setting, the ℓ -th Fourier coefficient isthe p -adic logarithm of a Gross-Stark ℓ -unit in a class field of F when ℓ is an inert primein F/ Q . Otherwise, it is zero. In Section 6.2, we will rewrite c ϕ ( n ) to show that thearchimedean and non-archimedean settings are in some sense complementary to each other. Remark . In [24], it is shown that certain Z [ ϕ ]-linear combinations of the c + ϕ ( n )’s arethe values of Hilbert modular functions at big CM points. Thus, getting a handle on theindividual c + ϕ ( n ) allows us to give explicit factorization formula of the CM values in the spiritof Gross and Zagier [17]. Furthermore the CM values are defined over F . We hope to pursuethis line of investigation in the future.In order to best reflect the nature of the coefficients c + ϕ ( n ), we have stated Theorem 1.1 asan existence result. Its proof is through explicit construction and it is possible to write downa closed formula for c + ϕ ( n ) as a finite sum of Q -linear combination of logarithms of algebraicnumbers in F . We will do this numerically for η ( τ ) at the end.The construction starts by deforming Hecke’s theta integral with a spectral parameter s ∈ C (see (5.0.2)). The derivative of this deformed integral at s = 0 differs from thedesired harmonic Maass form by a non-modular contribution − ˜Θ ∗ ( τ, L ) from the boundary.We then construct a modular object ˜Θ( τ, L ) that differs from this boundary contribution ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 5 by a translation-invariant holomorphic function ˜Θ + ( τ, L ), whose Fourier coefficients are in κ m Z [ ϕ ] · log ε F . Their difference is then the desired harmonic Maass form.The outline of the paper is as follows. In Section 2, we will recall some basic informationabout vector-valued automorphic forms and theta functions following [21] and [2]. In Sec-tion 3, we will define a continuous function g τ ∈ L ( R ) ∩ L ∞ ( R ) with nice properties (seeProposition 3.3) and use it in Section 4 to construct ˜Θ( τ, L ), which offsets the effect of theboundary contribution to the deformed theta integral. Finally in the last two sections, wewill state and prove the vector-valued and scalar-valued versions of our main result, beforegiving some numerical examples at the end. Acknowledgment
The authors thank Jan Bruinier for many illuminating conversations, and the anonymousreferee for detailed comments and several helpful suggestions, which greatly improved theexposition of the paper. 2.
Theta Lift from
O(1 , to SL . In this section, we will follow Kudla [21] to recall the work of Hecke [18] in terms of atheta lift from O(1 ,
1) to SL .2.1. Indefinite, anisotropic Z -lattice of rank 2. An even, integral lattice is a Z -module L equipped with a quadratic form Q : L → Z . It is anisotropic if L does not contain annonzero isotropic vector, i.e. λ ∈ L \{ } with Q ( λ ) = 0. If such lattice is indefinite and hasrank 2, then it can be described by L = Z and a quadratic form Q ( m, n ) = Am + Bmn + Cn with A, B, C ∈ Z such that the discriminant D := B − AC > D is fundamental, then A Z + B + √ D Z is an integral ideal in the real quadraticfield Q ( √ D ) with norm | A | 6 = 0 andNm Am + B + √ D n ! = A · Q ( m, n ) . In fact, there is a correspondence between indefinite binary quadratic forms and ideals inreal quadratic fields (see [8]). We will use the latter language throughout this work.Let Q be an algebraic closure of Q and fix an embedding Q ֒ → C throughout. For adiscriminant D ≥ F = Q ( √ D ) be the correspondingreal quadratic field with ring of integers O F . The Z -lattice O D := Z + Z D + √ D is a subringof O F . Its dual with respect to the norm form Nm as the quadratic form is d − D O F , where d D := √ D O D . If D is fundamental, then O D = O F and d D = d F is the different. The groupof units O × F is generated by − ε F > PIERRE CHAROLLOIS AND YINGKUN LI
For an integral ideal a ⊂ O D with A := [ O D : a ] and a positive integer M ∈ N , considerthe Z -lattice(2.1.1) ( L a ,M , Q a ,M ) := (cid:18) M a , Nm F/ Q AM (cid:19) . It is anisotropic and has rank 2. The induced bilinear form is given by B a ,M ( λ, µ ) := Tr F/ Q ( λ · µ ′ ) AM for all λ, µ ∈ L a ,M with ′ the non-trivial automorphism in Gal( F/ Q ). The dual lattice L ∗ a ,M is given by ad − D and the finite quadratic modular L ∗ a ,M /L a ,M is isomorphic to O D /M d D .Finally, notice that − L a ,M is isometric to L ad D ,M via λ λ √ D .2.2. Weil representation and automorphic forms.
Let (
L, Q ) be an indefinite, even,integral lattice of rank 2 with bilinear form ( , ) : L × L → Z . As usual, let L ∗ be the duallattice of L and { e h : h ∈ L ∗ /L } denote the canonical basis of the vector space C [ L ∗ /L ]and e ( a ) := e πia for any a ∈ C . Then Γ := SL ( Z ) acts on C [ L ∗ /L ] through the Weilrepresentation ρ L as (see e.g. [2, § ρ L ( T )( e h ) = e ( Q ( h )) e h , ρ L ( S )( e h ) = 1 p | L ∗ /L | X δ ∈ L ∗ /L e ( − ( δ, h )) e δ , where T = (cid:18) (cid:19) , S = (cid:18) −
11 0 (cid:19) . Note that ρ − L = c ◦ ρ L ◦ c on C [ L ∗ /L ], where c : C → C denotes complex conjugation. If P ⊂ L is a finite index sublattice, then L ∗ ⊂ P ∗ and denote s : L ∗ /P → L ∗ /L the natural surjection. There is a linear map ψ : C [ P ∗ /P ] → C [ L ∗ /L ] defined by(2.2.2) ψ ( e h ) := ( e s ( h ) , h ∈ L ∗ /P , otherwise.It is straightforward to check that ψ is in fact Γ-linear (with respect to ρ P and ρ L ).Let d L be the level of L and Γ( d L ) ⊂ Γ the principal congruence subgroup of level d L .Then ρ L is trivial on Γ( d L ) and can be viewed as a representation of SL ( Z /d L Z ) (see e.g.Proposition 4.5 in [28]). Let ζ d L be a primitive d th L root of unity. For a ∈ ( Z /d L Z ) × ,let σ a ∈ Gal( Q ( ζ d L ) / Q ) be the element that sends ζ d L to ζ ad L . Then σ a acts naturally on W L := Q ( ζ d L )[ L ∗ /L ]. Let ς a ∈ GL( W L ) be the left action given by(2.2.3) ς a · w := σ − a ( w ) , w ∈ W L . In [27], McGraw extended ρ L to a unitary representation of GL ( Z /d L Z ) on W L , where theaction of(2.2.4) J a := ( a ) ∈ GL ( Z /d L Z )is ς a . This is the main ingredient used to prove the rationality of basis of vector-valuedmodular forms. The result we need can be stated as follows. ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 7
Proposition 2.1. [27, Theorem 4.3] The map that sends γ to ρ L ( γ ) when γ ∈ SL ( Z /d L Z ) and J a to ς a is a unitary representation of GL ( Z /d L Z ) on W L . In other words, if weview ρ L ( γ ) ∈ GL | L ∗ /L | ( Q ( ζ d L )) with respect to the standard basis of W L , then σ a ( ρ L ( γ )) = ρ L ( J − a γJ a ) . Now, we will quickly recall some facts about automorphic forms. Let k ∈ Z be an integer, V an n -dimensional C -vector space, Γ ′ ⊂ Γ a finite index subgroup and ρ : Γ ′ → GL( V )a representation. Then a real-analytic function f = ( f j ) ≤ j ≤ n : H → V is a vector-valuedautomorphic form on Γ ′ with weight k and representation ρ if it satisfies(2.2.5) ( f | k,ρ γ )( τ ) := ρ ( γ ) − · (cid:18) ( cτ + d ) − k f j (cid:18) aτ + bcτ + d (cid:19)(cid:19) ≤ j ≤ n = f ( τ )for all γ = ( a bc d ) ∈ Γ ′ and τ ∈ H . If ρ is trivial, then we may omit it in the slash operator.We denote the space of such functions by A k,ρ (Γ ′ ). The subspace of A k,ρ (Γ ′ ) consisting offunctions holomorphic on H is denoted by M ! k,ρ (Γ ′ ), which is usually called the space ofweakly holomorphic modular forms. Let M k,ρ (Γ ′ ) and S k,ρ (Γ ′ ) be the usual space of modularforms and cusp forms respectively. A function f ∈ A k,ρ (Γ ′ ) is called a harmonic weak Maassform if ∆ k f = 0 and f has at most linear exponential growths at all the cusps. It satisfies ξ k f ∈ M !2 − k,ρ (Γ ′ ) by the definition of ∆ k in (1.0.1). Furthermore, if ξ k f vanishes at allthe cusps, then we call f a harmonic Maass form. We use H k,ρ (Γ ′ ) to denote the space ofharmonic Maass forms on Γ ′ of weight k and representation ρ .Since f ∈ H k,ρ (Γ ′ ) satisfies ∆ k f = 0, it can be written canonically as the difference of aholomorphic part f + and non-holomorphic part f ∗ . Let B be a basis of V . Then f + and f ∗ have the following Fourier expansions. f + ( τ ) = X e ∈ B X n ∈ Q n ≫−∞ c + ( n, e ) q n e , f ∗ ( τ ) = (4 π ) k − X e ∈ B X n ∈ Q n> c ( n, e )Γ(1 − k, πnv ) q − n e . It is readily checked that ξ k f ( τ ) = ξ k ( − f ∗ ( τ )) = P e ∈ B (cid:18)P n ∈ Q n> n − k c ( n, e ) q n (cid:19) e ∈ S − k,ρ (Γ ′ ).2.3. Vector-valued theta functions.
Let ( V R , Q ) denote the quadratic space of signature(1 ,
1) with V R = R and for X = ( x , x ) , Y = ( y , y ) ∈ V R (2.3.1) Q ( X ) := x x , B ( X, Y ) := x y + x y . Note that ( V R , Q ) ∼ = ( V R , − Q ) via the map(2.3.2) ι (( x , x )) := ( x , − x ) . PIERRE CHAROLLOIS AND YINGKUN LI
The symmetric domain attached to V R is the hyperbola D := { Z − ⊂ V R | ( Z − , Z − ) = − } ,which is parametrized by R × viaΦ : R × → D t Z − t := (cid:16) t √ , − t − √ (cid:17) . We denote by D + the connected component of D , which is parametrized by R × + under Φ.Define(2.3.3) Z + t := 1 √ (cid:0) t, t − (cid:1) ∈ ( Z − t ) ⊥ . Then d Φ (cid:0) t ddt (cid:1) = Z + t ∈ R and { Z + t , Z − t } is an orthogonal basis of V R with Q ( Z + t ) = − Q ( Z − t ) = . We can write X = X + t − X − t with(2.3.4) X αt := B ( X, Z αt ) Z αt = αx t − + x t √ Z αt for any X = ( x , x ) ∈ V R and α ∈ { + , −} . In this basis, the quadratic space V R becomes( R , , Q ) with R , = R and(2.3.5) Q (( x, y )) := x − y . Let F denote the Fourier transform on R , with respect to Q , i.e. F ( φ )( x, y ) := Z ∞−∞ Z ∞−∞ φ ( w, r ) e ( xw − yr ) dwdr for any Schwartz function φ on R , .An important ingredient in forming the theta kernel is the archimedean part of theSchwartz function. In the setting of Hecke, this is given by(2.3.6) φ τ ( x, y ) := √ v · x · e (cid:18) x τ − y τ (cid:19) . As a function on the quadratic space ( R , , Q ), φ τ satisfies(2.3.7) φ τ +1 ( W ) = e ( Q ( W )) φ τ ( W ) , φ − /τ ( W ) = − τ F ( φ τ )( W ) , W ∈ R , . Now for any even, integral lattice L ⊂ V R , the vector-valued theta function Θ( τ, L ; t ) := P h ∈ L ∗ /L Θ h ( τ, L ; t ) e h with(2.3.8) Θ h ( τ, L ; t ) := X X ∈ L + h φ τ ( B ( X, Z + t ) , B ( X, Z − t )) ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 9 transforms on SL ( Z ) with weight 1 and representation ρ L in the variable τ by Theorem 4.1in [2]. Similarly, the image of − L under the involution ι : − V R → V R is an even, integrallattice in ( V R , Q ), and we define(2.3.9) Θ( τ, − L ; t ) := Θ( τ, ι ( − L ); t ) . Suppose L is the image of the embedding( L a ,M , Q a ,M ) ֒ → ( V R , Q ) λ λ := 1 √ AM ( λ, λ ′ ) . (2.3.10)for some D, a , M . Then ( L, Q ) ∼ = ( L a ,M , Q a ,M ) and Θ h ( τ, L ; t ) becomes(2.3.11) Θ h ( τ, L ; t ) = √ v √ AM X λ ∈ ad − D λ − h ∈ M a (cid:0) λ ′ t + λt − (cid:1) e (cid:18) ( λt − + λ ′ t ) τ − ( λt − − λ ′ t ) τ AM (cid:19) . Similarly, the expression above becomes Θ h ( τ, − L ; t ) after changing λ ′ to − λ ′ . If P ⊂ L isa sublattice of finite index and ψ : C [ P ∗ /P ] → C [ L ∗ /L ] the Γ-linear map defined in (2.2.2),then it is straightforward to check that(2.3.12) ψ (Θ( τ, P ; t )) = Θ( τ, L ; t )for all τ ∈ H and t ∈ R × + .2.4. Theta integral.
As in the end of last subsection, suppose (
L, Q ) = ( L a ,M , Q a ,M ). Thediscriminant kernel Γ L is the subgroup of SO + ( L ) ∼ = Z consisting of those units which arecongruent to 1 modulo M √ D . Notice that ε ∈ Γ L if and only if ε ′ ∈ Γ L . Also, Nm( ε ) = 1for all ε ∈ Γ L . Let Γ ′ L ⊂ Γ L be subgroup of totally positive element and ε L > ϑ ( τ, L ) := Z ε L Θ( τ, L ; t ) dtt = Z log ε L Θ( τ, L ; e ν ) dν. Since B ( λε, Z ± t ) = B ( λ, Z ± ε ′ t ) for any totally positive ε ∈ O × F , we can unfold the integralto obtain Z ε L Θ h ( τ, L ; t ) dtt = X λ ∈ Γ ′ L \ L + h, λ =0 Z ∞ φ τ ( B ( λ, Z + t ) , B ( λ, Z − t )) dtt = √ v √ AM X λ ∈ Γ ′ L \ L + h, λ =0 e (cid:18) λλ ′ uAM (cid:19) Z ∞−∞ (cid:0) λ ′ e ν + λe − ν (cid:1) e (cid:18) ( λe − ν ) + ( λ ′ e ν ) AM iv (cid:19) dν.
Using the identity e − πy = 2 √ y R ∞−∞ e ν e − πy ( e ν + e − ν ) dν , we can evaluate Z ∞−∞ λ ′ e ν e (cid:18) ( λe − ν ) + ( λ ′ e ν ) AM iv (cid:19) dν = 12 r AMv sgn( λ ′ ) e (cid:18) | λλ ′ | ivAM (cid:19) , λ = 0 . (2.4.2)If λ ∈ L + h has negative norm, then the integral will vanish. Thus(2.4.3) ϑ ( τ, ± L ) = X h ∈ L ∗ /L e h X λ ∈ Γ ′ L \ L + h, ± Q ( λ ) > sgn( λ ) e ( | Q ( λ ) | τ )is in S ,ρ ± L (SL ( Z )). Hecke noticed that if there exists ε < L , then ϑ ( τ, L ) vanishesidentically [18, Satz 1]. Thus, we can suppose Γ ′ L = Γ L .Another way to show that ϑ ( τ, ± L ) is holomorphic without explicitly computing the inte-gral in equation (2.4.1) is to apply the lowering operator ξ to Θ( τ, ± L ; t ) dtt and show thatit is an exact form on R × . Then its integral over Γ L \ R × + would vanish since this locallysymmetry domain has no boundary. Let d t := t ∂∂t be the invariant vector field on R × + , i.e.an element in the Lie algebra of O( V R ). We have the following proposition, which is a veryspecial case of a result by Kudla and Millson (see section 8 of [22]). Proposition 2.2.
For all τ ∈ H and t ∈ R × , we have (2.4.4) ξ L τ Θ( τ, ± L ; t ) = − d t Θ( τ, ∓ L ; t ) . Proof.
We will show the equality with L on the left hand side and − L on the right handside. The other combination of signs can be proved similarly. Straightforward calculationsshow that2 ξ φ τ ( x, y ) = xy (1 − πvy ) φ τ ( y, x ) , x ∂φ τ ∂x = (1 + 2 πix τ ) φ τ , ∂φ τ ∂y = − πiyτ φ τ , d t Z ± t = Z ∓ t . This implies that for all X ∈ V R ,(2.4.5) 2 ξ φ τ ( B ( X, Z + t ) , B ( X, Z − t )) = d t φ τ ( B ( X, Z − t ) , B ( X, Z + t )) . Since φ τ ( B ( ι ( X ) , Z + t ) , B ( ι ( X ) , Z − t )) = − φ τ ( B ( X, Z − t ) , B ( X, Z + t )), equation (2.4.4) followsimmediately from the definition of Θ( τ, ± L ; t ) in equations (2.3.8), (2.3.9) and (2.4.5). (cid:3) Remark . The calculations above can be made to work when L is isotropic (see [21]). Inthat case, the resulting modular form is an Eisenstein series of weight one.3. A Special Function.
In this section, we will introduce a continuous function g τ ∈ L p ( R ) for 1 ≤ p ≤ ∞ and useit later as a replacement for the usual Schwartz function to form a real-analytic theta series.The inspiration of this function comes from Zwegers’ work on the Appell-Lerch sum [32].After adding a non-holomorphic function, the Appell-Lerch sum becomes a modular object (areal-analytic Jacobi form). Jacobi theta functions are prototypical examples of such modular ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 11 objects and are constructed by averaging a Schwartz function over a lattice. In view of this,it is very natural to expect that the completed Appell-Lerch sum can be decomposed intothe average of a special function over a lattice. This line of thought eventually broughtus to the function g τ , which satisfies properties similar to the function W z in Proposition1 of Section 2.3 of [19] (there z is in the upper half plane, as our τ here). Using W z ,Hirzebruch and Zagier constructed a non-holomorphic modular form of weight 2, which isessentially the product of the weight 3 / / g τ willbe used to produce a harmonic Maass form of weight 1 /
2. In Section 4, we will use it toconstruct a preimage ˜Θ( τ, L ) of Θ( τ, L ; 1) under ξ , which turns out to be a sum of productsof weight harmonic Maass forms and weight theta functions (see (4.1.1)).3.1. Fourier transform and distribution.
For 1 ≤ p ≤ ∞ , let L p ( R ) be the space ofbounded functions on R with respect to the L p -norm, and S ( R ) ⊂ L p ( R ) the subspace ofSchwartz functions. Denote the Fourier transform of g ∈ L ( R ) by(3.1.1) F ( g )( x ) := Z R g ( w ) e ( wx ) dw, with inverse F − : S ( R ) → S ( R ) given by F − ( ϕ )( x ) := F ( ϕ )( − x ) for ϕ ∈ S ( R ). Inaddition, we also have the following standard linear operators ∂, µ : S ( R ) → S ( R )(3.1.2) ∂ ( ϕ )( x ) := dϕ ( x ) dx , µ ( ϕ )( x ) := 2 πixϕ ( x ) , which commutes with F as follows(3.1.3) F ◦ µ = ∂ ◦ F , F ◦ ∂ = − µ ◦ F . The Fourier transform of the Gaussian e − απx is e − πx /α / √ α for any α ∈ C with Re( α ) > τ ∈ H , we define another linear operator E τ : S ( R ) → S ( R )(3.1.4) E τ ( ϕ )( x ) := e (cid:18) x τ (cid:19) Z x e (cid:18) − w τ (cid:19) ϕ ( w ) dw. It is a bijection with the inverse given by(3.1.5) E − τ ( ϕ )( x ) := e (cid:18) x τ (cid:19) ddx (cid:18) e (cid:18) − x τ (cid:19) ϕ ( x ) (cid:19) = ( − τ · µ + ∂ )( ϕ )( x ) . It turns out that the conjugate of E − /τ by F equals to τ E τ . Lemma 3.1.
For any τ ∈ H and ϕ ∈ S ( R ) , we have (3.1.6) F ◦ E − /τ ◦ F − ◦ E − τ ( ϕ ) = τ · ϕ. Proof.
By definition and basic properties of Fourier transform in Eq. (3.1.3), we have F − ◦ E − τ = F − ◦ ( − τ µ + ∂ ) = ( τ ∂ + µ ) ◦ F − = τ E − − /τ ◦ F − . from which the claim follows immediately. (cid:3) The space of continuous functionals on S ( R ) is the space of tempered distributions anddenoted by S ′ ( R ). For a distribution T ∈ S ′ ( R ), we can define its derivative ∂T ∈ S ′ ( R )and Fourier transform F ( T ) ∈ S ′ ( R ) to be the distributions satisfying ∂T ( ϕ ) := T ( − ∂ϕ ) , F ( T )( ϕ ) := T ( F ( ϕ ))for all ϕ ∈ S ( R ). We can embed L ∞ ( R ) into S ′ ( R ) via the map I : L ∞ ( R ) → S ′ ( R ) g
7→ I ( g ) : ϕ Z R ϕ ( x ) g ( x ) dx, (3.1.7)whose kernel consists of functions that are zero almost everywhere. We can expand theinput of I to include any measurable function g on R such that the integral R R ϕ ( x ) g ( x ) dx converges absolutely for any ϕ ∈ S ( R ). It is easy to check that I ( ∂g ) = ∂ I ( g ) if g isdifferentiable almost everywhere on R with jump singularities, and F ( I ( g )) = I ( F ( g )) if g ∈ L ( R ) ∩ L ∞ ( R ).Let δ be the Dirac delta distribution, which is characterized by the property that(3.1.8) δ ( ϕ ) = ϕ (0)for all ϕ ∈ S ( R ). Equivalently, we have δ = ∂ I ( H ), where H ( x ) := sgn( x )+12 ∈ L ∞ ( R ) is theHeaviside step function. One frequently used distribution is the Dirac comb distribution(3.1.9) C ( ϕ ) := X n ∈ Z ϕ ( n ) , which satisfies F ( C ) = C by the Poisson summation formula. For our purpose, the interestingdistribution is in fact the shifted version of C defined by(3.1.10) C + ( ϕ ) := X m ∈ Z + 12 ϕ ( m ) e (cid:16) m (cid:17) . It is easy to check that F ( C + ) = i · C + .For any τ ∈ H , we can define the following even function in L ∞ ( R )(3.1.11) ω τ ( x ) := e (cid:18) − x τ (cid:19) X m ≥| x | , m ∈ Z + 12 e (cid:18) m τ + m (cid:19) , ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 13 which is holomorphic as a function in τ ∈ H and satisfies the following key property(3.1.12) Z R ϕ ( x ) e (cid:18) − x τ (cid:19) ddx (cid:18) e (cid:18) x τ (cid:19) ω τ ( x ) (cid:19) dx = C + ( ϕ )for any ϕ ∈ S ( R ). Simple manipulations give us the following lemma. Lemma 3.2.
For any τ ∈ H and ϕ ∈ S ( R ) , we have (3.1.13) I ( ω τ )( ϕ ) = −C + ( E τ ( ϕ )) , F ( I ( ω τ )) = ( − iτ ) − I ( ω − /τ ) . Proof.
By definition, we can rewrite I ( ω τ )( ϕ ) = Z R (cid:18) ∂ x Z x e (cid:18) − w τ (cid:19) ϕ ( w ) dw (cid:19) e (cid:18) x τ (cid:19) ω τ ( x ) dx = lim x →∞ E τ ( ϕ ) ω τ | x − x − Z R E τ ( ϕ )( x ) e (cid:18) − x τ (cid:19) ddx (cid:18) e (cid:18) x τ (cid:19) ω τ ( x ) (cid:19) dx = −C + ( E τ ( ϕ )) , where the last step follows from Eq. (3.1.12). The second claim in (3.1.13) is then a conse-quence of the formal calculations F ( I ( ω τ ))( ϕ ) = I ( ω τ )( F ( ϕ )) = −C + ( E τ ◦ F ( ϕ )) = −C + ( F ◦ E − /τ ( ϕ )) /τ = − ( F ( C + ))( E − /τ ( ϕ )) /τ = − ( − iτ ) − C + ( E − /τ ( ϕ )) = ( − iτ ) − I ( ω − /τ )( ϕ )using Lemma 3.1. (cid:3) A special function in L p ( R ) . Before constructing g τ , we first recall the complemen-tary error function(3.2.1) erfc( x ) := 2 √ π Z ∞ x e − t dt = 1 − erf( x ) , which decays square exponentially, and a special modular form(3.2.2) η ( τ ) := − i · X m ∈ Z + 12 m · e (cid:18) m τ + m (cid:19) = q / Y n ≥ (1 − q n ) . It satisfies the transformation properties(3.2.3) η ( τ + 1) = e (cid:18) (cid:19) η ( τ ) , ( − iτ ) − / η ( − /τ ) = η ( τ )and is a modular form of weight 3 / Now for τ = u + iv ∈ H , we define the following functions g + τ ( x ) := e (cid:18) − x τ (cid:19) sgn( x ) i · η ( τ ) X m> | x | , m ∈ Z + 12 ( m − | x | ) e (cid:18) m τ + m (cid:19) ,g ∗ τ ( x ) := e (cid:18) − x τ (cid:19) sgn( x )2 erfc( √ πv | x | ) ,g τ ( x ) := g + τ ( x ) − g ∗ τ ( x ) . (3.2.4)The function g ∗ τ decays as a Schwartz function, but is discontinuous at 0. On the other hand, g + τ has the same type of discontinuity at 0, but is only piecewise differentiable and does notdecay quickly enough. Under the operator τ µ + ∂ , the function g + τ is closely related to ω τ .By themselves alone, these two functions would behave strangely under Fourier transform.Fortunately, their difference g τ enjoys extremely nice properties. Proposition 3.3.
The function g τ is in L p ( R ) for ≤ p ≤ ∞ . It is odd, continuous on R and differentiable on R \ ( Z + ) . Furthermore for all x ∈ R , it satisfies(1) ξ g τ ( x ) = √ v x e (cid:16) x τ (cid:17) ,(2) g τ +1 ( x ) = e (cid:16) − x (cid:17) g τ ( x ) ,(3) F ( g − /τ )( x ) = − i √− iτ g τ ( x ) .Proof. From the definition, it is easy to see that g τ ∈ L ∞ ( R ) is odd and continuous on R .Also, the functions 2 e ( x τ ) g ∗ τ ( x ) − sgn( x ) = − erf( √ πvx ) and2 e (cid:18) x τ (cid:19) g + τ ( x ) − sgn( x ) = 2 ixη ( τ ) e (cid:18) x τ (cid:19) ω τ ( x ) + 2 i sgn( x ) η ( τ ) X
0, where(3.2.5) m w := ⌈| w | + ⌉ − ∈ Z + is the least half integer greater than or equal to | w | . By writing t := m w − | w | with t ∈ [0 , Z ∞ ( m w − w ) e − α ( m w − w ) dw < X m ∈ Z + 12 ,m> Z te − αt (2 m − t ) dt < ζ (2) e α α . ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 15
Therefore, g τ ∈ L ( R ) ∩ L ∞ ( R ) ⊂ L p ( R ) for all 1 ≤ p ≤ ∞ and F ( g τ ) ∈ L ∞ ( R ) ∩ C ( R ).For the last three claims, (1) and (2) are consequences of straightforward calculations fromthe definition. The last claim takes a little more calculations to see and we use the notationsin section 3.1. Since g τ is differentiable on R \ ( Z + ), we have(3.2.6) e (cid:18) − x τ (cid:19) ddx (cid:18) e (cid:18) x τ (cid:19) g τ ( x ) (cid:19) = − ω τ ( x ) iη ( τ ) + √ v e (cid:18) − x τ (cid:19) =: γ τ ( x ) ∈ L ∞ ( R )for x ∈ R \ ( Z + ). By the same argument in the proof of Lemma 3.2, we know that(3.2.7) I ( g τ )( ϕ ) = −I ( γ τ )( E τ ( ϕ )) . Now, Lemma 3.2 and (3.2.3) implies(3.2.8) F ( I ( γ τ )) = √− iτ I ( γ − /τ ) . Proceeding as in the formal argument in the proof of Lemma 3.2 gives us I ( F ( g τ ))( ϕ ) = −I ( γ τ )( E τ ◦ F ( ϕ )) = −I ( γ τ )( F ◦ E − /τ ( ϕ )) /τ = −F ( I ( γ τ ))( E − /τ ( ϕ )) /τ = − i ( − iτ ) − / I ( γ − /τ )( E − /τ ( ϕ )) = − i ( − iτ ) − / I ( g − /τ )( ϕ )for any ϕ ∈ S ( R ). This implies that F ( g τ ) + i ( − iτ ) − / g − /τ ∈ L ∞ ( R ) vanishes almosteverywhere. Since this difference is also continuous, it must be identically zero. Changing τ to − /τ then proves the last claim. (cid:3) Harmonic Maass form of weight . In this section, we will use the function g τ from the previous section to construct some vector-valued harmonic Maass forms of weight . The basic idea is to treat g τ as a Schwartz function and average it over the elements in(a translate of) a lattice. To make this idea work, we need the following results. Lemma 3.4. If x ∈ b Z for some b ∈ N , then b · e (cid:16) x τ (cid:17) g + τ ( x ) ∈ Z J q K with q = e ( τ ) and g + τ ( x ) is holomorphic in the interior of the upper half plane with ord ∞ ( g + τ ( x )) ≥ | x | b + b − .Proof. The first claim is clear from the expression of g + τ ( x ) in (3.2.4). For the second claim,notice that the inequality m > | x | implies m ≥ | x | + b when m ∈ Z + and x ∈ b Z with2 | b . Thus, m − x ≥ | x | +1 /bb and we obtain the bound on ord ∞ ( g + τ ( x )). The − comesfrom the η ( τ ) in the denominator, and is responsible for a possible pole at i ∞ . (cid:3) Proposition 3.5.
For any fixed N ∈ N and τ ∈ H , the infinite sum (3.3.1) G τ ( x ) := X n ∈ Z g τ ( N n + x ) converges absolutely for all x ∈ R and uniformly for x in compact subsets of R \ ( Z + ) . For x ∈ R , it satisfies (3.3.2) lim x → x τ − / G − /τ ( x ) = e (1 / N X n ∈ Z g τ (cid:16) nN (cid:17) e (cid:16) nx N (cid:17) . For x ∈ Z + , the function G τ ( x ) satisfies (3.3.3) lim x → x G τ ( x ) = G τ ( x ) + e ( x / πN η ( τ ) . Remark . If N ∈ R \ N , then the argument of the proof will not be sufficient to show theabsolute convergence of the sum defining G τ . Proof.
By definition, g τ = g + τ − g ∗ τ and g ∗ τ decays like a Schwartz function. For x ∈ R , recall m x ∈ Z + as defined in (3.2.5) and denote f τ ( x ) := sgn( x ) iη ( τ ) ( m x − | x | ) e (cid:18) m x − x τ + m x (cid:19) . Since P n ∈ Z ( g τ − f τ )( N n + x ) converges absolutely and uniformly for x ∈ R , it suffices toanalyze the convergence of P n ∈ Z f τ ( N n + x ). When x ∈ Z + , f τ ( N n + x ) = 0 for all n ∈ Z .Otherwise, the minimum distance between x and Z + is ǫ >
0, and | f τ ( N n + x ) | ≪ τ,x e − ǫnv .Therefore, the sum defining G τ ( x ) converges absolutely and uniformly for x in a compactsubset of R \ ( Z + ). Note that this argument does not work for N ∈ R \ N .Applying the Poisson summation formula and Prop. 3.3 now gives us G τ ( x ) = iN √− iτ X n ∈ Z g − /τ (cid:16) nN (cid:17) e (cid:16) nxN (cid:17) for x ∈ R \ ( Z + ). The sum on the right hand side converges absolutely and uniformly for x ∈ R and defines a continuous function in x . Therefore the equation still holds when takingthe limit x → x for x ∈ Z + . Changing τ to − /τ then gives us (3.3.2).To understand the removable discontinuity of G τ , notice that both sides of (3.3.3) areperiodic with period N ∈ N . So we can suppose that x ∈ ( Z + ) ∩ [ − N,
0) and obtainlim x → x +0 G τ ( x ) = G τ ( x ) + lim ǫ → + X n ≤ f τ ( N n + x + ǫ ) = G τ ( x ) + e ( x / iη ( τ ) lim ǫ → + ǫ − e ( ǫN τ )= G τ ( x ) + e ( x / πN η ( τ ) . The limit from the left can be calculated analogously. (cid:3)
ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 17
Now, we are ready to construct some harmonic Maass forms of weight . For any N ∈ R such that N ∈ N , the lattice(3.3.4) P N := N Z is even integral (with dual lattice P ∗ N = N Z ) with respect to the quadratic form Q ( x ) := x .We denote by ρ N the Weil representation associated to ( P N , Q ). The functions θ N ( τ ) := X h ∈ P ∗ N /P N θ N,h ( τ ) e h , θ N,h ( τ ) := X r ∈ P N + h e (cid:18) r τ (cid:19) , θ N ( τ ) := X h ∈ P ∗ N /P N θ N,h ( τ ) e h , θ N,h ( τ ) := X r ∈ P N + h r e (cid:18) r τ (cid:19) (3.3.5)are vector-valued modular forms in M / ,ρ N and S / ,ρ N respectively.Suppose from now on that N ∈ N . Then P ∗ N ⊂ Q contains Z + , which is translationinvariant under P N . Therefore, it makes sense to consider the cosets ( Z + ) /P N . We define(3.3.6) ˜ θ N,h ( τ ) := (P r ∈ P N + h g τ ( r ) , if h ∈ ( P ∗ N \ ( Z + )) /P N , e ( h/ E ( τ )12 Niη ( τ ) + P r ∈ P N + h g τ ( r ) , if h ∈ ( Z + ) /P N . Here, E ( τ ) := 1 − P n ≥ σ ( n ) q n is the holomorphic Eisenstein series of weight 2 thatsatisfies the cocycle relation(3.3.7) E ( τ ) − τ − E ( − /τ ) = 6 iπτ . Since g τ is odd, we have ˜ θ N, − h ( τ ) = − ˜ θ N,h ( τ ). The main result of this section is as follows. Theorem 3.7.
Suppose that N ∈ N . Then the function ˜ θ N := P h ∈ P ∗ N /P N ˜ θ N,h e h satisfies ξ / ˜ θ N = √ θ N and is a harmonic Maass form in H / ,ρ N . Furthermore, the holomorphicpart of ˜ θ N has rational Fourier coefficient with denominator bounded by N .Proof. Since the sum defining ˜ θ N converges absolutely by Prop. 3.5, we can apply ξ / to ˜ θ N termwisely and conclude that ξ / ˜ θ N = √ θ N from Prop. 3.3. The bound on the denominatorthen follows from Lemma 3.4, which also implies that ˜ θ N is regular on H and has at mostlinear exponential growth near the cusps. To show that ˜ θ N ∈ H / ,ρ N , we just need to check(3.3.8) ˜ θ N,h ( τ + 1) = e ( − h /
2) ˜ θ N,h ( τ ) , τ − / ˜ θ N,h ( − /τ ) = e (1 / N X t ∈ P ∗ N /P N e ( ht ) ˜ θ N,t ( τ )for all h ∈ P ∗ N /P N . Since g τ +1 ( r ) = e ( − r / g τ ( r ) by Prop. 3.3 and e ( − h /
2) = e ( − /
8) = η ( τ ) /η ( τ + 1) if h ∈ Z + , the first equation holds. For the second equation, we start with identity (3.3.2). Suppose h ∈ ( P ∗ N \ ( Z + )) /N Z .Then substituting x = h into identity (3.3.2) gives us τ − / ˜ θ N,h ( − /τ ) = e (1 / N X t ∈ P ∗ N /P N e ( ht ) ˜ θ N,t ( τ ) − E ( τ )12 N iη ( τ ) X t ∈ ( Z + 12 ) /P N e (( h + ) t ) . By choosing t = n + with 0 ≤ n ≤ N − Z + ) /P N ,we see that the second sum above vanishes and the equation above becomes the second claimin (3.3.8). Suppose h ∈ ( Z + ) /N Z and we also denote h to be the representative in [0 , N ).Then (3.3.2) implies that(3.3.9) τ − / lim x → h G − /τ ( x ) = e (1 / N X t ∈ P ∗ N /P N e ( ht ) ˜ θ N,t ( τ ) − e ( h/ E ( τ )12 η ( τ ) . Using (3.3.3), we can evaluate the left hand side above aslim x → h G − /τ ( x ) = ˜ θ N,h ( − /τ ) − e ( h/ E ( − /τ )12 N iη ( − /τ ) + e ( h/ iη ( − /τ ) τ πiN = ˜ θ N,h ( − /τ ) − e ( h/ E ( τ ) − iπτ ) τ N iη ( − /τ ) + e ( h/ iη ( − /τ ) τ πiN = ˜ θ N,h ( − /τ ) − e ( h/ E ( τ ) τ N i ( − iτ ) / η ( τ ) . Substituting this into (3.3.9) and using the fact that τ − / τ i ( − iτ ) / = e (1 /
8) then finishes theproof. (cid:3)
Example 3.8.
In the simplest case N = 2, P ∗ /P = Z / Z and θ ( τ ) = η ( τ )( e / − e / ).Then the holomorphic part of ˜ θ ,h ( τ ), which we denote by ˜ θ +2 ,h ( τ ), vanishes for h = 0 , θ +2 , / ( τ ) = E ( τ )24 η ( τ ) + X r ∈ Z +1 / g + τ ( r )= E ( τ )24 η ( τ ) + X r ∈ Z +1 / m ∈ Z +1 / ,m> | r | sgn( r )( m − | r | ) iη ( τ ) e (cid:18) m − r τ + m (cid:19) = E ( τ ) / − F (2)2 ( τ ) η ( τ ) = − q − /
24 ( − q + 231 q + 770 q + O ( q )) = − ˜ θ +2 , / ( τ ) , where F (2)2 ( τ ) := P b>a> ,b − a odd a ( − b q ab/ comes out of the substitution a = m − | r | , b = m + | r | . The function ˜ θ +2 , / plays an important role in Matthieu Moonshine [9, 15]. ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 19 Construction of ˜Θ( τ, L ) . In this section, we will construct a modular preimage of Θ( τ, L ; t ) under ξ at t = 1, andwe denote it by ˜Θ( τ, L ). Recall from (2.3.8) that Θ( τ, L ; t ) is constructed from φ τ defined in(2.3.6). If we let˜Θ ∗ ( τ, L ; t ) := X h ∈ L ∗ /L ˜Θ ∗ h ( τ, L ; t ) e h , ˜Θ ∗ h ( τ, L ; t ) := X X ∈ L + h ˜ φ ∗ τ ( B ( X, Z + t ) , B ( X, Z − t )) , ˜ φ ∗ τ ( x, y ) := 2 e (cid:18) y τ (cid:19) g ∗ τ ( x ) = e (cid:18) y − x τ (cid:19) sgn( x )erfc( √ πv | x | ) , (4.0.1)then Prop. 3.3 implies that ξ ˜Θ ∗ ( τ, L ; t ) = − Θ( τ, L ; t ). However, ˜Θ ∗ ( τ, L ; t ) is not modularsince ˜ φ ∗ τ does not behave well under the Fourier transform F . In view of Prop. 3.3, it is morenatural to replace g ∗ τ in (4.0.1) with g τ to form a modular form, but it might not convergefor all L and t (see Remark 3.6).The situation becomes much better at the special point t = 1, where the theta kernelΘ( τ, L ; t ) becomes a finite sum of products of weight holomorphic unary theta series withanti-holomorphic theta functions. Even though ˜Θ( τ, L ) is not harmonic, it still breaks natu-rally into the sum of a holomorphic part and a non-holomorphic part ˜Θ ∗ ( τ, L ) := ˜Θ ∗ ( τ, L ; 1).The holomorphic part is a vector-valued Laurent series in q /d L , which will have rationalFourier coefficients with explicitly bounded denominators. This holomorphic function is alsocalled a “mixed-mock modular form” in the literature [9]. Our method should work for any t ∈ F ∩ R × + , and it would be very interesting to construct the preimage for all t . We split theconstruction into a special case and the general case, and deduce the latter from the former.4.1. Special case.
Suppose that L = L a , AN for some integral ideal a ⊂ O D and positiveinteger N . Let P := 2 A N ( Z ⊕ √ D Z ) ⊂ L be a sublattice. Via the map a + b √ D ( aAN , √ DbAN ), the lattice P is isometric to P AN ⊕ ( − P AN √ D ) and P ∗ /P ∼ = P ∗ AN /P AN × P ∗ AN √ D /P AN √ D as finite abelian groups (see (3.3.4)). Therefore,Θ( τ, P ; 1) = 2 √ v √ X h ∈ P ∗ /P X a + b √ D ∈ P + h aAN e (cid:18) ( a/ ( AN )) τ (cid:19) e − ( √ Db/ ( AN )) τ ! = √ v θ AN ( τ ) ⊗ θ AN √ D ( − τ )by (2.3.11), where θ and θ are defined in (3.3.5). Recall that there is a Γ-linear map ψ : C [ P ∗ /P ] → C [ L ∗ /L ] defined in (2.2.2) and a harmonic Maass form ˜ θ AN constructed insection 3.3. We can now define(4.1.1) ˜Θ( τ, L ) := 2 ψ ( ˜ θ AN ( τ ) ⊗ θ AN √ D ( τ ))and prove the following result Proposition 4.1.
The function ˜Θ( τ, L ) is a real-analytic modular form in A ,ρ − L such that ξ ( ˜Θ( τ, L )) = Θ( τ, L ; 1) . Furthermore, its holomorphic part (4.1.2) ˜Θ + ( τ, L ) := 2 ψ ( ˜ θ +2 AN ( τ ) ⊗ θ AN √ D ( τ )) is a formal Laurent series in AN Z [ L ∗ /L ](( q /d L )) Proof.
Since ψ is defined over Z , it commutes with ξ and is Γ-linear with respect to ρ − P and ρ − L as well. Therefore, ˜Θ( τ, L ) ∈ A ,ρ − L and ξ ˜Θ( τ, L ) = 2 √ vψ (cid:16) ξ / ( ˜ θ AN ( τ )) ⊗ θ AN √ D ( − τ ) (cid:17) = ψ (Θ( τ, P ; 1)) = Θ( τ, L ; 1)by (2.3.12) and Theorem 3.7, which also implies the bound on the denominator. (cid:3) General case.
For any even, integral lattice (
L, Q ) and N ∈ N , denote the scaledlattice ( N L, QN ) by N L . Note that they have the same dual lattice and there is a naturalprojection L ∗ / ( N L ) → L ∗ /L. The following simple lemma then relates Θ( τ, L ; t ) and Θ( τ, N L ; t ). Lemma 4.2.
Fix τ ∈ H and t ∈ R × + . For any h ∈ L ∗ /L , we have Θ h ( τ, L ; t ) = X δ ∈ L ∗ /NL, δ ≡ h mod L Θ δ ( N τ, N L ; t ) . Equivalently, let C L,N be the | L ∗ /L | × | L ∗ /N L | matrix defined by (4.2.1) C L,N := ( L ( h − δ )) h ∈ L ∗ /L,δ ∈ L ∗ /NL , where L is the characteristic function of L . Then (4.2.2) Θ( τ, L ; t ) = C L,N · Θ( N τ, N L ; t ) . Remark . This already appeared in Hecke’s work [18, Eq. III in § L = L a ,M with a ⊂ O D an integral ideal and M ∈ N any natural number. Define N ∈ N to be the smallest positive integer such that(4.2.3) N M = 2 A ( N ′ ) for some N ′ ∈ N . In particular, we can always choose N = 2 AM and N ′ = M . Using thefunction ˜Θ( τ, N L ) in (4.1.1), we can construct ˜Θ( τ, L ) with the proposition below. Proposition 4.4.
The function C L,N · ˜Θ( N τ, N L ) maps to Θ( τ, L ; 1) under ξ and is in A ,ρ − L (Γ ( N )) . ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 21
Proof.
The first claim follows from equation (4.2.2) and ξ ˜Θ( N τ, N L ) = Θ(
N τ, N L ). Forthe second claim, notice that for γ ∈ Γ ( N ), we have(4.2.4) (cid:16) C L,N · ˜Θ( N τ, N L ) (cid:17) | ,ρ − L γ = ρ − − L ( γ ) · C L,N · ρ − NL ( γ N ) · ˜Θ( N τ, N L ) , where γ N := ( N ) · γ · (cid:0) /N (cid:1) ∈ Γ. Thus, it suffices to show the matrix identity(4.2.5) ρ − L ( γ ) · C L,N = C L,N · ρ − NL ( γ N )for all γ ∈ Γ ( N ). Denote M γ := ρ L ( γ ) · C L,N − C
L,N · ρ NL ( γ N ) . It suffices to prove that e h is in the right kernel of M γ for all h ∈ L ∗ /N L . For t ∈ R × + ,consider the theta series θ ( τ, N L ; t ) := √ v X h ∈ L ∗ /NL e h X λ ∈ NL + h e (cid:18) ( λt − + λ ′ t ) AM N τ − ( − λt − + λ ′ t ) AM N τ (cid:19) ∈ A ,ρ NL (Γ) . Then C L,N · θ ( τ, N L ; t ) = θ ( τ, L ; t ) and for all γ ∈ Γ ( N ) , t ∈ R × + M γ · θ ( N τ, N L ; t ) = 0as a C [ L ∗ /L ]-valued function on H . This follows from the same calculations that producedequation (4.2.4). This power series identity necessarily becomes an identity between theFourier coefficients, which are functions of v . From the asymptotic behavior with respectto v , we can deduce that e h is in the kernel of M γ for all h ∈ L ∗ /N L , i.e. M γ vanishesidentically. Applying complex conjugation then gives us equation (4.2.5). (cid:3) Now, we can average C L,N · ˜Θ( N τ, N L ) over Γ ( N ) \ Γ to define(4.2.6) ˜Θ( τ, L ) := 1[Γ : Γ ( N )] X γ ∈ Γ ( N ) \ Γ (cid:16) C L,N · ˜Θ( N τ, N L ) (cid:17) | ,ρ − L γ. The main result of this section is as follows.
Theorem 4.5.
Let L = L a ,M . The function ˜Θ( τ, L ) ∈ A ,ρ − L (Γ) is a real-analytic auto-morphic form such that ξ ( ˜Θ( τ, L )) = Θ( τ, L ; 1) and ˜Θ + ( τ, L ) := ˜Θ( τ, L ) + ˜Θ ∗ ( τ, L ) is in κ L Z [ L ∗ /L ](( q )) with (4.2.7) κ L := 12 A ( N ′ ) · φ ( N ) , where φ ( N ) := [Γ : Γ ( N )] = N Q p | N prime (1 + p ) . In particular κ L can be chosen to divide AM ) φ (2 AM ) . Proof.
Since ξ commutes with | and conjugates ρ − L to ρ L , we obtain the first claim fromProposition 4.4.To prove the second claim, we will first show that ˜Θ + ( τ, L ) ∈ Q [ L ∗ /L ](( q )), then give abound of the denominator. For γ = ( ∗ ∗ c ∗ ) ∈ Γ, we can write N γ := gcd( N, c ) and(4.2.8) (cid:18) N (cid:19) γ = γ N · (cid:18) N γ b N/N γ (cid:19) , b ∈ Z , γ N ∈ Γ . Then for every γ ∈ Γ , τ ∈ H and f ∈ A ,ρ (Γ), we have(4.2.9) f ( N τ ) | γ = N γ N ρ ( γ N ) · f ( τ γ ) , τ γ := ( γ − N ( N ) γ ) · τ = N γ τ + bN/N γ ∈ H . Now applying this to f ( τ ) = ˜Θ( τ, N L ) ∈ A ,ρ − NL (Γ) gives us˜Θ( N τ, N L ) | γ = N γ N ρ − NL ( γ N ) · ˜Θ + ( τ γ , N L ) . Substituting this into the definition of ˜Θ( τ, L ) gives us(4.2.10) ˜Θ( τ, L ) = 1[Γ : Γ ( N )] X γ ∈ Γ ( N ) \ Γ N γ N ρ − − L ( γ ) · C L,N · ρ − NL ( γ N ) ˜Θ( τ γ , N L ) . Notice that we have the following (rather cute) linear algebra identity relating the non-holomorphic parts(4.2.11) ˜Θ ∗ ( τ, L ) = N γ N ρ − − L ( γ ) · C L,N · ρ − NL ( γ N ) ˜Θ ∗ ( τ γ , N L ) , γ ∈ Γ . To prove this identity, we first apply equation (4.2.9) to f ( τ ) = Θ( τ, N L ; 1) ∈ A ,ρ NL (Γ) toobtain Θ( τ, L ; 1) = N γ N ρ − L ( γ ) · C L,N · ρ NL ( γ N )Θ( τ γ , N L ; 1)for all γ ∈ Γ. Since ξ ( ˜Θ ∗ ( τ γ , N L )) = Θ( τ γ , N L ; 1), we see that the difference between the twosides of equation (4.2.11) vanishes under ξ , implying that it is holomorphic. Furthermore,this difference is stable under τ τ + N and vanishes as v → ∞ , so has a Fourier expansion P n ≥ a n e ( nτ /N ). However, integrating this difference against e ( − nu/N ) over u ∈ [0 , N ]equals to zero for all n ≥
1, which means a n = 0 for all n ≥
1. Thus the difference vanishesidentically. Now substituting ˜Θ( τ γ , N L ) = ˜Θ + ( τ γ , N L ) − ˜Θ ∗ ( τ γ , N L ) and identity (4.2.11)into equation (4.2.10), we can write(4.2.12) ˜Θ + ( τ, L ) = 1[Γ : Γ ( N )] X γ ∈ Γ ( N ) \ Γ N γ N ρ − − L ( γ ) · C L,N · ρ − NL ( γ N ) · ˜Θ + ( τ γ , N L ) . The Weil representations ρ − L and ρ − NL are defined over Q ( ζ N † ) / Q with N † = ADM N and ζ N † a primitive ( N † ) th root of unity. For a ∈ ( Z /N † Z ) × , recall that J a = ( a ) ∈ GL ( Z /N † Z )and σ a ∈ Gal( Q ( ζ N † ) / Q ) be the corresponding element as in Section 2.2. Let γ ′ ∈ Γ be any
ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 23 element such that its image in Γ( N † ) \ Γ ∼ = SL ( Z /N † Z ) is J − a γJ a . Then N γ = N γ ′ and wecan write (cid:18) N (cid:19) γ ′ = γ ′ N · (cid:18) N γ ab N/N γ (cid:19) with the image of γ ′ N ∈ Γ in SL ( Z /N † Z ) being J − a γ N J a .Since ˜Θ + ( τ, N L ) ∈ Q [ L ∗ /N L ](( q /d NL )) and τ γ = N γ τ + bN/N γ with b ∈ Z , we have σ a ˜Θ + ( τ γ , N L ) = ˜Θ + (cid:18) N γ τ + abN/N γ , N L (cid:19) = ˜Θ + ( τ γ ′ , N L ) , τ γ ′ := (cid:18) ( γ ′ N ) − (cid:18) N (cid:19) γ ′ (cid:19) · τ. Since d − L | d − NL | N † , the representations ρ − L and ρ − NL are trivial on Γ( N † ). By Proposi-tion 2.1, we have σ a (cid:16) N γ ρ − − L ( γ ) · C L,N · ρ − NL ( γ N ) · ˜Θ + ( τ γ , N L ) (cid:17) = N γ ′ ρ − − L ( γ ′ ) · C L,N · ρ − NL ( γ ′ N ) · ˜Θ + ( τ γ ′ , N L ) . Thus, σ a permutes the summands on the right hand side of equation (4.2.12), which meansthat ˜Θ + ( τ, L ) has Fourier coefficients in Q .From the explicit formula of Weil representation in [28, Theorem 4.7], we know thatthe denominator of every entry in ρ − L , resp. ρ − NL , is bounded by √ AM , resp. N √ AM .Proposition 4.1 and the choice of N in equation (4.2.3) tells us that the denominator of˜Θ + ( τ ′ , N L ) is bounded by 6 AN ′ . Thus, the denominator of ˜Θ + ( τ, L ) is bounded by κ L . (cid:3) Deformation of Theta Integral.
In this section, we will construct a harmonic Maass form with the following property.
Theorem 5.1.
In the notation of Section 2, let M ∈ N , D ≥ a discriminant, a ⊂ O D an integral ideal with A := [ O D : a ] , and ϑ ( L, τ ) the vector-valued cusp form associated to ( L, Q ) = ( M a , Nm AM ) . There exists a harmonic Maass form ˜ ϑ ( τ, L ) = P h ∈ L ∗ /L ˜ ϑ h ( τ, L ) e h in H ,ρ − L (SL ( Z )) such that ξ ( ˜ ϑ ( τ, L )) = ϑ ( τ, L ) and its holomorphic part ˜ ϑ + h ( τ, L ) has theFourier expansion ˜ ϑ + h ( τ, L ) = X n ∈ Q ,n ≫−∞ c + L ( n, h ) q n , with the Fourier coefficient c + L ( n, h ) satisfying c + L ( n, h ) − X λ ∈ Γ L \ L + h − Q ( λ )= n> sgn( λ ) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) ∈ κ Z · log ε L (5.0.1) for an explicit constant κ ∈ N depending on D and M only. In particular, when a is a proper O -ideal and gcd( A, M ) = 1 , then one can choose κ to divide M φ (2 M ) , where φ is themultiplicative function defined in Theorem 1.1. Remark . The summation in equation (5.0.1) is finite and the choice of representative λ ∈ Γ L \ L + h does not affect the statement of the result. Remark . Using a trick with Stokes’ theorem, the theorem above immediately impliesthat the Petersson norm of ϑ ( τ, L ) is in κ Z · log ε L . Using the Fourier-Jacobi expansion oftheta integrals by Kudla [20], the second author has shown that this norm is in fact alwaysin Z · log ε L [26]. Remark . When L is isotropic, a similar statement holds, and the Fourier coefficients arelogarithms of rational numbers (see [25]).The starting point of the construction is the deformed integral I ( τ, − L, s ) defined by(5.0.2) I ( τ, − L, s ) := Z ε L t s Θ( τ, − L ; t ) dtt . Since [1 , ε L ] is compact, I ( τ, − L, s ) is holomorphic for s ∈ C and has the following Taylorseries expansion at s = 0(5.0.3) I ( τ, − L, s ) = ϑ ( τ, − L ) + I ′ ( τ, − L ) s + O ( s ) , where I ′ ( τ, − L ) = ∂∂s I ( τ, − L, s ) | s =0 . By Proposition 2.2, applying ξ to I ( τ, L, s ) gives us ξ I ( τ, − L, s ) = − Z ε L t s d t Θ( τ, L ; t ) dtt = 1 − ε sL τ, L ; 1) + s I ( τ, L, s )after using integration by parts, which implies(5.0.4) 2 ξ ( I ′ ( τ, − L )) + log ε L · Θ( τ, L ; 1) = ϑ ( τ, L ) . Recall that ˜Θ( τ, L ) ∈ A ,ρ − L (Γ) is the preimage of Θ( τ, L ; 1) under ξ as in Theorem 4.5.Then 2 I ′ ( τ, − L ) + log ε L · ˜Θ( τ, L ) ∈ A ,ρ − L (Γ) is a preimage of ϑ ( τ, L ) under ξ . In the restof this section, we will calculate its Fourier expansion and prove Theorem 5.1.5.1. Fourier expansion of I ′ ( τ, − L ) . Now, we will calculate the Fourier expansion of(5.1.1) I ′ h ( τ, − L ) := Z ε L log t · Θ h ( τ, − L ; t ) dtt for each h ∈ L ∗ /L . To state the main result, we will first setup a few notations and makesome choices. For simplicity, we write ε = ε L . For λ = 0, let(5.1.2) r ( λ ) := (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) . For each orbit Λ ∈ Γ L \ L + h with Q (Λ) = 0, we fix a representative λ ∈ Λ such that(5.1.3) 1 ≤ r ( λ ) < ε . ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 25
After such a representative has been fixed, we will set sgn(Λ) := sgn( λ ),(5.1.4) a (Λ) := ( sgn( λ ) log r ( λ ) , r ( λ ) = 1 , − log ε, r ( λ ) = 1 , and use the convenient notation(5.1.5) λ n := λ ε n . Notice that − ε < a (Λ) < ε . We can now state the main result of this section. Proposition 5.5.
The function I ′ h ( τ, − L ) has the Fourier expansion (5.1.6) 2 I ′ h ( τ, − L ) = X Λ ∈ Γ L \ L + hQ (Λ) < a (Λ) e ( − Q (Λ) τ ) − ˜ ϑ ∗ h ( τ, L ) − log ε · ˜Θ ∗ h ( τ, L ) , where ˜ ϑ ∗ h ( τ, L ) and ˜Θ ∗ h ( τ, L ) are given by ˜ ϑ ∗ h ( τ, L ) = X λ ∈ Γ L \ L + hQ ( λ ) > sgn( λ )Γ(0 , πQ ( λ ) v ) q − Q ( λ ) , ˜Θ ∗ h ( τ, L ) = X λ ∈ L + h ˜ φ ∗ τ (cid:18) λ + λ ′ √ AM , λ − λ ′ √ AM (cid:19) . Proof.
Substituting in equation (2.3.11) gives us I ′ h ( τ, − L ) = r vAM X Λ ∈ Γ L \ L + hQ (Λ) =0 e ( − Q (Λ) u ) X λ ∈ Λ Z ε (cid:0) λt − − λ ′ t (cid:1) e (cid:18) (( λt − ) + ( λ ′ t ) ) iv AM (cid:19) log t dtt Since Λ = { λ ε n : n ∈ Z } , each sum over Λ becomes X n ∈ Z Z ε (cid:0) λ n t − − λ ′ n t (cid:1) e (cid:18) (( λ n t − ) + ( λ ′ n t ) ) iv AM (cid:19) log t dtt . In each summand, let ν := − log r ( λ n )2 + log t . Then, we can write log t = ν + log r ( λ )2 + n log ε and break I ′ h ( τ, − L ) into three pieces(5.1.7) 2 I ′ h ( τ, − L ) = X Λ ∈ Γ L \ L + hQ (Λ) =0 sgn(Λ) e ( − Q (Λ) τ ) ( J (Λ) + J (Λ) + J (Λ)) where J (Λ) , J (Λ) and J (Λ) are defined by J (Λ) := log r ( λ ) Z ∞−∞ p | Q (Λ) | v (cid:0) e − ν − sgn( Q (Λ)) e ν (cid:1) e | Q (Λ) | ( e ν + sgn( Q (Λ)) e − ν ) iv ! dν,J (Λ) := 2 Z ∞−∞ p | Q (Λ) | v (cid:0) e − ν − sgn( Q (Λ)) e ν (cid:1) e | Q (Λ) | ( e ν + sgn( Q (Λ)) e − ν ) iv ! νdν,J (Λ) := sgn(Λ)2 log ε √ v √ AM X n ∈ Z n Z ε (cid:0) λ n t − − λ ′ n t (cid:1) e (cid:18) ( λ n t − + λ ′ n t ) iv AM (cid:19) dtt . Note that J and J comes naturally out of the unfolding process Hecke used, which doesnot work directly for J . Like equation (2.4.2), we can evaluate the terms J (Λ) and J (Λ)as − Q (Λ) > − Q (Λ) < J (Λ) sgn(Λ) log r ( λ ) e ( − Q (Λ) iv ) 0 J (Λ) 0 − sgn(Λ) e (2 Q (Λ) iv )Γ(0 , πQ (Λ) v )Using the identity d t erf( at + bt − ) = √ π ( at − bt − ) e − ( at + bt − ) , we obtain2 r vAM Z ε (cid:0) λ ′ n t − λ n t − (cid:1) e (cid:18) ( λ n t − + λ ′ n t ) iv AM (cid:19) dtt = erf (cid:18)r πvAM ( λ ′ n t + λ n t − ) (cid:19) | ε . Applying this and the identity erf( x ) = sgn( x ) − sgn( x )erfc( | x | ) to J (Λ) gives ussgn(Λ) J (Λ) = log ε X n ∈ Z n (cid:18) erf (cid:18)r πvAM ( λ ′ n − + λ n − ) (cid:19) − erf (cid:18)r πvAM ( λ ′ n + λ n ) (cid:19)(cid:19) = log ε (sgn( λ + λ ′ ) − sgn( λ + λ ′ )) + log ε X n ∈ Z sgn( λ n + λ ′ n )erfc (cid:18)r πvAM | λ ′ n + λ n | (cid:19) . The first term is − sgn(Λ) log ε if λ = − λ ′ and zero otherwise. After substituting these intoequation (5.1.7), we obtain equation (5.1.6). (cid:3) Proof of Theorem 5.1.
Suppose (
L, Q ) = ( L a ,M , Q a ,M ) for some a ⊂ O D ⊂ O F and M ∈ N . Define(5.2.1) θ ( τ, L ) := 2 I ′ ( τ, − L ) + log ε L ˜Θ( τ, L ) . By Theorem 4.5 and Proposition 5.5, θ ( τ, L ) ∈ H ,ρ − L (Γ) and the holomorphic part θ + ( τ, L ) := θ ( τ, L ) + ˜ ϑ ∗ ( τ, L ) is given by θ + ( τ, L ) = X h ∈ L ∗ /L e h X Λ ∈ Γ L \ L + h, Q (Λ) < sgn(Λ) log r (Λ) q − Q (Λ) + log ε L ˜Θ + ( τ, L ) . Theorem 4.5 implies that θ ( τ, L ) satisfies the Theorem 5.1 with κ = κ L . ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 27
Now, we can reduce κ as follows. Let b be another ideal of O D such that a = b · ( µ O D ) witha totally positive element µ ∈ F satisfying µ − ∈ M d D . This is an equivalence relation,under which there are only finitely many equivalence classes of O D -ideals. Let A := [ O D : b ]and N ∈ N such that N M = 2 A ( N ′ ) for some N ′ ∈ N . Then ϑ ( τ, L ) = ϑ ( τ, L b ) and θ ( τ, L ) := θ ( τ, L b ) + log (cid:12)(cid:12)(cid:12) µµ ′ (cid:12)(cid:12)(cid:12) ϑ ( τ, − L ) also satisfies Theorem 5.1 with κ = κ L b . Suppose κ ′ = gcd( κ L , κ L b ) = c κ L + c κ L b with c , c ∈ Z , then c κ L · θ ( τ,L )+ c κ L b · θ ( τ,L ) κ will satisfy thestatement in Theorem 5.1 with κ = κ ′ . Since there are only finitely many equivalence classes,we can repeated this process finitely many times to find the minimal κ , which only dependson the data O D and M . This proves the last part of Theorem 5.1.In particular when the O D -ideal a is proper and gcd( A, M ) = 1, we can choose a totallypositive µ ∈ F such that µ − ∈ M d D and µ O D = ab − for some integral, proper O D -ideal b relatively prime to a . Then it is possible to reduce κ to at least 24 M φ (2 M ).6. Scalar-Valued Result and Numerical Examples.
In this section, we will use the result in [31] to produce a scalar-valued version of ϑ ( τ, L ),and prove the scalar-valued version of Theorem 5.1.6.1. Reducing the level.
We first need to reduce the level of certain vector-valued auto-morphic forms. Fix a fundamental discriminant
D >
1, an integral ideal m ⊂ O F ⊂ F := Q ( √ D ) ⊂ R and denote M = Nm( m ), N = M D and χ D ( · ) = (cid:0) D · (cid:1) the quadratic Dirichletcharacter. Let a ⊂ O F be an arbitrary integral ideal relatively prime to m with A := Nm( a ),and denote L = L ad ,M . Then L ∗ = a and there is a canonical surjection map of finite abeliangroups(6.1.1) L ∗ /L = a /M da → a / ma ∼ = ֒ → O F / m . This induces a natural, linear map from C [ L ∗ /L ] to C [ a / ma ]. Let { e σ : σ ∈ a / ma } be thecanonical basis of C [ a / ma ]. Under this and the canonical basis of C [ L ∗ /L ], the linear mapis given by the matrix(6.1.2) C a , m := ( ma ( h − σ )) h ∈ L ∗ /L,σ ∈ a / ma , where ma is the characteristic function of ma ⊂ a .Define a representation ρ m of Γ ( N ) on C [ O F / m ] by(6.1.3) ρ m ( γ ) e σ = χ D ( d ) e dσ , γ = ( ∗ ∗∗ d ) ∈ Γ ( N ) . Since a and m are relatively prime to each other, there is a canonical isomorphism between C [ a / ma ] and C [ O F / m ] induced by a / ma ∼ = ֒ → O F / m . Conjugating ρ m by this isomorphismgives rise to a representation of Γ ( N ) on C [ a / ma ], which we also denote by ρ m . We can nowuse Theorem 3 in [31] to study precisely the effect of C a , m on the level of the vector-valuedautomorphic forms such as Θ( N τ, L ; t ) in equation (2.3.11). Proposition 6.1.
In the notations above, we have C a , m · Θ( N τ, ± L ; t ) ∈ A ,ρ m (Γ ( N )) forany t ∈ R × + .Proof. For each σ ∈ a / ma , we denote the e σ -component of C a , m Θ( τ, L ; t ) by Θ σ ( τ, a , m ; t ).From the definition, it is easy to check that(6.1.4) Θ σ ( AN τ, a , m ; t ) = √ v X λ ∈ am + σ ( λ ′ t + λt − ) e (cid:18) Nm( λ ) u + 12 (( λt − ) + ( λ ′ t ) ) iv (cid:19) . Theorem 3 in [31] with the choice of integral ideal I = am and σ ∈ a then implies C a , m Θ( AN τ, L ; t )) ∈ A ,ρ m (Γ ( AN )). It follows readily from equation (6.1.4) that C a , m Θ( N ( τ +1) , L ; t ) = C a , m Θ( N τ, L ; t ) . Since Γ ( N ) is generated by ( ) and the matrices ( a bc d ) ∈ Γ ( N )satisfying A | b , we obtain the desired result. The same argument works for − L . (cid:3) Remark . Integrating over t, it is easy to verify from Proposition 6.1 and equation (2.4.1)that(6.1.5) C a , m · ϑ ( N τ, ± L ) = X λ ∈ Γ L \ ( am + σ ) , ± Nm( λ ) > sgn( λ ) q | Nm( λ ) | /A σ ∈ a / ma ∈ S ,ρ m (Γ ( N )) . Recall from Section 2.3 that Θ( τ, ± L ; t ) ∈ A ,ρ ± L (SL ( Z )) for all t ∈ R × + . It turns out thatone can bootstrap a more general result out of the proposition above. Proposition 6.3.
In the notations above, C a , m · f ( N τ ) is in A ,ρ m (Γ ( N )) for all f ∈A ,ρ ± L (SL ( Z )) .Proof. The proof is analogous to that of Proposition 4.4. For γ = ( a bNc d ), recall that γ N =( a Nbc d ) as in Proposition 4.4. For an arbitrary f ∈ A ,ρ L (SL ( Z )), we want to show that( C a , m · f ( N τ )) | γ = ρ m ( γ ) · C a , m · f ( N τ ) . Since f ( N τ ) | γ = ρ L ( γ N ) · f ( N τ ), it suffices to prove the identity M γ · f ( N τ ) = 0, where M γ := C a , m · ρ L ( γ N ) − ρ m ( γ ) · C a , m . By Proposition 6.1, the equality above holds with f ( τ ) = Θ( τ, L ; t ) for any t ∈ R × + . AllFourier coefficients of the corresponding identity do vanish and therefore, arguing as inProposition 4.4, we know that the right kernel of M γ contains { e h − e − h : h ∈ L ∗ /L } . Fromequation (2.2.1), we have ρ L ( S ) e h = − e − h for all h ∈ L ∗ /L . Then any f = P h ∈ L ∗ /L f h e h ∈A ,ρ L (SL ( Z )) satisfies f h = − f − h . Thus, 2 f ( N τ ) = P h ∈ L ∗ /L f h ( N τ )( e h − e − h ) is in the rightkernel of M γ . The same argument works for − L . (cid:3) ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 29
Ray class group character.
Let m = m · ∞ be a modulus with M = Nm( m ), I m bethe group of fractional ideals of O F relatively prime to m and P m be the group of principalfractional ideals generated by elements µ ∈ F such that µ ≡ × m and µ >
0. Considera ray class group character of conductor m (6.2.1) ϕ : Cl m := I m /P m → C × . The class group Cl F of F is a natural quotient of Cl m . The presence of ∞ in m is equivalentto(6.2.2) ϕ (( µ )) = ϕ f ( µ ) · sgn( µ ) . with ϕ f a character on ( O F / m ) × satisfying ϕ f ( ε ) = sgn( ε ) for all ε ∈ O × F . After extendingby zero, we can view ϕ f , resp. ϕ , as a map on F , resp. fractional ideals of O F . By a slightabuse of notation, we write ϕ ( λ ) := ϕ (( λ )).Let N = DM and χ be the product of χ D and the restriction of ϕ to Q . Associated to ϕ is a Hecke eigenform f ϕ ∈ S ,χ (Γ ( N )) given by(6.2.3) f ϕ ( τ ) := X b ⊂O F ϕ ( b ) q Nm( b ) = X [ a ] ∈ Cl F ϕ ( a − ) f ϕ, a ( τ ) , f ϕ, a ( τ ) := X ( λ ) ⊂ a ϕ ( λ ) q Nm(( λ ) a − ) . Even though ϕ ( a ) depends on the representative of [ a ] ∈ Cl F , the product ϕ ( a ) f ϕ, a is inde-pendent of such choice.For fixed integral ideal a ∈ I m , let A = Nm( a ) and L = L ad ,M . Then we can write f ϕ, a ( τ ) = f ϕ, a , + ( τ ) + f ϕ, a , − ( τ ), where(6.2.4) f ϕ, a , ± ( τ ) := 1[ O × F : Γ L ] X λ ∈ Γ L \ a , ± Nm( λ ) > ϕ f ( λ )sgn( λ ) q | Nm( λ ) | /A . Now, we can express f ϕ, a as a linear combination of ϑ σ, ± as follows. Proposition 6.4.
Let C ϕ := ( ϕ f ( σ )) σ ∈O F / m be a row vector. Then left multiplication by C ϕ is a linear map from A ,ρ m (Γ ( N )) to A ,χ (Γ ( N )) . In particular, (6.2.5) f ϕ, a , ± ( τ ) = C ϕ · C a , m · ϑ ( N τ, ± L )[ O × F : Γ L ] ∈ S ,χ (Γ ( N )) . Proof.
This is a consequence of Remark 6.2 above and Section 5 of [31]. (cid:3)
The scalar-valued version of Theorem 5.1 is as follows.
Theorem 6.5.
For each class in Cl F , fix a representative a ∈ I m with A = Nm( a ) . Let ϕ and f ϕ ∈ S ,χ (Γ ( N )) be as above. There exists a harmonic Maass form ˜ f ϕ ∈ H ,χ (Γ ( N )) suchthat ξ ˜ f ϕ = f ϕ and the holomorphic part of ˜ f ϕ has the Fourier expansion P n ≫−∞ c + ϕ ( n ) q n satisfying (6.2.6) c + ϕ ( n ) − X [ a ] ∈ Cl F ϕ ( a ) X ( λ ) ⊂ a , Nm(( λ ) a − )= n ϕ ( λ ) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) ∈ κ m Z [ ϕ ] · log ε F with κ m | M φ (2 M ) .Proof. For each representative a ∈ I m , let L = L ad ,M and ˜ ϑ ( τ, ± L ) ∈ H ,ρ ± L (SL ( Z )) be theharmonic Maass form constructed in Theorem 5.1. Consider(6.2.7) ˜ f ϕ, a ( τ ) := 1[ O × F : Γ L ] C ϕ · (cid:16) C a , m · ˜ ϑ ( N τ, L ) + C a , m · ˜ ϑ ( N τ, − L ) (cid:17) . By Propositions 6.3 and 6.4, ˜ f ϕ, a ∈ H ,χ (Γ ( N )) and ξ ˜ f ϕ, a = f ϕ, a . Its holomorphic part ˜ f + ϕ, a has the Fourier expansion ˜ f + ϕ, a ( τ ) = X n ≫ c + ϕ, a ( n ) q n . As a consequence of Theorem 5.1, the coefficient c + ϕ, a ( n ) satisfies c + ϕ, a ( n ) − X ( λ ) ⊂ a , Nm(( λ ) a − )= n ϕ (( λ )) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) ∈ κ · [ O × F : Γ L ] Z [ ϕ ] · log ε L . Since O × F = {± ε nF : n ∈ Z } , we have log ε L [ O × F :Γ L ] = log ε F . By Theorem 5.1, we can choose κ = 24 M φ (2 M ). Summing over the representatives of all the classes in Cl F with respect to ϕ ( a − ) = ϕ ( a ) finishes the proof. (cid:3) When F has class number one, we can choose a = O F as the only summand in the sumover Cl F and arrange to have X ( λ ) ⊂O F Nm(( λ ))= n ϕ ( λ ) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) = 12 X ( λ ) ⊂O F Nm(( λ ))= n ϕ ( λ ) log (cid:12)(cid:12)(cid:12)(cid:12) λλ ′ (cid:12)(cid:12)(cid:12)(cid:12) + X ( λ ) ⊂O F Nm(( λ ))= n ϕ ( λ ′ ) log (cid:12)(cid:12)(cid:12)(cid:12) λ ′ λ (cid:12)(cid:12)(cid:12)(cid:12) = c ϕ ( n ) ∈ C / ( Z [ ϕ ] log ε F ) . Theorem 1.1 then follows from Theorem 6.5.To compare this result with the p -adic version in [10], we first define the character ϕ ♥ := ϕ/ϕ ′ with ϕ ′ the composition of ϕ and the conjugation on F . The kernel of ϕ ♥ fixes anumber field H , which is a class field of F . When ℓ is an inert prime in F/ Q , Darmon,Lauder and Rotger defined in [10] an element u ( ϕ ♥ , ℓ ) ∈ Z [ ϕ ] ⊗ O H [1 /ℓ ] × and showed thatits p -adic logarithm is the ℓ th Fourier coefficient of a generalized overconvergent eigenformof weight one.
ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 31
When ℓ = λλ ′ is a split prime in F/ Q with σ λ , σ λ ′ ∈ Gal(
H/F ) the respective Frobeniuselements, the ℓ th Fourier coefficient in the p -adic setting is zero, whereas we can define anelement u ( ϕ ♥ , ℓ ) ∈ Z [ ϕ ] ⊗ ( F × / O × F ) by(6.2.8) u ( ϕ ♥ , ℓ ) := X σ ∈ Gal( F/ Q ) ϕ − ♥ ( σσ λ σ − ) ⊗ σ ( λ ) ∈ Z [ ϕ ] ⊗ ( O F [1 /ℓ ] × / O × F ) . After extending the complex logarithm by Z [ ϕ ]-linearity, we have(6.2.9) c ϕ ( ℓ ) = log | u ( ϕ ♥ , ℓ ) | ∈ C / ( Z [ ϕ ] log ε F ) . This might be a complicated way to write c ϕ ( ℓ ) since there are only two elements inGal( F/ Q ). Nevertheless, it shows that the archimedean and non-archimedean situationscomplement each other. In the latter, the element u ( ϕ ♥ , ℓ ) is well-defined in Z [ ϕ ] ⊗ O H [1 /ℓ ] × and can be used to generate class fields of F . In the former, the ℓ -unit u ( ϕ ♥ , ℓ ) lies in theground field F and is only well-defined up to Z [ ϕ ] log ε F . This reflects the difficulty inchoosing a canonical harmonic Maass form with a given ξ -image.6.3. Examples.
The first example was studied in detail in Section 7 of [24], where F = Q ( √ m = (cid:16) √ (cid:17) and ϕ (2) = i . The cusp form f ϕ is the unique normalized eigenformin S ,χ (Γ (145)) with χ = (cid:0) · (cid:1) · ϕ | Q . The second author showed loc. cit. that there exists aunique harmonic Maass form F ϕ in H ,χ (Γ (145)) that maps to f ϕ and has only a simple poleat i ∞ . Using a modularity-based algorithm, the second author numerically calculated theFourier coefficients c + ϕ ( n ) of the holomorphic part of F ϕ as complex numbers. Then they wereidentified as Z [ ϕ ]-linear combinations of logarithm of numbers in F × . When n = ℓ = λλ ′ isa split prime in F , there was a precise conjecture about the shape of c + ϕ ( ℓ ), which is impliedby Theorem 1.1 if κ m = 1.The second example is about a rather classical weight one cusp form studied by Hecke[18]. Set D = 12 , F = Q ( √ D ) , L = a = d F = 2 √ O F , M = 1, and L ∗ = O F . Thediscriminant kernel Γ L is generated by ε L = 7 + 4 √
3, which is the square of the fundamentalunit ε F = 2 + √
3. In this case, the theta series ϑ ( τ, L ) is a 12-dimensional, holomorphicvector-valued weight one cusp form on SL ( Z ). The components correspond to L ∗ /L = O F / d F . It turns out 8 of the 12 components vanish identically. The other 4 componentscorrespond to h = ± , ± (2 + √ ∈ O F / d F and satisfy ϑ = − ϑ − = ± ϑ ± (2+ √ . So e := e + e √ − e − − e − (2+ √ is an eigenvector of ρ L ( T ) and ρ L ( S ) with eigenvalues e (1 /
12) and − i respectively. This implies ϑ (cid:18) − τ , L (cid:19) = − iτ ϑ ( τ, L ) , ϑ ( τ + 1 , L ) = e (cid:18) (cid:19) ϑ ( τ, L ) . Since η ( τ ) := q / Q ∞ n =1 (1 − q n ) also satisfies this transformation property and has zeroonly at the cusps, they are equal up to a multiplicative constant. By comparing the first non-vanishing Fourier coefficient, Hecke obtained ϑ ( τ, L ) = X λ ∈ Γ L \O F λ ≡ √ λ ) > sgn( λ ) q Nm( λ ) / = η ( τ ) ∈ S ,ρ sc L (Γ) , with Γ = SL ( Z ) and ρ sc L the restriction of ρ L on the eigenspace spanned by e .The representation ρ sc L is a multiplier system of Γ and can be expressed in terms ofDedekind sums. Similarly, ρ sc − L is the restriction of ρ − L on e , and the conjugate of ρ sc L .The space S ,ρ sc L (Γ) is spanned by ϑ ( τ, L ) since the 12 th power of any form in that space isin the one dimensional space S (Γ). On the other hand, the space S ,ρ sc − L (Γ) is trivial and ϑ ( τ, − L ) vanishes identically. Therefore the elements λ ∈ O F with negative norm will nevercontribute to any holomorphic modular object. It is worth noting that η ( τ ) can also be con-structed from lattices of signature (2 , Q ( √−
1) and Q ( √−
3) (see e.g. [18, 29]). So the methodsof [14] and [16] could also be used to produce a ξ preimage of η ( τ ).Before constructing the harmonic Maass form ˜ ϑ ( τ, L ), notice that ϑ ( τ, L ) = ϑ ( τ, L d F / , ).So we can suppose that L = L d / , . First, we need to construct ˜Θ( τ, L ). This can be done asin Section 4.2 with N = 6 in equation (4.2.3). Then | L ∗ / L | = 432 = 6 | L ∗ /L | . Using theprocedures in Section 4.1, we can construct ˜Θ( τ, L ). Its holomorphic part ˜Θ + ( τ, L ) hasrational Fourier coefficients with denominators bounded by 6. Then equation (4.2.6) defines˜Θ( τ, L ), whose holomorphic part is given by equation (4.2.12). There, the sum over Γ (6) \ Γhas 12 summands, each of which is the product of a 12 ×
432 matrix N γ N ρ − − L ( γ ) ·C L,N · ρ − NL ( γ N )and a vector ˜Θ + ( τ γ , N L ) of size 432. By Theorem 4.5, all the components of ˜Θ + ( τ, L )have rational Fourier coefficients with bounded denominator. Using SAGE [12], we havenumerically implemented this procedure and calculated the Fourier coefficients of ˜Θ + h ( τ, L )for each h ∈ L ∗ /L . Since ρ − L ( (cid:0) − − (cid:1) ) e h = e − h and the weight is odd, we have ˜Θ h ( τ, L ) = − ˜Θ − h ( τ, L ) for all h ∈ L ∗ /L . When h = 0 , √ , √ h = − h and ˜Θ h ( τ, L ) = 0. Forthe other h , the Fourier expansions are listed in the table below. h · [Γ : Γ (6)] · ˜Θ + h ( τ, L )1 4 q − / − q / − q / − q / + O ( q / )1 + √ q / − q / − q / − q / + O ( q / )2 q − / + 192 q / + 4736 q / + 51052 q / + 365634 q / + O ( q / )2 + √ q − / + 380 q / + 8714 q / + 85060 q / + O ( q / )Now as in (5.2.1), we define ˜ ϑ ( τ, L ) := 2 I ′ ( τ, − L ) + log ε L · ˜Θ( τ, L ) ∈ H ,ρ − L (Γ), whichsatisfies ξ ˜ ϑ ( τ, L ) = ϑ ( τ, L ). Then the eigen-component(6.3.1) ˜ f ( τ ) := 14 (cid:16) ˜ ϑ ( τ, L ) + ˜ ϑ √ ( τ, L ) − ˜ ϑ − ( τ, L ) − ˜ ϑ − (2+ √ ( τ, L ) (cid:17) ∈ H ,ρ sc − L (Γ) ARMONIC MAASS FORMS ASSOCIATED TO REAL QUADRATIC FIELDS 33 maps to η ( τ ) under ξ . Note that ˜ f = ˜ ϑ ( τ,L )+ ˜ ϑ √ ( τ,L )2 . Its holomorphic part ˜ f + can bewritten explicitly as˜ f + ( τ ) = 12 X h ∈{ , √ } ˜Θ + h ( τ, L ) · log ε L + X Λ ∈ Γ L \ L + hQ (Λ) < a (Λ) q | Q (Λ) | using Prop. 5.5. Suppose ˜ f + has the Fourier expansion P n ≥− ,n ≡
11 mod 12 c + ( n ) q n/ , thenthe coefficients have the shape(6.3.2) c + ( n ) = −
112 log (cid:12)(cid:12)(cid:12)(cid:12) u ( n ) u ( n ) ′ (cid:12)(cid:12)(cid:12)(cid:12) , where u ( n ) ∈ O F is a unit outside primes dividing n . For example, c + ( −
1) = · · [Γ:Γ (6)] log ε L = log ε F and u ( −
1) = ε − F . The next coefficient c + (11) has two parts. One contribution comesfrom ˜Θ + h , which is − · · [Γ:Γ (6)] log ε L = − log( ε F /ε ′ F ). The other contribution is12 X h ∈{ , √ } Λ ∈ Γ L \ L + hQ (Λ)= − a (Λ) = 12 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ − √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − √ −
17 + 10 √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − √ − √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ − √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! = 2 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ − √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12)(cid:12) ε F ε ′ F (cid:12)(cid:12)(cid:12)(cid:12) . Therefore, c + (11) = 2 log | (1+2 √ / (1 − √ |− / ε F /ε ′ F ) and u (11) = (1 − √ · ε F .Some of the other u ( n ) with n ≤
300 are listed below. n
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Equipe de th´eorie des nombres, Institut de Math´ematiques de Jussieu-PRG, Case 247. 4,Place Jussieu, 75252 Paris Cedex, France
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