Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler-Selberg type relations
aa r X i v : . [ m a t h . N T ] F e b HIGHER SIEGEL THETA LIFTS ON LORENTZIAN LATTICES ANDHARMONIC MAASS FORMS
JOSHUA MALES
Abstract.
We investigate so-called “higher” Siegel theta lifts on Lorentzian lattices in the spiritof Bruinier–Ehlen–Yang and Bruinier–Schwagenscheidt. We give an series representation of the liftin terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourierseries involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. We alsonote how one could obtain the Fourier expansion. We discuss one explicit example, and show howone could obtain infinite families of relationships between Hurwitz class numbers and divisor powersums. Introduction
In recent years, there have been many investigations into certain theta lifts and their relationshipto modular objects. Perhaps one of the most striking applications is in realising rationality andalgebraicity results. In a recent breakthrough paper, Bruinier, Ehlen, and Yang [5] made a majoradvance towards the Gross–Zagier conjecture by proving that a certain two-variable Green functionevaluated at CM points in one variable and an average over CM points in the other variable takesalgebraic values. A pivotal result that the authors there used was the connection between the Greenfunction and a “higher” Siegel theta lift on a lattice of signature (2 , on lattices of signature (1 ,
2) to investigate traces of cycle integrals of a certain cusp form. Byrelating the lift to the Fourier coefficients of certain modular objects called harmonic Maass forms,the results also gave an elegant proof of the rationality of such traces of cycle integrals, which hadpreviously been shown in [3].More recently, Bruinier and Schwagenscheidt [7] investigated the Siegel theta lift on a Lorentzianlattice L (that is, of signature (1 , n )). For τ = u + iv ∈ H and z ∈ Gr( L ), the Grassmanian of L ,this is given by Z reg F D f ( τ ) , Θ L ( τ, z ) E v k dµ ( τ ) , where the regularised integral is defined by R reg F := lim T →∞ R F T , where F T denotes the standardfundamental domain for Γ truncated at height T . Here, F is the standard fundamental domain forSL ( Z ), f is a vector-valued harmonic Maass form of appropriate weight, h , i is the natural bilinearpairing on the group ring C [ L ′ /L ] that is antilinear in the second variable, Θ L is the Siegel thetafunction associated to the lattice L , and dµ = dudvv is the usual invariant measure on H .In the present paper we consider an extension of this lift, first considered by Bruinier, Ehlen,and Yang in [5] for the n = 2 case. Borrowing their terminology, we call this a higher Siegel thetalift. Let k := − n and j ∈ N . For a vector-valued harmonic Maass form f ∈ H k − j,L we considerΛ reg j ( f, z ) := Z reg F D R jk − j ( f )( τ ) , Θ L ( τ, z ) E v k dµ ( τ ) , The paper [2] also used the higher Millson theta lift where R nκ := R κ +2 n − ◦ · · · ◦ R κ with R κ := id is an iterated version of the Maass raising operator R κ := 2 i ∂∂τ + κv . By the results of [4, Section 2.3] for the Siegel theta function the integral convergesfor every z ∈ H .This situation lies at the interface of [2, 5] which included iterated raising operators but restrictedto the case of n = 2, and [7] where no iterated raising operators were included, but n was unre-stricted. We borrow heavily techniques from each of these papers, and obtain a description of thelift Λ reg as a series involving Gauss hypergeometric functions F , as well as a constant term of aFourier expansion involving coefficients of f , a theta function, and a harmonic Maass form. Finally,we note how one could obtain the Fourier expansion of the lift. Note that for j = 0 we recover thelift studied by Bruinier [4] and Bruinier–Schwagenscheidt [7], and for n = 2 the lift used centrallyin [2].For the rest of the introduction, we restrict to the n = j = 1 case and the isotropic lattice L described explicitly in Section 4. Let f t be the unique weakly holomorphic modular form on SL ( Z )whose Fourier expansion starts q − t + O (1). By unfolding the integral and recognising the resultingGauss hypergeometric function in terms of more elementary functions, we obtain the followingtheorem. Theorem 1.1.
Let z ∈ Gr( L ) . Then Λ reg1 ( f t , z ) = 2 π X a,b ∈ N ab = t min( a, b ) . Next, we describe the evaluation of the theta lift at a special point w (see Section 2.6). There aretwo central arguments that aid us here. Firstly, that when evaluated at a CM point the Siegel thetafunction on L splits as the tensor product Θ P ⊗ Θ N where P is one-dimensional positive-definite,and N is n -dimensional negative-definite. The second crucial observation is that using techniquesof Bruinier–Ehlen–Yang [5] we are able to shift the iterated Maass raising operator acting on f to instead act on Θ L . Combining these arguments with an application of Stokes’ theorem yieldsRankin-Cohen brackets [ · , · ] ℓ of Θ N and G P , which denotes a preimage of Θ P under ξ κ := 2 iv κ ∂∂τ .We obtain a description of the lift as the constant term of a Fourier expansion, denoted by CT, ofthese objects (see Theorem 3.2 for a precise formulation of the results in the general setting). Theorem 1.2.
Let f be a weakly holomorphic modular form of weight k − for L . We have Λ reg1 ( f, w ) = − π h f ( τ ) , [ G P ( τ ) , Θ N − ( τ )] i ) . Since Θ P is a unary theta function it has weight , and its preimage G P has weight . In fact,via [13] we can recover examples involving the Hurwitz class numbers H (each of whose generatingfunction has weight ). For example, combining Theorems 1.1 and 1.2 for the input function f t ,and evaluating at the special point described in Section 4 yields the classical formula X n ∈ Z (cid:0) t − n (cid:1) H (cid:0) t − n (cid:1) = X a,b ∈ N ab = t min ( a, b ) . Further examples could be concluded for other objects whose generating functions are (theholomorphic part of) weight harmonic Maass forms, e.g. the level N Hurwitz class numbers,the classical spt partition function of Andrews, or the small divisor functions studied in [10, 11].For the sake of succinctness, we omit them here.
Outline.
We begin in Section 2 by recalling preliminary results needed for the rest of the paper.In Section 3 we prove the main results of the paper, which are extensions of Theorems 1.1 and 1.2.Finally, in Section 4 we compute some examples and prove those stated in the introduction.
IGHER SIEGEL THETA LIFTS ON LORENTZIAN LATTICES AND HARMONIC MAASS FORMS 3
Acknowledgments.
The author would like to thank Andreas Mono for helpful comments on anearlier draft of the paper. 2.
Preliminaries
We collect some preliminary results needed for the sequel.2.1.
The Weil representation.
The metaplectic extension of SL ( Z ) is defined as e Γ := Mp ( Z ) := (cid:26) ( γ, φ ) : γ = (cid:18) a bc d (cid:19) ∈ SL ( Z ) , φ : H → C holomorphic , φ ( τ ) = cτ + d (cid:27) , with generators e T := (( ) ,
1) and e S := (cid:0)(cid:0) −
11 0 (cid:1) , √ τ (cid:1) . We let e Γ ∞ denote the subgroup generatedby e T .Let L be an even lattice of signature ( r, s ) with quadratic form q and associated bilinear form( · , · ). Let L ′ denote its dual lattice, C [ L ′ /L ] be the group ring of L ′ /L with standard basis elements e µ for µ ∈ L ′ /L , and h· , ·i be the natural bilinear form on C [ L ′ /L ] given by h e µ , e ν i = δ µ,ν . TheWeil representation ρ L associated with L is the representation of e Γ on C [ L ′ /L ] defined by ρ L (cid:16) e T (cid:17) ( e µ ) := e ( q ( µ )) e µ , ρ L (cid:16) e S (cid:17) ( e µ ) := e (cid:0) ( s − r ) (cid:1)p | L ′ /L | X ν ∈ L ′ /L e ( − ( ν, µ )) e ν . Here and throughout e ( x ) := e πix . The Weil representation ρ L − associated to the lattice L − =( L, − q ) is called the dual Weil representation associated to L .2.2. Harmonic Maass forms.
Let κ ∈ Z . We define the slash-operator by f | κ,ρ L ( γ, φ )( τ ) := φ ( τ ) − κ ρ − L ( γ, φ ) f ( γτ ) , for a function f : H → C [ L ′ /L ] and ( γ, φ ) ∈ e Γ. Following [6], a smooth function f : H → C [ L ′ /L ]is called a harmonic Maass form of weight κ with respect to ρ L if it is annihilated by the weight κ Laplace operator ∆ κ := − v (cid:18) ∂ ∂u + ∂ ∂v (cid:19) + iκv (cid:18) ∂∂u + i ∂∂v (cid:19) , if it is invariant under the slash-operator | κ,ρ L , and if there exists a C [ L ′ /L ]-valued Fourier polyno-mial (the principal part of f ) P f ( τ ) := X µ ∈ L ′ /L X n ≤ c + f ( µ, n ) e ( nτ ) e µ such that f ( τ ) − P f ( τ ) = O ( e − εv ) as v → ∞ for some ε >
0. We denote the vector space ofharmonic Maass forms of weight κ with respect to ρ L by H κ,L , and we let M ! κ,L be the subspaceof weakly holomorphic modular forms. Every f ∈ H κ,L can be written as a sum f = f + + f − of aholomorphic and a non-holomorphic part. These have Fourier expansions of the form f + ( τ ) = X µ ∈ L ′ /L X n ≫−∞ c + f ( µ, n ) e ( nτ ) e µ , f − ( τ ) = X µ ∈ L ′ /L X n< c − f ( µ, n )Γ(1 − κ, π | n | v ) e ( nτ ) e µ , where Γ( s, x ) := R ∞ x t s − e − t dt denotes the incomplete Gamma function.The antilinear differential operator ξ κ = 2 iv κ ∂∂τ from the introduction maps a harmonic Maassform f ∈ H κ,L to a cusp form of weight 2 − κ for ρ L − . We further require the Maass loweringand raising operators L κ := − iv ∂∂τ and R κ = 2 i ∂∂τ + κv , which lower (resp. raise) the weight of asmooth function transforming like a modular form of weight κ for ρ L by two. JOSHUA MALES
Maass–Poincar´e series.
Let κ ∈ Z with κ <
0, and denote by M µ,ν the usual M -Whittakerfunction (see [1, equation 13.1.32]). We define, for s ∈ C and y ∈ R \{ } , M κ,s ( y ) := | y | − κ M sgn( y ) κ ,s − ( | y | ) . (2.1)Following [4], for µ ∈ L ′ /L and m ∈ Z − q ( µ ) with m > F µ, − m,κ,s ( τ ) := 12Γ(2 s ) X ( γ,φ ) ∈ e Γ ∞ \ e Γ ( M κ,s ( − πmv ) e ( − mu ) e µ ) | κ,ρ L ( γ, φ )( τ ) . The series converges absolutely for Re( s ) >
1, and at the point s = 1 − κ , the function F µ, − m,κ ( τ ) := F µ, − m,κ, − κ ( τ )defines a harmonic Maass form in H κ,L with principal part e ( mτ )( e µ + e − µ ) + c for some constant c ∈ C [ L ′ /L ]. Let H cusp κ,L be the subspace of H κ,L that maps to a cusp form under ξ κ . Then it is aclassical fact that for κ < f ∈ H cusp κ,L may be written as f ( τ ) = 12 X h ∈ L ′ /L X m ≥ c + f ( h, − m ) F h,m,κ ( τ ) + c , where c is a constant in C [ L ′ /L ] which may be non-zero only if κ = 0.The following lemma is [2, Lemma 2.1], and follows inductively from [5, Proposition 3.4]. Lemma 2.1.
For n ∈ N we have that R nκ ( F µ, − m,κ,s ) ( τ ) = (4 πm ) n Γ (cid:0) s + n + κ (cid:1) Γ (cid:0) s + κ (cid:1) F µ, − m,κ +2 n,s ( τ ) . Operators on vector-valued modular forms.
For an even lattice L we let A κ,L be thespace of C [ L ′ /L ]-valued smooth modular forms (i.e., modular forms which possess derivatives of allorders) of weight κ with respect to the representation ρ L .Let K ⊂ L be a sublattice of finite index. Since we have the inclusions K ⊂ L ⊂ L ′ ⊂ K ′ wetherefore have L/K ⊂ L ′ /K ⊂ K ′ /K , hence the natural map L ′ /K → L ′ /L, µ ¯ µ . For µ ∈ K ′ /K and f ∈ A κ,L , and g ∈ A κ,K , define( f K ) µ := ( f ¯ µ if µ ∈ L ′ /K, µ L ′ /K, (cid:0) g L (cid:1) ¯ µ = X α ∈ L/K g α + µ , where µ is a fixed preimage of ¯ µ in L ′ /K . The following lemma may be found in [8, Section 3]. Lemma 2.2.
There are two natural maps res
L/K : A κ,L → A κ,K , f f K , tr L/K : A κ,K → A κ,L , g g L , such that for any f ∈ A κ,L and g ∈ A κ,K , we have h f, g L i = h f K , g i . Rankin–Cohen brackets.
Let K and L be even lattices. For n ∈ N and functions f ∈ A κ,K and g ∈ A ℓ,L with κ, ℓ ∈ Z the n -th Rankin–Cohen bracket is defined by[ f, g ] n := 1(2 πi ) n X r,s ≥ r + s = n ( − r Γ( κ + n )Γ( ℓ + n )Γ( s + 1)Γ( κ + n − s )Γ( r + 1)Γ( ℓ + n − r ) f ( r ) ⊗ g ( s ) , where f ( r ) = ∂ r ∂τ r f . The tensor product of two vector-valued functions f = P µ f µ e µ ∈ A κ,K and g = P ν g ν e ν ∈ A ℓ,L is defined by f ⊗ g := X µ,ν f µ g ν e µ + ν ∈ A κ + ℓ,K ⊕ L . IGHER SIEGEL THETA LIFTS ON LORENTZIAN LATTICES AND HARMONIC MAASS FORMS 5
The following proposition is [5, Proposition 3.6].
Proposition 2.3.
Let f ∈ H κ,K and g ∈ H ℓ,L be harmonic Maass forms. For n ∈ N we have ( − π ) n L κ + ℓ +2 n ([ f, g ] n ) = Γ( κ + n )Γ( n + 1)Γ( κ ) L κ ( f ) ⊗ R nℓ ( g ) + ( − n Γ( ℓ + n )Γ( n + 1)Γ( ℓ ) R nκ ( f ) ⊗ L ℓ ( g ) . Theta functions.
For a positive definite lattice (
K, q ) of rank n we define the vector-valuedtheta function Θ K ( τ ) := X µ ∈ K ′ /K X X ∈ K + µ e ( q ( X ) τ ) e µ . The function Θ K is a holomorphic modular form of weight n for the Weil representation ρ K .For an even lattice L of signature ( s, r ) the Siegel theta function is defined byΘ L ( τ, z ) := v r X µ ∈ L ′ /L X X ∈ L + µ e ( q ( X z ) + q ( X z ⊥ )) e µ . (2.2)The Siegel theta function transforms in τ like a modular form of weight s − r for ρ L and is invariantin z under the subgroup Γ L of the orthogonal group O( L ) which fixes the classes of L ′ /L .If K ⊂ L is a sublattice of finite index, Lemma 2.2 implies thatΘ L = (Θ K ) L . (2.3)As in [7], we call w ∈ Gr( L ) a special point if it is defined over Q (i.e. w ∈ L ⊗ Q ). Then theorthogonal complement of w in V is defined over Q as well, and we denote it by w ⊥ . We havethe splitting L ⊗ Q = w ⊕ w ⊥ . Let P = L ∩ w and N = L ∩ w ⊥ be the corresponding positivedefinite one-dimensional and negative definite n -dimensional sublattices of L , and note that P ⊕ N has finite index in L . A direct computation shows that the evaluation of the Siegel theta functionat w splits as Θ P ⊕ N ( τ, w ) = Θ P ( τ ) ⊗ v Θ N − ( τ ) . (2.4)2.7. Unary theta functions.
Let d ∈ N . Then the lattice Z ( d ) = ( Z , dx ) is one-dimensionalpositive-definite, has level 4 d , and its discriminant group is isomorphic to Z / d Z with the quadraticform x x / d . We define the unary theta series by θ ,d ( τ ) := X r (mod 2 d ) X n ∈ Z n ≡ r (mod 2 d ) q n d e r . It is a holomorphic modular form of weight for the Weil representation of Z ( d ).3. The higher Siegel theta lift
In this section we compute the lift Λ reg j in two different ways (as well as offering a pathway as tohow one could compute its Fourier expansion).3.1. A series representation.
We first obtain an expression for Λ reg j ( F µ, − m,k − j , z ) as a series.Recall that k = − n . Theorem 3.1.
Assume that q ( X z ) = 0 for every X ∈ L + µ with q ( X ) = − m . We have Λ reg j ( F µ, − m,k − j , z ) = 2(4 πm ) n +12 +2 j Γ(1 + j )Γ( n + j )Γ (cid:0) n +32 + 2 j (cid:1) × X X ∈ L + µq ( X )= − m ( − πq ( X z ⊥ )) − n − j F (cid:18) n j, j ; n + 32 + 2 j ; m q ( X z ⊥ ) (cid:19) . The series on the right-hand side converges absolutely.
JOSHUA MALES
Proof.
We consider the regularized theta lift of the Maass Poincar´e series F µ, − m,k − j,s . ApplyingLemma 2.1 we obtainΛ reg ( F µ, − m,k − j,s , z ) = (4 πm ) j Γ (cid:0) s + k (cid:1) Γ (cid:0) s + k − j (cid:1) Z reg F D F µ, − m,k,s ( τ ) , Θ L ( τ, z ) E v k dµ ( τ ) . The usual unfolding argument yields the above expression as2(4 πm ) j Γ (cid:0) s + k (cid:1) Γ(2 s )Γ (cid:0) s + k − j (cid:1) Z ∞ Z M k,s ( − πmv ) e ( − mu )Θ L,µ ( τ, z ) v k − dudv, where Θ L,µ denotes the µ -th component of Θ L . Inserting the Fourier expansion of Θ L given in(2.2) and the definition of M − k,s given in (2.1), after evaluating the integral over u , this is2(4 πm ) j − k Γ (cid:0) s + k (cid:1) Γ(2 s )Γ (cid:0) s + k − j (cid:1) X X ∈ L + µq ( X )= − m Z ∞ M − k ,s − (4 πmv ) v − − k e − πv ( q ( X z ) − q ( X z ⊥ )) dv. The integral is an inverse Laplace transform and can be computed using equation (11) on page 215of [9]. We obtain2(4 πm ) s + j − k Γ (cid:0) s + k (cid:1) Γ( s − − k )Γ(2 s )Γ (cid:0) s + k − j (cid:1) X X ∈ L + µq ( X )= − m ( − πq ( X z ⊥ )) k + − s F (cid:18) s − − k , s + k s ; m q ( X z ⊥ ) (cid:19) . To obtain the statement of the theorem, we plug in the point s = 1 − k + j . Convergence followssimilarly to that of [7, Theorem 3.1] (cid:3) Remark.
In the case of j = 0, we recover the results of [7]. For small values of n, j the Gausshypergeometric function on the right-hand side can be evaluated explicitly.For example, choose n = 1. Then by [12, Eq. 15.4.17] and using that F is symmetric in thefirst two arguments, we have that F (cid:18) j,
12 + j ; 2 + 2 j ; m q ( X z ⊥ ) (cid:19) =
12 + 12 (cid:18) − m q ( X z ⊥ ) (cid:19) ! − − j . For other values of n and j , one can use transformations of F along with the contiguous hyper-geometric functions to obtain a (lengthy) description in terms of linear combinations of elementaryfunctions. The coefficients in the linear combination are rational.Using Theorem 3.1 and the fact that any f ∈ H cusp k − j,L can be written as a linear combination ofthe Maass Poincar´e series as in Section 2.3, one directly obtains a series representation of the liftΛ reg j ( f, z ) as (4 π ) + j Γ(1 + j )Γ( n + j )Γ (cid:0) n +32 + 2 j (cid:1) X X ∈ L ′ q ( X ) < c + f ( X, q ( X )) q ( X ) n +12 +2 j ( − q ( X z ⊥ )) − n − j × F (cid:18) n j, j ; n + 32 + 2 j ; m q ( X z ⊥ ) (cid:19) . (3.1)3.2. Evaluating the theta lift at special points.
We now evaluate the theta integral at specialpoints. Recall that at a special point w we have the positive-definite one-dimensional lattice P = L ∩ w and negative-definite n dimensional lattice N = L ∩ w ⊥ .Recall that G P denotes a harmonic Maass form of weight for ρ P that maps to Θ P under ξ .We denote its holomorphic part by G + P . For simplicity, we now assume that the input f for theregularized theta lift is weakly holomorphic (if this is not the case, then one obtains another integralon the right-hand side). IGHER SIEGEL THETA LIFTS ON LORENTZIAN LATTICES AND HARMONIC MAASS FORMS 7
Theorem 3.2.
Let f ∈ M ! k − j,L . We have Λ reg j ( f, w ) = π Γ (1 + j )2Γ (cid:0) + j (cid:1) ( − π ) j CT (cid:16)D f P ⊕ N ( τ ) , (cid:2) G + P ( τ ) , Θ N − ( τ ) (cid:3) j E(cid:17) . Remark.
For fixed j , one can evaluate the constant term as sums over lattice vectors as in [7, Remark3.6]. However, the terms arising from the Rankin-Cohen bracket quickly become unwieldy for largevalues of j . Proof of Theorem 3.2.
The proof is similar to those of [5, Theorem 5.4] and [2, Theorem 4.1] andso we only sketch the details, for the convenience of the reader. Note that Lemma 2.2 and (2.3)imply that h f, Θ L i = (cid:10) f, (Θ P ⊕ N ) L (cid:11) = h f P ⊕ N , Θ P ⊕ N i . Thus we may assume that L = P ⊕ N if f is replaced by f P ⊕ N . For simplicity, we write just f instead of f P ⊕ N throughout.First, using the self-adjointness of the raising operator (see [4, Lemma 4.2]) we obtain Z reg F D R jk − j ( f )( τ ) , Θ( τ, w ) E v k dµ ( τ ) =( − j Z reg F D f ( τ ) , R j − k (cid:16) v k Θ L ( τ, w ) (cid:17)E dµ ( τ ) . The apparent boundary term appearing disappears in the same way as in the proof of [4, Lemma 4.4].Using the splitting (2.4) of the Siegel theta function and the formula R ℓ − κ (cid:16) v κ g ( τ ) ⊗ h ( τ ) (cid:17) = v κ g ( τ ) ⊗ R ℓ ( h )( τ )which holds for every holomorphic function g , every smooth function h , and κ, ℓ ∈ R , we obtain R j − k (cid:16) v − k Θ P ⊕ N ( τ, w ) (cid:17) = L ( G P )( τ ) ⊗ R j − k (Θ N − )( τ ) . Since L (Θ N − ) = 0, Proposition 2.3 implies that L ( G P )( τ ) ⊗ R j − k (Θ N − )( τ ) = π Γ ( j + 1)2Γ (cid:0) + j (cid:1) ( − π ) j L − k +2 j (cid:16) [ G P ( τ ) , Θ N − ( τ )] j (cid:17) . Therefore we have that Z reg F D R jk − j ( f )( τ ) , Θ L ( τ, w ) E v k dµ ( τ )= π Γ ( j + 1)2Γ (cid:0) + j (cid:1) ( − π ) j Z reg F D f ( τ ) , L − k +2 j (cid:16) [ G P ( τ ) , Θ N − ( τ )] j (cid:17)E dµ ( τ ) . Applying Stokes’ Theorem as in the proof of [6, Proposition 3.5] yields the result, noting thatapparent boundary terms arising vanish because f is weakly holomorphic. (cid:3) Remark.
Note that one can obtain similar formulae by considering similar lifts but replacing Θ L by the modified theta function Θ ∗ L as in [7]. In particular, one can obtain formulae involving theclassical mock theta functions of Ramanujan. The arguments run analogously to those presentedhere.3.3. The Fourier expansion.
To end this section, we remark how one could obtain the Fourierexpansion of the lift Λ reg j ( f, z ) for any f ∈ H k − j,L . This, however, would be a lengthy butcompletely clear calculation that we do not require in the present paper, and so we omit thedetails.Firstly, note that Lemma 2.1 implies that one can rewrite the lift as a linear combination of thelift Z reg F D F µ, − m,k,s ( τ ) , Θ L ( τ, z ) E v k dµ ( τ ) . JOSHUA MALES
Recall that we evaluate at s = 1 − k + j . In the case of j = 0 the Fourier expansion of this lift wascalculated by Bruinier in his celebrated habilitation [4]. One could then proceed as in the proofof [4, Theorem 2.15 and Proposition 3.1] to obtain a Fourier expansion for Λ reg ( f, z ) which is of asimilar shape to the j = 0 case. Finally, one could also translate this to the hyperbolic model ofthe Grassmanian in the same fashion as [7, Theorem 3.3].4. An example on an isotropic lattice of signature (1 , n = 1 and compute an explicit example in the spirit of Bruinier–Schwagenscheidt.For any f ∈ H cusp k − j,L by (3.1) we have thatΛ reg j ( f, z ) = 4 j π + j Γ(1 + j )Γ (cid:0) + j (cid:1) Γ(2 + 2 j ) X X ∈ L ′ q ( X ) < c + f ( X, q ( X )) (cid:16)p q ( X z ⊥ ) − p q ( X z ) (cid:17) j . (4.1)Take the rational quadratic space Q with quadratic form Q ( a, b ) = ab . Then a lattice L in Q has signature (1 ,
1) and is isotropic. For a special point w we follow [7] and let y = ( y , y ) ∈ L be its primitive generator with y , y >
0. Let y ⊥ ∈ L be the primitive generator of w ⊥ withpositive second coordinate. Then y ⊥ is a positive multiple of ( − y , y ). Defining d P := y y and d N := y ⊥ y ⊥ we have P ∼ = ( Z , d P x ) and N ∼ = ( Z , − d N x ). As in [7], one may identifyΘ P = θ ,d P , Θ N − = θ ,d N . We elucidate the theorem in the case that j = 1 - for other j similar theorems hold, althoughcomputing the Rankin-Cohen bracket explicitly becomes a lengthy exposition. Theorem 4.1.
Suppose that L is as above and that f is weakly holomorphic. Choose j = 1 . Theevaluation of the higher theta lift at w = R ( y , y ) ∈ Gr( L ) is given by Λ reg1 ( f, w ) = 2 π y y X X =( X ,X ) ∈ L ′ X X < c + f ( X, X X ) min ( | X y | , | X y | ) = 4 π X α ∈ P ′ /Pβ ∈ N ′ /Nα + β ∈ L ′ X λ ∈ N + β X n ∈ Z − q ( λ ) c + f ( α + β, − n ) c + G P ( α, n + q ( λ ))( n + q ( λ ))+ 4 π X α ∈ P ′ /Pβ ∈ N ′ /Nα + β ∈ L ′ X λ ∈ N + β X n ∈ Z − q ( λ ) c + f ( α + β, − n ) c + G P ( α, n + q ( λ )) q ( λ ) . Proof of Theorem 4.1.
To obtain the first line of the theorem, we note the following explicit for-mulae (recalled from [7] for the convenience of the reader). We have(
X, y ) = X y + X y , − X X | y | + ( X, y ) = ( X y − X y ) , along with q ( X w ) = 12 ( X, y ) | y | , | q ( X w ⊥ ) | = − X X + 12 ( X, y ) | y | . Combining these yields p q ( X w ⊥ ) − p q ( X w ) = 2 min ( | X y | , | X y | ) . IGHER SIEGEL THETA LIFTS ON LORENTZIAN LATTICES AND HARMONIC MAASS FORMS 9
We then plug this into (4.1) with j = 1. To obtain the second line, we first need to compute theRankin-Cohen bracket. First note that[ G + P ( τ ) , Θ N − ( τ )] = 14 πi (cid:18) − ∂∂τ G + P ( τ ) ⊗ Θ N − ( τ ) + 3 G + P ( τ ) ⊗ ∂∂τ Θ N − ( τ ) (cid:19) . We have that ∂∂τ G + P ( τ ) = 2 πi X α ∈ P ′ /P X n ∈ Q c + G P ( α, n ) ne ( nτ ) e α ,∂∂τ Θ N − ( τ, w ) = − πi X β ∈ N ′ /N X λ ∈ N + β q ( λ ) e ( − q ( λ ) τ ) e β . Plugging in Fourier expansions, we find that[ G + P ( τ ) , Θ N − ( τ )] = − X α ∈ P ′ /P X n ∈ Q c + G P ( α, n ) ne ( nτ ) e α ⊗ X β ∈ N ′ /N X λ ∈ N + β e ( − q ( λ ) τ ) e β − X α ∈ P ′ /P X n ∈ Q c + G P ( α, n ) e ( nτ ) e α ⊗ X β ∈ N ′ /N X λ ∈ N + β q ( λ ) e ( − q ( λ ) τ ) e β = − X α ∈ P ′ /Pβ ∈ N ′ /N X λ ∈ N + β X n ∈ Q − q ( λ ) c + G P ( α, n + q ( λ ))( n + q ( λ )) e ( nτ ) e α + β − X α ∈ P ′ /Pβ ∈ N ′ /N X λ ∈ N + β X n ∈ Q − q ( λ ) c + G P ( α, n + q ( λ )) e ( nτ ) q ( λ ) e α + β Along with the Fourier expansion of f and the definition of h , i , this yields the second line of thetheorem. (cid:3) Example . Let L be a hyperbolic plane, so it is spanned by the vectors (1 ,
0) and (0 , L is unimodular, and vector-valued forms here are actually scalar-valued forms for SL ( Z ). Let y = ( y , y ) ∈ N with gcd( y , y ) = 1. This means that y ⊥ = ( − y , y ) and d := d N = d P = y y .Recall that Θ P = θ ,d , Θ N − = θ ,d . By results of Zagier [13], the generating function G + P ( τ ) = − π X n ≥ n ≡ H ( n ) e (cid:16) nτ (cid:17) e − π X n ≥ n ≡ H ( n ) e (cid:16) nτ (cid:17) e is the holomorphic part of a harmonic Maass form G P of weight for the dual Weil representation ρ L − which maps to Θ P under ξ . Using Theorem 4.1 at the special point y = (1 , f = f t , some simplification yields the classical relation X n ∈ Z (cid:0) t − n (cid:1) H (cid:0) t − n (cid:1) = X a,b ∈ N ab = t min ( a, b ) Further examples could also be computed by combining the techniques in [7] with those in thecurrent paper. As remarked previously, for small values of j , these can be written explicitly, butfor larger values of j the terms arising from the Rankin–Cohen brackets quickly become unwieldy. With x = ( x , . . . , x j ) ∈ Z j , p some polynomial and Q a quadratic form, one could obtain formulaefor sums of the shape X x ∈ Z j p ( x ) H ( t − Q ( x ))in terms of linear combinations (with rational coefficients) of divisor power sums. It is also clearthat one could find examples for other generating functions of weight , for examples those listedin the introduction. References [1] M. Abramowitz and I. Stegun,
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