A representation theoretic approach to an eisenstein series

A REPRESENTATIONTHEORETIC APPROACH TO AN EISENSTEIN SERIES by AI XINGHUAN B.Sc (Hons), NUS A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE NATIONAL UNIVERSITY OF SINGAPORE 2010 i Summary In [1], the authors have developed a representation-theoretic approach to reveal the correspondence between certain automorphic representation and Kohnen’s plus space. Though the quoted article focuses on cusp forms, the methodology applies to a larger audience, including the Eisenstein series of half-integral weight. The current writing will carry out this approach, with necessary adjustments, to establish a similar link between certain induced representation and the Eisenstein series Hr/2 of half-integral weight. ii Acknowledgements I would to express my thanks and appreciation to Associate Professor LOKE Hung Yean, for his academic guidance and patience; Prof ZHU Cheng Bo, Prof LEE Soo Teck, Mr. TANG U-Liang for their kindness to answer to my questions; last but not least, my parents and friends who have been encouraging and supporting me during my graduate study. Contents Number-Theoretic Approach to Hr/2 1.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Modular group and congruence subgroups . . . 1.1.2 Modular functions and modular forms . . . . . 1.1.3 Eisenstein series . . . . . . . . . . . . . . . . . . 1.1.4 Slash operator . . . . . . . . . . . . . . . . . . . 1.1.5 Double coset and Hecke operator . . . . . . . . 1.2 Modular Forms of Half-Integral Weight . . . . . . . . . 1.2.1 Modular form of weight r/2 . . . . . . . . . . . 1.2.2 A 4-sheeted covering of GL+ (Q) . . . . . . . . . 1.2.3 Eisenstein series . . . . . . . . . . . . . . . . . . 1.2.4 Hecke operators on forms of half-integral weight 1.2.5 The Shimura map . . . . . . . . . . . . . . . . . 1.2.6 Kohnen’s plus space . . . . . . . . . . . . . . . Representation-theoretic approach 2.1 Induction over Adele . . . . . . . 2.2 Inductions over local fields . . . . 2.3 Main Theorem . . . . . . . . . . 2.4 Proof . . . . . . . . . . . . . . . . iii to . . . . . . . . Hr/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 10 11 12 13 . . . . 15 16 20 25 27 iv CONTENTS Chapter Number-Theoretic Approach to Hr/2 In this chapter, we introduce the basic notion of modular forms and related concepts. The Hecke eigenform Hr/2 will be discussed in its number-theoretic background. The main references are [2], [3]. 1.1 1.1.1 Modular Forms Modular group and congruence subgroups Let H = {z|Im(z) > 0} be the upper half plane of C. The special linear group SL2 (Z) acts on H by linear fractional transformation, i.e. for all γ =   a b  . Let Γ be a subgroup of   ∈ SL2 (Z), z ∈ H, we define γ · z = az+b cz+d c d 1. NUMBER-THEORETIC APPROACH TO HR/2 SL2 (Z) of finite index, then Γ is called a congruence subgroup if it contains         a b a b  1    Γ(N ) =  ≡   ∈ SL2 (Z)    c d c d    (mod N )   for some positive integer N . For our purpose, the following two congruence subgroups are the most important. Namely,     a b   Γ0 (N ) =   ∈ SL2 (Z) c ≡   c d (mod N )      and         a b   a b  1 ∗  Γ1 (N ) =   ∈ SL2 (Z)  ≡    c d c d 1.1.2    (mod N ) .   Modular functions and modular forms Let k be an integer. A complex-valued function f on H is weakly modular of weight 2k if f is meromorphic on H and satisfies the relation: f (z) = (cz + d)−2k f ( az + b ) cz + d 1.1. MODULAR FORMS   a b  for all   ∈ SL2 (Z). A meromorphic function is weakly modular of c d weight 2k if and only if f (z + 1) = f (z) and f (− ) = z 2k f (z). z In particular, since f has period 1, we can rename f to f˜ as a function of q = e2πiz . Then f (z) = f˜(q) = +∞ −∞ an q n . A weakly modular function f is called modular if it is meromorphic at infinity, which means that f˜ extends to a meromorphic function at the origin. A modular function is called a modular form if it is holomorphic everywhere, including at infinity. This means that an are zero for all n smaller than a certain integer. Furthermore, if a modular form is zero at infinity, it is called a cusp form. Here “holomorphic at infinity” means an are zero for n < and “zero at infinity” means an are zero for n ≤ 0. 1.1.3 Eisenstein series For any integer k greater than 1, an Eisenstein series of weight 2k is defined as follows Gk (z) = m,n (mz + n)2k where the summation is over all pairs of integers (m, n) other than (0, 0). The Eisenstein series Gk (z) is a modular form of weight 2k. We have Gk (∞) = 2ζ(2k) where ζ denotes the Riemann zeta function. 1. NUMBER-THEORETIC APPROACH TO HR/2 Denote by Mk the C-vector space of modular forms of weight 2k, Mk0 the C-vector space of cusp forms of weight 2k. Then dim(Mk /Mk0 ) ≤ 1. In fact, Mk = Mk0 ⊕ CGk for k ≥ 2. When k = 0, 2, 3, 4, 5, Mk is a vector space of dimension with basis 1, G2 , G3 , G4 , G5 respectively, and of course, Mk0 = for these values of k. Denote g2 = 60G2 , and g3 = 140G3 . Let ∆ = g23 − 27g32 , then multiplication by ∆ defines an isomorphism from Mk−6 onto Mk0 . 1.1.4 Slash operator Let GL+ (Q) denote the subgroup of GL2 (Q) consisting of matrices with positive determinant. Suppose f is a modular form of weight 2k. We define a “slash operator |[γ]2k ” as follows, (f |[γ]2k )(z) = (det γ)k (cz + d)−2k f (γz)   a b  + where γ =   ∈ GL2 (Q). It can be verified that (f |[γ1 ]2k )|[γ2 ]2k = c d f |[γ1 γ2 ]2k , for γ1 , γ2 ∈ GL+ (Q)]. See [2] 1.1. MODULAR FORMS 1.1.5 Double coset and Hecke operator Let ∆ be any subgroup of any group G, and let ξ ∈ G be any element of G such that ∆ and ξ −1 ∆ξ are commensurable, i.e., their intersection ∆ = ∆ ∩ ξ −1 ∆ξ is of finite index in either group. Let [∆ : ∆ ] = d and write ∆ = ∪dj=1 ∆ δj , where δj ∈ ∆. Then ∆ξ∆ = ∪dj=1 ∆ξδj . Suppose that Γ is a congruence subgroup of SL2 (Z), and let α ∈ GL+ (Q). Denote Γ = Γ ∩ α−1 Γα, d = [Γ : Γ ], and Γ = ∪dj=1 Γ γj . Suppose that f (z) is a function invariant under |[γ]2k for all γ ∈ Γ. Then we may define the action of the double coset ΓαΓ on f by d (f |[ΓαΓ]2k )(z) = (f |[αγj ]2k )(z). j=1 Let Γ = Γ1 (N ), and let n be a positive integer. Denoteby ∆ the set of 1 ∗  integer matrices of determinant n which are congruent to   modulo n N . Then we define Hecke operator Tn on a modular form f ∈ Mk (Γ1 (N )) by Tn (f ) = nk−1 f |[Γ1 (N )αΓ1 (N )]2k , where the sum runs over all double cosets of Γ1 (N ) in ∆. If n is squarefree, there is only one double coset of Γ1 (N ) in ∆. It follows that if (m, n) = 1, Tm and Tn commute and for prime number 2.2. INDUCTIONS OVER LOCAL FIELDS 21 Then          a b      ×   × {±1} a ∈ Z , b ∈ Z if F = Qp  p p     a−1 , B∩K =      {(±I , ±1)} ∼ if F = R = Z/4Z where I2 denotes the × identity matrix, and          b        if F = Qp  b ∈ Zp     N ∩B =      (I , 1) if F = R Let χ be a character of T , i.e., χ : T → C× . We may extend χ to be a character of B = T N by setting χ(tn) = χ(t), for all t ∈ T , n ∈ N . The principal series representation π of SL2 (F) is defined as Ind SL2 (F) B(F) χ=      f : SL2 (F) → C f (tng) = χ(t)f (g),     ∀t ∈ T , n ∈ N , g ∈ SL2 (F)  . 22 2. REPRESENTATION-THEORETIC APPROACH TO HR/2 In our context, the functions in this collection are C ∞ -functions if F = R, and locally constant with a compact support modulo B(F) if F = Qp , p is prime. The group SL2 (F) acts on the principal series by right translation, i.e., (π(g)f ) (x) = f (xg). The Iwasawa decomposition SL2 (F) = B(F)K(F) implies that Ind SL2 (F) B(F) χ is completely determined by χ and the values of f on K(F). In other words, as a vector space, Ind SL2 (F) B(F) χ=    f : K(F) → C   f (tng) = χ(t)f (g),    .  ∀t ∈ T ∩ K, n ∈ N ∩ K, g ∈ K  SL2 (F) We now proceed to investigate Ind B(F) χ over different fields, so that we may prepare for further discussions on an induced representation over a ring of Adeles in the later sections. Case We now choose F = Qp , where p = 2. Let Kp = SL2 (Zp )×1, then Kp is a subgroup of Kp . We have SL2 = BKp 2.2. INDUCTIONS OVER LOCAL FIELDS and 23           α β   × B ∩ K p =  , α ∈ Z , β ∈ Z .   p p   −1   α Let χ be a character trivial on B ∩ Kp , then Ind SL2 (Qp ) B(Qp ) χ=      f : Kp → C f (tng) = χ(t)f (g),    .  ∀t ∈ T ∩ Kp , n ∈ N ∩ Kp , g ∈ Kp  Note that the constant function fp (k) = 1, ∀k ∈ Kp , is a function in Ind SL2 (Qp ) B(Qp ) χ, and moreover, it is invariant under the action of Kp . We call such a vector a Kp -spherical vector and the representation a Kp -spherical representation. The spherical vector is unique. Case Let F = Q2 . In this case, we may define a γ-spherical vector. This has been explained extensively in [1]. Case If F = R, then B ∩ K = T ∩ K = {(±1, ±1)} ∼ = Z/4Z. 24 2. REPRESENTATION-THEORETIC APPROACH TO HR/2 Since K with  K = SO2 (R), we  identify       cos θ sin θ     , 1 −π < θ ≤ π ∪     − sin θ cos θ           cos θ sin θ    , −1 −2π < θ ≤ −π or π < θ ≤ 2π .      − sin θ cos θ We again have an induced representation SL2 (R) Ind B(R) χ=      f :K→C   f (xg) = χ(x)f (g),  .  ∀x ∈ T ∩ K, g ∈ K  Suppose that χ(−I2 , −1) = i (or − i), Indχ has a basis of functions {e2n+1/2 |n ∈ Z}, where e2n+1/2 (k(θ)) = e(2n+1/2)θi for k(θ) ∈ K. In fact, if s + ∈ / 2Z + 1/2, the induced representation is irreducible. If s + ∈ Z + 1/2 and s ≥ 0, the induced representation has a submodule spanned by {e2n+1/2 |2n + 1/2 ≥ s + 1}. The vector es+1 is then the lowest weight vector. The submodule is called the discrete series representation of SL2 (R). 2.3. MAIN THEOREM 2.3 25 Main Theorem    a b   For g =   ,  ∈ G(R), we define a holomorphic function on the c d upper half plane H by J(g, z) = (cz + d)1/2 , where for any complex number w, arg(w) ∈ (−π/2, π/2]. In fact, J(g1 g2 , z) = J(g1 , g2 z)J(g2 , z) for g1 , g2 ∈ G(R) by lemma 3.3 in [5]. According to Proposition 3.1 in [5] and [1], there exists a bijection Q that maps a certain space Ar/2 of functions on SL2 (Q)\G(A) to the space of cusp forms Sr/2 (Γ0 (4)) by (Qϕ)(z) = ϕ(g∞ )J(g∞ , i)r ,  √ √   y x/ y  so that g∞ i = where given z = x + iy ∈ H, g∞ =  √  1/ y x + iy = z. √ √ yi+x/ y √ 0i+1/ y = Now we revert to the series E(g, s, ϕ) defined in the first section of this chapter. Although being a function on SL2 (Q)\G(A), this function does not belong to the space Ar/2 mentioned above. However, we try to apply the same bijection map on E(g, s, ϕ). For a specific ϕ, we will arrive at Hr/2 which is of interest to many mathematicians. In the definition of E(g, s, ϕ), the summand is a function on G(A). So given a ϕ, we can express the summand as a direct product of functions on 26 2. REPRESENTATION-THEORETIC APPROACH TO HR/2 each local field, i.e. ϕχs = ϕ∞ ϕp . In order to prove the theorem, we p choose ϕ in the following way. For p = 2, let ϕp be the spherical vector we found in Case of section 2.2; let ϕ2 be a γ-spherical vector; and ϕ∞ is associated with the lowest weight vector in the induced representation in Case of section 2.2. Let r ≥ be a positive odd integer, and set s = r/2 − 1. With such ϕ, we define Hr/2 (z) = E(g∞ , s, ϕ)J(g∞ , i)r ϕ((x, sA (x))g∞ )χs+1 (xg∞ )J(g∞ , i)r , = x∈∆B(Q)\G(Q) In [6], an operator W4 commuting with Hecke operators (except T4 ) is defined by r (W4 ϕ)(z) = (−2iz)− ϕ − 4z . Theorem. Let r ≥ be a positive odd integer, then (W4 Hr/2 )(z) is a scalar multiple of Hr/2 . 2.4. PROOF 2.4 27 Proof In the definition of Hr/2 (z), x runs through the coset representatives of B(Q)\G(Q). We now consider the action of G(Q) on the collection of lines in Q2 . Then B(Q) is the stablizer of these lines. So we may represent each line by a point (c, d) it passes through, such c, d ∈ Z and gcd(c, d) = 1. This  that     ∗ ∗    enables us to identify B(Q)\G(Q) as   c, d ∈ Z, gcd(c, d) = 1, d > .    c d  For simplicity, we write ϕp ((x, 1)) as ϕp (x) for all p. Then we have Hr/2 (z) = sA (x)ϕ2 (x)ϕ∞ (xg∞ ) ( ac db )∈SL2 (Z), J(xg∞ , i)r . J(x, g∞ i)r x= (c,d)=1,d>0 We may choose a convenient scalar multiple of the lowest weight vector as ϕ∞ so that ϕ∞ (xg∞ )J(xg∞ , i)r = 1. In fact, according to the Iwasawa decomposition,    m n   cos θ sin θ  xg∞ =   , −1 m − sin θ cos θ r for some m, n ∈ R and radian θ. So J(xg∞ , i)r = m− e− irθ . On the other 28 2. REPRESENTATION-THEORETIC APPROACH TO HR/2 hand, if we set ϕ(1) = 1, then   m n  irθ ϕ(xg∞ ) = ϕ  e −1 m = ϕ(1)χs (m)e = |m|s+1 e r = m2 e irθ irθ irθ This gives ϕ∞ (xg∞ )J(xg∞ , i)r = 1. Now we examine the simplified expression sA (x)ϕ2 (x)J(x, z)−r , Hr/2 (z) = ( ac db )∈SL2 (Z), x= (c,d)=1,d>0 before applying the operator W4 to it. Note that ϕ2 is invariant on left translation by any element in B(Q). We consider the following three cases, depending on the nature of c for general x = ( ac db ). Case 1: 4|c.     −a −b  a b  If d ≡ (mod 4), then we use   instead of  . So it −c −d c d 2.4. PROOF 29   a b  suffices to consider only d ≡ (mod 4). In this case,   ∈ Γ1 (4) and c d   a b  ϕ2   = 1. c d Case 2: c ≡ (mod    4).    b   a  0 a b  Now   ∈ K0 , where K0 =   =  −2a + c −2b + d −2 c d           a b  a b  1    ∈ SL (Z ) c ∈ 4Z  K0 . ∈   2 . So     c d  c d   a b  Then ϕ2   = because the function is not supported on the c d   1  coset   K0 by theorem 16 in [1]. Case 3: c is odd.     a b  −a −b  If c ≡ (mod 4), then we change   to  . It suffices to c d −c −d assume that (mod  4).   c ≡1   1 n a b  a + nc b + nd Since   =  we may choose an appro0 c d c d 30 2. REPRESENTATION-THEORETIC APPROACH TO HR/2 priate n so that a + nc is divisible by 4. Thus we may also assume that 4|a without loss of generality.        a b   1 −c −d  1 On the other hand,  =  ∈  K0 . c d −1 a b −1 Again by the remarks in [1], we have   + ir a b  + γ(−1) √ γ(−1) = − , ϕ2   = 2 c d   a b  for all   such that c ≡ (mod 4) and 4|a. Here the Weil character c d γ(−1) = −ir . Computation In the following computations, x = ( ac db ) ∈ SL2 (Z). (W4 Hr/2 )(z) r =(−2iz)− Hr/2 − 4z  r =(−2iz)−   sA (x)ϕ2 (x)J(x, − (c,d)=1,d>0 −r  ) 4z   r  =(−2iz)−  sA (x)ϕ2 (x)J(x, − (c,d)=1 4|c,d>0 −r ) + 4z sA (x)ϕ2 (x)J(x, − (c,d)=1,d>0 c≡1 or 3(mod 4) Now we handle the two summations separately. −r  )  4z 2.4. PROOF 31 For the first summation, we have r (−2iz)− sA (x)ϕ2 (x)J(x, − (c,d)=1 4|c,d>0 r =(−2iz)− sA (x)ϕ2 (x)(− (c,d)=1 4|c,d>0 r −r ) 4z r c + d)− 4z r r =(−2iz)− (4z) sA (x)ϕ2 (x)(−c + 4dz)− (c,d)=1 4|c,d>0 r r sA (x)(−c + 4dz)− =(2i) (c,d)=1,4|c d≡1(mod 4) because when 4|c, ϕ2 (x) = 1. Let c = −4c1 , we get  r r (2i) 4− (c1 ,d)=1 d≡1(mod 4)   b   a −r s A   (c1 + dz) −4c1 d  b  a Since   ∈ K1 , by proposition 2.16 of [5], −4c1 d   b   a s A   = −4c1 d −c1 d . (2.1) 32 2. REPRESENTATION-THEORETIC APPROACH TO HR/2 Then (2.1) becomes r (−2i)− (c1 ,d)=1 d≡1(mod 4) −c1 d (c1 + dz)− −d c (d + cz)− r or r (−2i)− (c,d)=1 c≡1(mod 4) r (2.2) Go back to the second summation, r (−2iz)− sA (x)ϕ2 (x)J(x, − (c,d)=1,d>0 c≡1 or 3(mod 4) r =(−2iz)− sA (x)ϕ2 (x)(− (c,d)=1 c≡1(mod 4) r r −r ) 4z r c + d)− 4z r =(−2iz)− (4z) sA (x)ϕ2 (x)(−c + 4dz)− (c,d)=1 c≡1(mod 4) r sA (x) − =(2i) (c,d)=1 c≡1(mod 4) + ir r (−c + 4dz)− r because if c is odd, ϕ2 (x) = − 1+i . Let d1 = −4d, we get r (2i) − + ir r r (−1)− sA (x)(c + d1 z)− (2.3) (c,d1 )=1,d1 =−4d c≡1(mod 4) In this case, sA may be computed using proposition 2.16 of [5] and a nice 2.4. PROOF 33 property of Hilbert symbol, namely,  p (a, b)p = 1. We have  a b  sA (x) = sA   c d     −b a 0 −1 = s A    −d c          −b a 0 −1  −b a 0 −1 = s A   sA   σA  ,  −d c −d c = = −d c (−d, c)∞ −d c . (−d, c)v v[...]... ) χ, and moreover, it is invariant under the action of Kp We call such a vector a Kp -spherical vector and the representation a Kp -spherical representation The spherical vector is unique Case 2 Let F = Q2 In this case, we may define a γ-spherical vector This has been explained extensively in [1] Case 3 If F = R, then B ∩ K = T ∩ K = {(±1, ±1)} ∼ Z/4Z = 24 2 REPRESENTATION- THEORETIC APPROACH TO HR/2... (4)) to Mr−1 (SL2 (Z)) and from Sr/2 (Γ0 (4)) to Sr−1 (SL2 (Z)) Kohnen has also modified the Hecke operator T4 , and defined a slightly + + different operator T4 on Mr/2 (Γ0 (4)), then proved that Hr/2 is a Hecke eigen+ form for the operator T4 14 1 NUMBER -THEORETIC APPROACH TO HR/2 Chapter 2 Representation- theoretic approach to Hr/2 Gelbart showed in [5] that there is a bijection mapping between certain... section of this chapter Although being a function on SL2 (Q)\G (A) , this function does not belong to the space Ar/2 mentioned above However, we try to apply the same bijection map on E(g, s, ϕ) For a specific ϕ, we will arrive at Hr/2 which is of interest to many mathematicians In the definition of E(g, s, ϕ), the summand is a function on G (A) So given a ϕ, we can express the summand as a direct product... [1] 2 1 Case 3: c is odd     a b   a −b  If c ≡ 3 (mod 4), then we change   to   It suffices to c d −c −d assume that c ≡  1 (mod  4)    1 n a b  a + nc b + nd Since   =  we may choose an appro0 1 c d c d 30 2 REPRESENTATION- THEORETIC APPROACH TO HR/2 priate n so that a + nc is divisible by 4 Thus we may also assume that 4 |a without loss of generality        a b ... certain automorphic representation and the space of cusp forms of half-integral weight Then in [1] one result is that a subspace of the automorphic functions corresponds to Kohnen’s plus space of cusp forms under the same bijection coupled with an operator introduced in [6] The objective of the current chapter is to show that, under the representation- theoretic methodology outlined by [5] and [1], we can... NUMBER -THEORETIC APPROACH TO HR/2 p and positive integer n, we have Tp Tpn = Tpn+1 + p2k−1 Tpn−1 1.2 MODULAR FORMS OF HALF-INTEGRAL WEIGHT 1.2 1.2.1 7 Modular Forms of Half-Integral Weight Modular form of weight r/2 It may be verified easily that Lemma For any odd integer r, if f (γz) = (cz + d)r/2 f (z), then f (z) = 0 So we need an alternative way to define the modular forms of half-integral weight... 4-sheeted covering of GL+ (Q) 2 Since the automorphy factor j(γ, z) is only defined for γ ∈ Γ0 (4), we do not 1 have a preferred branch of the square root (cz + d) 2 for an arbitrary element in GL+ (Q) Instead of GL+ (Q), we shall need a four-sheeted covering G, 2 2 which is sufficiently large to handle all possible branches of the square roots occurring in the automorphy factor j(γ, z) Let T denote the group... under [γ]r/2 for γ in a subgroup Γ of finite index in Γ0 (4) It is shown in [2] that such f admits a qh expansion similar to that of integral weight modular forms, where qh = e2πiz/h for some integer h In the same fashion, we say f is meromorphic at infinity if only finitely many negative powers of qh occur in the expansion, 10 1 NUMBER -THEORETIC APPROACH TO HR/2 and holomorphic if no negative powers of qh... 26 2 REPRESENTATION- THEORETIC APPROACH TO HR/2 each local field, i.e ϕχs = ϕ∞ ϕp In order to prove the theorem, we p choose ϕ in the following way For p = 2, let ϕp be the spherical vector we found in Case 1 of section 2.2; let ϕ2 be a γ-spherical vector; and ϕ∞ is associated with the lowest weight vector in the induced representation in Case 3 of section 2.2 Let r ≥ 5 be a positive odd integer, and... ϕ) = ϕ((x, sA (x))g)χs (xg) x∈∆B(Q)\G(Q) The sum converges absolutely and uniformly for g in any compact subset and s with Re(s) > 1 [8] This is a function on SL2 (Q)\G (A) 20 2 REPRESENTATION- THEORETIC APPROACH TO HR/2 2.2 Inductions over local fields Now we develop the theory for local places, namely, for SL2 (Qp ), where p is prime, and SL2 (R) In light of Iwasawa decomposition for SL2 (A) , we denote . weight. ii Acknowledgements I would to express my thanks and appreciation to Associate Pro- fessor LOKE Hung Yean, for his academic guidance and patience; Prof ZHU Cheng Bo, Prof LEE Soo Teck, Mr. TANG U-Liang for their. operator T + 4 . 14 1. NUMBER -THEORETIC APPROACH TO H R/2 Chapter 2 Representation- theoretic approach to H r/2 Gelbart showed in [5] that there is a bijection mapping between certain au- tomorphic. A REPRESENTATION- THEORETIC APPROACH TO AN EISENSTEIN SERIES by AI XINGHUAN B.Sc (Hons), NUS A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS FACULTY OF