HARMONIC EISENSTEIN SERIES OF WEIGHT ONE 7 YINGKUN LI 1 0 2 n a Abstract. Inthisshortnote,wewillconstructaharmonicEisensteinseriesofweightone, J whose image under the ξ-operator is a weight one Eisenstein series studied by Hecke [6]. 1 3 ] 1. Introduction T N In the theory of automorphic forms, Eisenstein series occupy an important place. Holo- . morphic Eisenstein series can be explicitly constructed and are usually the first examples of h t modular forms people encounter. Furthermore, their constant Fourier coefficients are special a m values of the Riemann zeta function, whereas the non-constant coefficients are the sums of [ the powers of divisors. Modularity then connects these two types of interesting quantities together. 1 v Holomorphic theta series constructed from positive definite lattices provide another source 0 of modular forms besides Eisenstein series. In [12], Siegel introduced non-holomorphic theta 4 9 series associated to indefinite lattices, and showed that they can be integrated to produce 8 Eisenstein series. Later in his seminal work [13], Weil studied this phenomenon for algebraic 0 groups, and deduced the famous Siegel-Weil formula. . 1 In the setting of theta correspondence between the orthogonal and sympletic groups, the 0 7 Siegel-Weil formula is an equality between the integral of a theta function on the orthogonal 1 side and an Eisenstein series on the symplectic side. With the knowledge of the theta kernel, : v one can then construct various symplectic Eisenstein series. A prototypical example of such i X a construction was already carried out by Hecke around 1926 [6], where he constructed a r theta kernel Θ(τ,t) from an indefinite lattice of signature (1,1) and integrated it to produce a a holomorphic modular form ϑ(τ) of weight one. This is an Eisenstein series if the lattice is isotropic and a cusp form otherwise. In [7], Kudla extended this construction to produce holomorphic Siegel modular forms of genus g and weight g+1, as a prelude to the important 2 works by Kudla and Millson later [9, 10]. In this note, we will consider a different theta kernel Θ˜(τ,t) for an isotropic, indefinite lattice of signature (1,1). Rather than holomorphic, its integral in t is a harmonic function Date: February 1, 2017. This work was partially supported by the DFG grant BR-2163/4-2and an NSF postdoctoral fellowship. 1 2 YINGKUNLI ˜ ϑ(τ) and related to the holomorphic Eisenstein series ϑ(τ) constructed by Hecke via ˜ ξϑ(τ) = ϑ(τ), where ξ = ξ is the differentiable operator introduced by Bruinier and Funke [4]. In the 1 notion loc. cit., ϑ˜(τ) is a harmonic Maass form of weight one. For any k 1Z, a harmonic ∈ 2 Maass formof weight k is a real analytic functions on the upper half-plane := τ = u+iv : H { v > 0 that transforms with weight k with respect to a discrete subgroup of SL (R), and is 2 } annihilated by the weight k hyperbolic Laplacian ∂f ∆ := ξ ξ , ξ (f) := 2ivk . k 2 k k k − − ◦ ∂τ HarmonicMaassformscanbewrittenasthesumofaholomorphicpartandanon-holomorphic part. The Fourier coefficients of their holomorphic parts are expected to contain interest- ing arithmetic information concerning the ξ -images of the non-holomorphic parts (see e.g. k [2, 5]). In [11], Kudla, Rapoport and Yang considered an Eisenstein series, which is harmonic. The Fourier coefficients of its holomorphic part are logarithms of rational numbers, and can be interpreted as the arithmetic degree of special divisors on an arithmetic curve. In view of their work and the Kudla program [8], we expect the Fourier coefficients of the harmonic Eisenstein series we construct to have a similar interpretation as well. The idea to construct ϑ˜(τ) is rather straightforward. If we can construct a function Θ˜(τ,t) such that it is modular in τ and satisfies ξΘ˜(τ,t) = Θ(τ,t) for each t, then simply integrating ˜ it in t will produce the desirable ϑ(τ). This idea has already been used in [3], where ξ 1/2 connected the theta kernels constructed from the Gaussian and the Kudla-Millson Schwartz form. In our setting, we will introduce an L function ϕ˜ , which is a ξ-preimage of the ∞ τ Schwartz function used in constructing Θ(τ,t) under ξ (see Prop. 3.4). We will then use this ˜ function to form a theta kernel Θ(τ,t) and integrate it to obtain the harmonic Eisenstein ˜ series ϑ(τ) in Theorem 4.3 in the last section. Acknowledgement. The idea of the function ϕ˜ came out of discussions with Pierre τ Charollois during a visit to Universit´e Paris 6 in November 2015. I am thankful for his encouragements that led to this note. 2. Theta lift from O(1,1) to SL 2 In this section, we will recall the construction of the Eisenstein series in [6] and [7]. For N N, letL = NZ2 bealatticewithquadraticformQ( a ) := ab andB( , ) : L L Zthe ∈ b N · · × → associated bilinear form. The dual lattice L V := L(cid:0) (cid:1)Q is then Z2 and the discriminant ∗ Q ⊂ ⊗ group is L /L = (Z/NZ)2. ∗ HARMONIC EISENSTEIN SERIES OF WEIGHT ONE 3 Let ρ be the Weil representation of SL (Z) on C[L /L]. As usual, let e : h L /L L 2 ∗ h ∗ { ∈ } denote the canonical basis of C[L /L] and e(a) := e2πia for any a C. Then the action of ∗ ∈ ρ on the generators T,S SL (Z) is given by (see e.g. [1, 4]) L 2 ∈ § 1 ρ (T)(e ) = e(Q(h))e , ρ (S)(e ) = e( (δ,h))e . L h h L h δ N − δXL∗/L ∈ The symmetric domain attached to V := L R is given by the hyperbola R ⊗ := Z V B(Z,Z) = 1 . R D { ∈ | − } We denote one of its two connected components by + and parametrize it by D Φ : R + ×+ → D N t t Z := . 7→ t r 2 (cid:18) 1/t(cid:19) − Let W := N/2 t Z . Then dΦ td = W V and W ,Z is an orthogonal basis t 1/t ∈ t⊥ dt t ∈ R { t t} of V . For pany X(cid:0)= (cid:1)x1 V , one can(cid:0)wri(cid:1)te X = X +X , where R x2 ∈ R Wt Zt (cid:0) (cid:1) t 1x +tx t 1x tx − 1 2 − 1 2 X := B(X,W )W = W , X := B(X,Z )Z = − Z . Wt t t √2N t Zt − t t √2N t Then the majorant of B( , ) associated to Z , denoted by B( , ) , is given by the positive t t · · · · definite quadratic form B(X,W )2 +B(X,Z )2 t 2x2 +t2x2 Q(X) := Q(X ) Q(X ) = t t = − 1 2. t Wt − Zt 2 2N Let R1,1 = (x,y) : x,y R be a quadratic space of signature (1,1) with the quadratic { ∈ } form Q((x,y)) = x2 y2 with associated bilinear form B ( , ). Given τ = u+iv in the ′ −2 ′ · · ∈ H upper half plane, we define the Schwartz function ϕ on R1,1 by τ ϕ : R1,1 C τ → (2.1) x2 y2 (x,y) √2v x e τ τ . 7→ · · (cid:18) 2 − 2 (cid:19) Now summing ϕ over any even, integral lattice M R1,1 of rank 2 would produce a real- τ ⊂ analytic theta series of weight 1 that transforms with respect to ρ . In our setting, we let M M be the image of L under the following isometry ι : V R1,1 t R (2.2) → X (B(X,W ),B(X,Z )) t t 7→ 4 YINGKUNLI for each t R . Now the vector-valued theta function ∈ ×+ (2.3) Θ(τ,t) := Θ (τ,t)e , Θ (τ,t) := ϕ (ι (X)) h h h τ t hXL∗/L XXL+h ∈ ∈ transforms with weight 1 and representation ρ in the variable τ by Theorem 4.1 in [1]. For L h = h1 Z2, we have explicitly h2 ∈ (cid:0) (cid:1) v x x t 2x2 +t2x2 Θ (τ,t) = (t 1x +tx )e 1 2u+ − 1 2iv h − 1 2 rN (cid:18) N 2N (cid:19) X x1 h1(N) ≡ x2 h2(N) ≡ Integrating over t R with respect to the invariant differential dt defines ∈ ×+ t 1 dt dt (2.4) ϑ (τ,s) := Θ (τ,t)ts + ∞Θ (τ,t)t s . h h h − Z t Z t 0 1 Here, the integral converges for s 0. As a function of s, it has analytic continuation to ℜ ≫ s C. Let ϑ (τ) be the constant term in the Laurent expansion of ϑ (τ,s) around s = 0. h h Th∈en ϑ (τ) is holomorphic and ϑ(τ) := ϑ (τ)e is an Eisenstein series of weight 1 h h L∗/L h h ∈ on SL (Z) and transforms with respect toPρ . It has the following Fourier expansion (see [7, 2 L Theorem 3.2]). Proposition 2.1. Write h = h1 Z2. Then ϑ (τ) has the Fourier expansion ϑ (τ) = h2 ∈ h h c (n)qn with (cid:0) (cid:1) n∈Q≥0 h P 1 h1 N ∤ h ,n h , 2 −hNi 1 | 2 c (0) := 1 h2 N h ,n ∤ h , h 2 −hNi | 1 2 (2.5) 0 otherwise,  c (n) :=  sgn(x ), n > 0. h 1 X X=(x1) L+h,Q(X)=n x2 ∈ Here x (0,1] is defined by the property x x Z. h i ∈ −h i ∈ HPr(oso,fx.)U:=se th∞ne=i0d(xen+titny)−es−2isπyth=e H√uyrRw0∞itze−zπeyt(at2+fut−n2c)t(ito+n.t−1)dtt and H(0,x) = 21 − x, wher(cid:3)e P 3. Some Special Functions In this section, we will introduce a special function ϕ˜ on R1,1 such that ξϕ˜ = ϕ . τ τ τ HARMONIC EISENSTEIN SERIES OF WEIGHT ONE 5 3.1. Non-holomorphic Part. Define the functions f : R R and ϕ : R1,1 C by τ∗ → ∗τ → (3.1) y2 x2 f (x) := sgn(x) erf √2πvx = sgn(x)erfc(√2πv x ), ϕ (x,y) := e − τ f (x). τ∗ − | | ∗τ (cid:18) 2 (cid:19) τ∗ (cid:16) (cid:17) where erf(x) := 2 xe r2dr and erfc(x) are the error and complementary functions respec- √π 0 − tively. StraightforwRard calculations show that (3.2) ξ(ϕ (x,y)) = ϕ (x,y) ∗τ − τ for all (x,y) R1,1. ∈ For each y R, the function ϕ (x,y) decays like a Schwartz function in x. Also, ϕ (x,y) ∈ ∗τ ∗τ satisfies y2 lim ϕ (x,y) lim ϕ (x,y) = 2e τ , x 0+ ∗τ −x 0− ∗τ (cid:18) 2 (cid:19) → → hence has a jump discontinuity at x = 0. Away from 0, it is smooth. Thus, we can view it as a tempered distribution on R1,1 and calculate its Fourier transform with respect to Q ′ − as follows. First, notice that as a distribution, f satisfies the differential equation τ∗ d (f (x)) = 2 δ(x) 2√2ve 2πvx2, dx τ∗ · − − where δ(x) is the Dirac delta function. This follows from d x = sgn(x), d sgn(x) = 2δ(x) dx| | dx as tempered distributions. Substituting in the definition of ϕ (x,y), we see that it satisfies ∗τ ∂ x2 y2 (3.3) (ϕ (x,y))+2πiτxϕ (x,y) = 2δ(x) 2√2ve τ e τ . ∂x ∗τ ∗τ (cid:18) − (cid:18)− 2 (cid:19)(cid:19) (cid:18) 2 (cid:19) Notice that e x2τ 2δ(x) = 2δ(x) as a distribution. − 2 (cid:16) (cid:17) For a Schwartz function φ on R1,1, we define its Fourier transform (φ) with respect to F the quadratic form Q by ′ − (3.4) (φ)(x,y) := φ(w,z)e( wx+yz)dwdz. F Z − R1,1 If φ is not a Schwartz function and the integral above converges, we also use it to denote its Fourier transform. Using the standard facts of Fourier transform (see e.g. [1, Lemma 3.1]), we have −τ∂∂xF(ϕ∗τ)(x,y)+2πixF(ϕ∗τ)(x,y) = (cid:18)2−2√2v(iτ)−1/2e(cid:18)−x22(−1/τ)(cid:19)(cid:19) e(cid:16)y2√2(−i1τ/τ)(cid:17). − 6 YINGKUNLI After dividing by τ on both sides and making the change of variable τ 1/τ, the − 7→ − equation becomes ∂ x2 y2 (ϕ )(x,y)+2πixτ (ϕ )(x,y) = 2τ √2ve τ √ iτ e τ , ∂xF ∗1/τ F ∗1/τ − (cid:18) (cid:18)− 2 (cid:19)− − (cid:19) (cid:18) 2 (cid:19) − − Now define (ϕ )(x,y) (3.5) Dτ∗(x,y) := ϕ∗τ(x,y)− F ∗−1/ττ . Then it satisfies the differential equation x2 d x2 y2 (3.6) e τ e τ (x,y) = 2 δ(x) √ iτ e τ . (cid:18)− 2 (cid:19) dx (cid:18) (cid:18) 2 (cid:19)Dτ∗ (cid:19) − − (cid:18) 2 (cid:19) (cid:16) (cid:17) We have the following result concerning the solutions to this differential equation. Proposition 3.1. For fixed τ ,y R, the only jump discontinuity of any piecewise 0 0 ∈ H ∈ continuous solution to the differential equation (3.6) is at x = 0. Suppose further that it is bounded in x. Then the solution agrees with the function (x,y ) defined by Dτ0 0 y2 x2 (3.7) (x,y ) := e 0 − τ sgn(x)erfc(√ iτ x ) Dτ0 0 (cid:18) 2 0(cid:19) − 0| | whenever the solution is continuous. In particular, (x,y) = (x,y) for all (x,y) R1,1 Dτ∗ Dτ ∈ and τ . ∈ H Remark 3.2. Here in τ = u + iv , the function erfc(√ iτ x ) is the unique holomorphic 0 0 0 0 − | | extension of erfc(√v x ). 0 | | Proof. The first claim is clear as a jump discontinuity at x = x of a piecewise solution would 0 produce a constant times δ(x x ). By the fundamental theorem of calculus, the solution, 0 − whenever continuous, would agree with (x,y ) up to a constant multiple of e x2τ , Dτ0 0 − 2 (cid:16) (cid:17) which is unbounded as x . Since (x,y ) is assumed to be bounded, the second → ∞ Dτ0 0 claim follows. Finally, for any fixed τ ,y R, we have ϕ (x,y ) L1(R). Thus its 0 ∈ H 0 ∈ ∗τ0 0 ∈ Fourier transform is continuous and bounded. That implies (x,z) is bounded and has no removable discontinuity on R, hence the third claim. Dτ∗ (cid:3) 3.2. Holomorphic Part. Now, we will define the holomorphic counterpart to ϕ as ∗τ y2 x2 (3.8) ϕ+(x,y) := e − τ sgn(x)1 , τ (cid:18) 2 (cid:19) y2>x2 where 1 is the characteristic function of the set (x,y) R1,1 : y2 > x2 . Even though y2>x2 { ∈ } ϕ+(x,y) is not a Schwartz function, it decays nicely enough such that we have the following τ result. HARMONIC EISENSTEIN SERIES OF WEIGHT ONE 7 Proposition 3.3. The following integral (ϕ+)(x,y) := ϕ+(w,z)e( wx+zy)dwdz F τ Z τ − R1,1 converges uniformly on compact subsets of (x,y) R1,1 : x2 = y2 . Furthermore, the { ∈ 6 } function (ϕ+) is bounded and continuously differentiable on (x,y) R1,1 : x2 = y2 . F τ { ∈ 6 } Proof. Let A = 1 ( 1 1) and make the rotational change of variables √2 1 1 − a w x x ′ := A , := A , (cid:18)b(cid:19) ·(cid:18)z(cid:19) (cid:18)y (cid:19) ·(cid:18)y(cid:19) ′ we can rewrite the integral above as T′ T (ϕ+)(x,y ) = lim 1 e(abτ)sgn(a b)e(ay +bx)dbda. F τ ′ ′ T,T′→∞Z T′Z T ab>0 − ′ ′ − − The integral over 0 < a < T ,0 < b < T can be evaluated explicitly as ′ T′ e(a2τ +a(x +y )) e((aτ +x)T +ay ) e(ay ) ′ ′ ′ ′ ′ da. Z πi(aτ +x) − 2πi(aτ +x) − 2πi(aτ +x) 0 ′ ′ ′ The same can be done for the region T < a < 0, T < b < 0. As T , the middle ′ − − → ∞ term vanishes, and we are left with T′ e(a2τ +a(x +y )) e(ay ) (3.9) (ϕ+)(x,y ) = lim ′ ′ ′ da. F τ ′ ′ T′→∞Z T′ πi(aτ +x′) − 2πi(aτ +x′) − The integral of the first term can be bounded with ∞ √re2−+r(2x′)2dr ≪ |x′|−1, which implies R−∞ that the integral converges uniformly and defines a continuously differentiable function away from xy = 0. Furthermore, it is bounded when x is large. On the other hand when x is ′ ′ ′ ′ | | | | close to zero, we can fix an absolute constant ǫ > 0 such that the integral over a (ǫ, ) | | ∈ ∞ converges absolutely independent of x,y . The rest of the integrand can be written as ′ ′ e(a2τ +a(x +y )) e(a2τ a(x +y )) x sin(ay ) ′ ′ ′ ′ ′ ′ + − = C (a,x,τ) +C (a,x,τ) aτ +x aτ +x 1 ′ (aτ)2 (x)2 2 ′ a ′ ′ ′ − − with C (a,x,τ) bounded above independently of a and x. Since ǫ x′ da and | j ′ | ′ | 0 (aτ)2 (x′)2 | ǫ sin(ay′)da are bounded independent of x and y , the integral of the firRst term−in Eq. (3.9) | 0 a | ′ ′ dRefines a bounded and continuously differentiable function on (x,y) R1,1 : x2 = y2 . { ∈ 6 } Away from xy = 0, the integral of the last term converges uniformly using integration by ′ ′ parts and defines a continuous function. Using standard formula in one dimensional Fourier transform, we can in fact evaluate it explicitly as e(ay ) ∞ ′ da = τ 1e(xy ( 1/τ))1 . − ′ ′ x′y′>0 Z 2πi(aτ +x) − ′ −∞ 8 YINGKUNLI (cid:3) From this, it is clear that it is bounded. Since ϕ+ is bounded, integrating against it defines a tempered distribution on R1,1. Thus, τ we can then study its Fourier transform (ϕ+) as we have done for ϕ . The analogue to F τ ∗τ equation (3.3) is as follows ∂ϕ+ y2 x2 (3.10) τ +2πiτxϕ+ = (2δ(x) δ(x y) δ(x+y))e − τ . ∂x τ − − − (cid:18) 2 (cid:19) Applying Fourier transform to both sides and making the change τ 1/τ yields 7→ − ∂ (ϕ+ )(x,y) y2 ( 1/τ) F −1/τ 2πix (ϕ+ )(x,y) = 2√ iτe τ δ(y x) δ(x+y) − ∂x − F 1/τ −(cid:18) − (cid:18) 2 (cid:19)− − − (cid:19) − Subtracting the previous two equations shows that the function defined by (ϕ+ )(x,y) (3.11) +(x,y) := ϕ+(x,y) F −1/τ Dτ τ − τ also satisfies the differential equation (3.6). Note that δ(y x) = δ(y x)e(y2 x2) and − ± ± 2 δ(x)e(x2τ) = δ(x). For each fixed τ , the function ϕ+ is bounded with only jump 2 0 ∈ H τ0 singularities when either x2 = y2 or x = 0. Proposition 3.3 implies that (ϕ+) has the same F τ property as well. So we can define (ϕ+)(x,y) on x2 = y2 such that +(x,y) is continuous F τ Dτ when y2 = x2 > 0. By Proposition3.1, weknow that + = andhave proved the following Dτ Dτ result. Proposition 3.4. For all (x,y) R1,1, the L (R1,1) function ∞ ∈ y2 x2 (3.12) ϕ˜ (x,y) := ϕ+(x,y) ϕ (x,y) = sgn(x)e − 1 erfc(√2πv x ) τ τ − ∗τ (cid:18) 2 (cid:19) y2>x2 − | | (cid:16) (cid:17) satisfies (1) ϕ˜ (x,y) = e(y2 x2)ϕ˜ (x,y), τ+1 −2 τ (2) (ϕ˜ )(x,y) = τϕ˜ (x,y), 1/τ τ F − (3) ξ(ϕ˜ (x,y)) = ϕ (x,y). τ τ 4. Real-Analytic Theta Series ˜ In this section, we will construct weight 1 real-analytic theta series ϑ(τ) that transforms with respect to ρ and maps to ϑ(τ) under ξ. Proposition 3.4 and the construction of ϑ (τ) L h − imply that we need to consider summing ϕ˜ (ι (X)) over X L+h and integrating over R τ t ∈ ×+ with respect to dt. However, the sum and integral are both divergent. The problem with t the sum is caused by isotropic elements in L. We will regularize the sum by considering a slight shift of the lattice L, and regularize the integral by adding the converging factor ts as usual. The ideas are simple, but the procedure to carry it out is a bit complicated. HARMONIC EISENSTEIN SERIES OF WEIGHT ONE 9 In the notations of sections 2 and 3, define the following series 1 Θ˜ (τ,t;ε,ε) := ϕ˜ (ι (X))e B X,ε (4.1) h ′ τ t ′ (cid:18) (cid:18) (cid:18)1(cid:19)(cid:19)(cid:19) X LX+h+ε(1) ∈ 1 for ε,ε ( 1, 1). This series converges for ε,ε ( 1, 1). It is modular for ε,ε ′ ∈ −2 2 ′ ∈ −2 2 ′ ∈ ( 1, 1) 0 , but not continuous at ε = 0. Define Θ and Θ+ as Θ˜ in (4.1) with ϕ˜ −2 2 \{ } ∗h h h τ replaced by ϕ and ϕ+ respectively. To preserve the modularity, we define the theta series ∗τ τ Θ˜(τ,t) = Θ˜ (τ,t)e by h L∗/L h h ∈ P (4.2) Θ˜ (τ,t) := c (0)+Θ˜ (τ,t;0,0) = c (0)+Θ+(τ,t;0,0)+Θ (τ,t;0,0), h h h h h ∗h where c (0) is defined in Proposition 2.1. They have the following relationship. h Proposition 4.1. For fixed τ , t R and h = h1 Z2, the series Θ˜ (τ,t;ε,ε) ∈ H ∈ ×+ h2 ∈ h ′ converges uniformly for (ε,ε) in compact subsets of ( 1,(cid:0)1)(cid:1) 0 ( 1, 1). It is continuous ′ −2 2 \{ }× −2 2 for (ε,ε) (0,min 1 , t2 ) ( 1, 1) and satisfies ′ ∈ {1+t2 1+t2} × −2 2 c (h) (4.3) lim Θ˜h(τ,t;ε, ε) −1 = Θ˜h(τ,t), ε 0+ ± − 2πi(τ 1)ε → ∓ where c (h) 0,1,2 is the number of h ,h that are divisible by N. 1 1 2 − ∈ { } Proof. Since ϕ decays like a Schwartz function, the series Θ converges absolutely and ∗τ ∗h uniformly for ε,ε R, except for h = 0 , in which case, we have lim Θ (τ,t;ε, ε) = Θ (τ,t;0,0)+1. ′F∈or Θ+, notice that B(cid:0)(0X(cid:1),Z )2 B(X,W )2 = 2x1x2εi→f 0X+ =∗h x1 V±. So ∗h h t − t − N x2 ∈ R we can write (cid:0) (cid:1) (4.4) Θ+(τ,t;ε,ε) = + h ′ X X n1 NZ+h1 n1 NZ+h1 n2∈NZ+h2 n2∈NZ+h2 n1∈n2 1 (n1+ε∈)(n2+ε)<0 ≤− n1n2=0 (n +ε)(n +ε) (n +n +2ε) sgn(t 1(n +ε)+t(n +ε))e 1 2 τ e 1 2 ε . − 1 2 ′ (cid:18)− N (cid:19) (cid:18) N (cid:19) Using the inequality (n +ε)(n +ε) > n1n2 for ε ( 1, 1), we see that the first sum in − 1 2 − 2 ∈ −2 2 equation (4.4) converges absolutely and uniformly for (ε,ε) in compact subset of ( 1, 1)2. ′ −2 2 Note that the second sum is empty if and only if N ∤ h for j = 1,2. Suppose N h and j 1 | ε > 0. Then n 1 and the summand becomes 2 ≤ − ετ ε 2εε ε2τ sgn(t 1ε+t(n +ε))e − ′( n ) e ′− − 2 2 (cid:18) N − (cid:19) (cid:18) N (cid:19) 10 YINGKUNLI Sincet > 0, ε < min 1 , t2 andn 1, we havet 1ε+t(n +ε) < t 1ε+t( 1+ε) < 0. {1+t2 1+t2} 2 ≤ − − 2 − − Then the second sum is just a geometric series and equals to 2εε ε2τ e (ετ ε) h2 e ′ − − ′ h−Ni . − (cid:18) N (cid:19) (cid:0)1 e(ετ ε) (cid:1) ′ − − Using the power series expansion eax = x 1 (1 a)+O(x), we see that we get a constant −1 ex − − 2− term 1 + h2 when ε = ε. W−hen h = 0 , this is just c (0). When h = 0 , there −2 h−Ni ′ ± 6 0 h 0 are twice this contribution, which sums to c (0(cid:0))+(cid:1) 1. Now we are done since Θ˜ (τ,(cid:0)t;(cid:1)ε,ε) = h h ′ Θ+(τ,t;ε,ε) Θ (τ,t;ε,ε). (cid:3) h ′ − ∗h ′ Proposition 4.2. The theta function Θ˜(τ,t) is a real-analytic modular form in τ of weight 1 with respect to ρ and satisfies ξ(Θ˜(τ,t)) = Θ(τ,t) and Θ˜ (τ,t) = O (1) for all t R . −L h τ ∈ ×+ ˜ Proof. The property ξ(Θ (τ,t)) = Θ (τ,t) and the modularity in T are clear from the h h definition. For the modularity in S, we can apply Poisson summation to obtain Θ˜ ( 1/τ,t;ε,ε) e(2εε) (4.5) h − ′ = ′ e((δ,h))Θ˜ (τ,t; ε,ε) δ ′ τ N − δXL∗/L ∈ with ε,ε ( 1, 1) 0 . Using the identity c (h) = Pδ∈L∗/Le((δ,h))c−1(δ), we obtain the ′ ∈ −2 2 \{ } −1 N desired modularity with respect to S after setting ε = ε, subtracting c−1(h) from both ′ − 2πi(τ 1)ε sides and taking the limit ε 0+. The asymptotic of Θ˜(τ,t) in t can be−seen from its definition, the decay of ϕ , an→d the expression 4.4. (cid:3) ∗τ Now to construct the preimage of ϑ (τ) under ξ, we consider the integral h dt 1 dt (4.6) ϑ˜ (τ;s) := ∞Θ˜ (τ,t)t s + Θ˜ (τ,t)ts , h h − h Z t Z t 1 0 which converges for (s) > 0 and can be analytically continued to s C via its Fourier ℜ ∈ expansion in τ. We are interested in the function ˜ ˜ (4.7) ϑ (τ) := Const ϑ (τ;s). h s=0 h It has the following desirable properties. Theorem 4.3. The function ϑ˜(τ) := ϑ˜ (τ)e is a harmonic Maass form of weight h L∗/L h h ∈ 1 with respect to ρ , and maps to thePEisenstein series ϑ(τ). It has the Fourier expansion L ϑ˜ (τ) = c˜−(n)qn + c (0)logv c (n)Γ(0,4πvn)q n, where c (n) Q are h n∈Q≥0 h h − n∈Q>0 h − h ∈ P P