A representation theoretic approach to an eisenstein series

Though the quoted arti-cle focuses on cusp forms, the methodology applies to a larger au-dience, including the Eisenstein series of half-integral weight.. Thecurrent writing will carry o

THEORETIC APPROACH

TO AN EISENSTEIN SERIES

by

AI XINGHUANB.Sc (Hons), NUS

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

FACULTY OF SCIENCE NATIONAL UNIVERSITY OF SINGAPORE

2010

Summary

In [1], the authors have developed a representation-theoretic proach to reveal the correspondence between certain automorphicrepresentation and Kohnen’s plus space Though the quoted arti-cle focuses on cusp forms, the methodology applies to a larger au-dience, including the Eisenstein series of half-integral weight Thecurrent writing will carry out this approach, with necessary ad-justments, to establish a similar link between certain induced rep-resentation and the Eisenstein series Hr/2 of half-integral weight

I would to express my thanks and appreciation to Associate fessor LOKE Hung Yean, for his academic guidance and patience;Prof ZHU Cheng Bo, Prof LEE Soo Teck, Mr TANG U-Liang fortheir kindness to answer to my questions; last but not least, myparents and friends who have been encouraging and supporting

Pro-me during my graduate study

1.1 Modular Forms 1

1.1.1 Modular group and congruence subgroups 1

1.1.2 Modular functions and modular forms 2

1.1.3 Eisenstein series 3

1.1.4 Slash operator 4

1.1.5 Double coset and Hecke operator 5

1.2 Modular Forms of Half-Integral Weight 7

1.2.1 Modular form of weight r/2 7

1.2.2 A 4-sheeted covering of GL+2(Q) 8

1.2.3 Eisenstein series 10

1.2.4 Hecke operators on forms of half-integral weight 11

1.2.5 The Shimura map 12

1.2.6 Kohnen’s plus space 13

2 Representation-theoretic approach to Hr/2 15 2.1 Induction over Adele 16

2.2 Inductions over local fields 20

2.3 Main Theorem 25

2.4 Proof 27

iii

Let H = {z|Im(z) > 0} be the upper half plane of C The special lineargroup SL2(Z) acts on H by linear fractional transformation, i.e for all γ =

SL2(Z) of finite index, then Γ is called a congruence subgroup if it contains

Let k be an integer A complex-valued function f on H is weakly modular ofweight 2k if f is meromorphic on H and satisfies the relation:

f (z) = (cz + d)−2kf (az + b

cz + d)

where the summation is over all pairs of integers (m, n) other than (0, 0) TheEisenstein series Gk(z) is a modular form of weight 2k We have Gk(∞) =2ζ(2k) where ζ denotes the Riemann zeta function.

Denote by Mk the C-vector space of modular forms of weight 2k, Mk0 theC-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 1with basis 1, G2, G3, G4, G5 respectively, and of course, Mk0 = 0 for thesevalues of k Denote g2 = 60G2, and g3 = 140G3 Let ∆ = g3

2 − 27g2

3, thenmultiplication by ∆ defines an isomorphism from Mk−6 onto M0

1.1 MODULAR FORMS 5

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 ∆0 =

∆ ∩ ξ−1∆ξ is of finite index in either group Let [∆ : ∆0] = d and write

∆ = ∪d

j=1∆0δj, where δj ∈ ∆ Then ∆ξ∆ = ∪d

j=1∆ξδj.Suppose that Γ is a congruence subgroup of SL2(Z), and let α ∈ GL+2(Q).Denote Γ0 = Γ ∩ α−1Γα, d = [Γ : Γ0], and Γ = ∪d

p and positive integer n, we have

TpTpn = Tpn+1+ p2k−1Tpn−1

1.2 MODULAR FORMS OF HALF-INTEGRAL WEIGHT 7

It may be verified easily that

Lemma For any odd integer r, if f (γz) = (cz + d)r/2f (z), then f (z) = 0

So we need an alternative way to define the modular forms of half-integralweight We first introduce the following notations

Let d be an odd integer, let c be any integer If d is a positive primenumber, the usual quadratic residue symbol (dc) is defined as

For an odd integer d, we define

j(γ, z) = (c

d)

−1

d (cz + d)1.Let k be an odd integer, a holomorphic function f on H is called a modularform of weight r2 if, for γ =

1.2 MODULAR FORMS OF HALF-INTEGRAL WEIGHT 9

det α for some t ∈ {±1} The group law of G is guaranteed

by the following proposition from [2]

Proposition Suppose (α, φ(z)), (β, ψ(z)) are two elements of G, then theproduct

(α, φ(z))(β, ψ(z)) = (αβ, φ(βz)ψ(z))defines a group law of G

Now, for any congruence subgroup Γ, we denote by eΓ the subgroup{(γ, j(γ, z)|γ ∈ Γ} of G

For ξ = (α, φ(z)) ∈ G and r ∈ Z, we define a “slash operator” |[ξ]r/2 onfunctions f defined on H by

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,

and holomorphic if no negative powers of qh occur.

Consider the action of elements in Γ0 on the cusps Q∪∞ Proposition 2 inchapter 4 section 1 of [2] implies that the above definitions of meromorphic orholomorphic functions depends only on the Γ0-equivalence class of the cusps

We say that f (z) is a modular form (cusp form) of weight r/2 for eΓ0 if it ismeromorphic (holomorphic) at every cusp of Γ0 We denote by Mr/2(eΓ0) thespace of modular forms of weight r/2 for eΓ0 and Sr/2(eΓ0) the space of cuspforms of weight r/2 for eΓ0

1.2 MODULAR FORMS OF HALF-INTEGRAL WEIGHT 11

presented as follows

Er/2= X

4|c,d>0 (c,d)=1

cd

where the sum is over all distinct right cosets of fΓ1(N ) in the double coset.

It is known that the Hecke operators on forms of half-integral weight arenontrivial only for perfect square n

For each positive squarefree integer, Shimura and Niwa constructed a lifting

of cusp forms of weight r/2 for Γ0(4N ) with character χ to cusp forms ofweight r − 1 for Γ0(2N ) with character χ2, where r ≥ 7 [2][6]

Theorem (Shimura) Let r ≥ 3 be an odd integer, λ = (r −1)/2, 4|N , χ be aDirichlet character modulo N Let f (z) =P∞

Then g ∈ Mr−1(N0, χ2) for some integer N0 which is divisible by the conductor

of χ2 If r ≥ 5, then g is a cusp form

Proposition (Niwa) In Theorem 1.2.5 one can always take N0 = N/2

The reuslts above are stated in their full generality However, we are onlyinterested in the simplest case when N equals to 4 and χ is trivial Then thetheorem works for f (z) ∈ Sr/2(eΓ0(4))

1.2 MODULAR FORMS OF HALF-INTEGRAL WEIGHT 13

au-to show that, under the representation-theoretic methodology outlined by [5]and [1], we can construct in certain induced representation a function thatcorresponds to the Hecke eigenform Hr/2.

The underlying field F is either Qp or R

15

2.1 Induction over Adele

Now, with the help of the Weil index, we construct an induced representationover the Adele A(Q)

G(A) = G(A) × {±1}, T(A) = T(A) × {±1}, B(A) = B(A) × {±1}

We equip G(A) with the group law in the proposition 2.6 of [5] Namely, atplace p, the group law on G(Qp) is

(g1, 1)(g2, 2) = (g1g2, 12σp(g1, g2)),

where σp(g1, g2) is a 2-cocycle defined as follows Let (∗, ∗)p be the Hilbert

2.1 INDUCTION OVER ADELE 17

symbol over Qp For all gp =

For all s ∈ C, we define the characters

b ∈ A





, t ∈ T(A), k ∈ K(A)

Let F denote the space of C∞-functions f on G(A) s.t

i) The right translates of f by K∞K2 span a finite dimensional subspace;

ii) f (ubtg) = γ(t)f (g), ∀u ∈ N(A), b ∈ B(Q), t ∈ T(A), g ∈ G(A), whereγ(t) = γ(t∞) ·Q γp(tp) is the Weil character defined in [7]

Then I(s) is an induced representation of G(A) by right translation

2.1 INDUCTION OVER ADELE 19

For ϕ ∈ I(s), we define

E(g, s, ϕ) = X

x∈∆B(Q)\G(Q)

ϕ((x, sA(x))g)χs(xg)

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)

2.2 Inductions over local fields

Now we develop the theory for local places, namely, for SL^2(Qp), where p isprime, and ^SL2(R)

In light of Iwasawa decomposition for ^SL2(A), we denote

where ∗ is an arbitrary element in the underlying field (or ring) With thesenotations, we only specify the underlying fields when necessary Now we havee

B = eT eN ⊆ gSL2 In particular, eB, eT , and eN are subgroups of gSL2

2.2 INDUCTIONS OVER LOCAL FIELDS 21

In our context, the functions in this collection are C∞-functions if F = R,and locally constant with a compact support modulo eB(F) if F = Qp, p isprime 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 IndSL^2 (F)

] B(F) χ is completely determined by χ and the values of f on]

We now proceed to investigate IndSL^2 (F)

] B(F) χ over different fields, so that wemay prepare for further discussions on an induced representation over a ring

of Adeles in the later sections

Case 1

We now choose F = Qp, where p 6= 2

Let Kp = SL2(Zp)×1, then Kp is a subgroup of fKp We have gSL2 = eBKp

2.2 INDUCTIONS OVER LOCAL FIELDS 23

α ∈ Z×p, β ∈ Zp







Let χ be a character trivial on eB ∩ Kp, then

Note that the constant function fp(k) = 1, ∀k ∈ Kp, is a function inIndSL^2 (Q p )

^

B(Q p ) χ, and moreover, it is invariant under the action of Kp We callsuch a vector a Kp-spherical vector and the representation a Kp-sphericalrepresentation The spherical vector is unique

Since K = SO2(R), we identify eK with

−2π < θ ≤ −π or π < θ ≤ 2π







We again have an induced representation

IndSL^2 (R)

^ B(R) χ =

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) ∈ eK

In fact, if s + 1 /∈ 2Z + 1/2, the induced representation is irreducible

If s + 1 ∈ Z + 1/2 and s ≥ 0, the induced representation has a submodulespanned by {e2n+1/2|2n + 1/2 ≥ s + 1} The vector es+1 is then the lowestweight vector The submodule is called the discrete series representation of

^

SL2(R)

x + iy = z.

Now we revert to the series E(g, s, ϕ) defined in the first section of thischapter Although being a function on SL2(Q)\G(A), this function does notbelong to the space Ar/2 mentioned above However, we try to apply thesame bijection map on E(g, s, ϕ) For a specific ϕ, we will arrive at Hr/2which is of interest to many mathematicians

In the definition of E(g, s, ϕ), the summand is a function on G(A) Sogiven a ϕ, we can express the summand as a direct product of functions on

each local field, i.e ϕχs = ϕ∞Q

p

ϕp In order to prove the theorem, wechoose ϕ in the following way For p 6= 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 inCase 3 of section 2.2

Let r ≥ 5 be a positive odd integer, and set s = r/2 − 1 With such ϕ,

.Theorem Let r ≥ 5 be a positive odd integer, then (W4Hr/20 )(z) is a scalarmultiple of Hr/2

2.4 PROOF 27

In the definition of Hr/20 (z), x runs through the coset representatives ofB(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 that c, d ∈ Z and gcd(c, d) = 1 Thisenables us to identify B(Q)\G(Q) as

c, d ∈ Z, gcd(c, d) = 1, d > 0







For simplicity, we write ϕp((x, 1)) as ϕp(x) for all p Then we have

Hr/20 (z) = X

x=(a b

c d)∈SL2(Z), (c,d)=1,d>0

sA(x)ϕ2(x)ϕ∞(xg∞)J (xg∞, i)

r

J (x, g∞i)r

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 Iwasawadecomposition,

hand, if we set ϕ(1) = 1, then

c ∈ 4Z2





 So

appro-priate n so that a + nc is divisible by 4 Thus we may also assume that 4|awithout loss of generality.

On the other hand,

sA(x)ϕ2(x)(−c + 4dz)−r

=(2i)r2

X

(c,d)=1,4|c d≡1(mod 4)

Then (2.1) becomes

(−2i)−r2

X

(c1,d)=1 d≡1(mod 4)

 −c1d

(c1 + dz)−r2

or

(−2i)−r X

(c,d)=1 c≡1(mod 4)

 −dc

sA(x)ϕ2(x)(−c + 4dz)−r2

=(2i)r2

X

(c,d)=1 c≡1(mod 4)

because if c is odd, ϕ2(x) = −1+i2r

X

(c,d1)=1,d1=−4d c≡1(mod 4)

sA(x)(c + d1z)−r2 (2.3)

In this case, s may be computed using proposition 2.16 of [5] and a nice

(c,d1)=1,d1=−4d c≡1(mod 4)

 −dc

(c + d1z)−r2

1 + ir

X

(c,d)=1,4|c

cd



Thus, combination of (2.2) and (2.4) gives

(d + cz)−r2 + (2i)r2

1

1 + ir

X

(c,d)=1,4|c d≡1(mod 4)

cd

(d + cz)−r2

 −dc

(d + cz)−r2 + X

(c,d)=1,4|c d≡1(mod 4)

cd

(d + cz)−r2

[1] H Y Loke and G Savin A representation theoretic approach to Kohnen’splus space of modular forms of half integral weight, arXiv:0902.1264v1,2009

[2] Neal Koblitz Introduction to Elliptic Curves and Modular Forms,Springer-Verlag, GTM 97, 1993

[3] J.P Serre A Course in Arithmetic, Springer-Verlag, GTM 7, 1973

[4] W Kohnen Modular forms of half-integral weight on Γ0(4), Math Ann.,

248 (3):246-266, 1980

[5] S S Gelbart Weil’s Representation and the Spectrum of the MetaplecticGroup, Springer-Verlag, Lecture Notes in Mathematics 530, 1976

[6] S Niwa On Shimura’s trace formula, Nagoya Math J., 66:183-202, 1977

[7] A Weil Sur certains groupes doperateurs unitaires, Acta Math.,

111:143-211, 1964

35

[8] D Bump Automorphic Forms and Representations, Cambridge sity Press, 1998.

[1] H Y Loke and G Savin A representation theoretic approach to Kohnen’splus... vector and the representation a Kp-sphericalrepresentation The spherical vector is unique

Since... fact, according to the Iwasawadecomposition,

hand, if we set ϕ(1) = 1, then