PDF On the image of Fourier-Jacobi expansion - 東京理科大学

Abstract Jacobi forms Our main theorem Proof Application

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On the image of Fourier-Jacobi expansion

Hiroki Aoki

Tokyo University of Science

March 2010

Borcherds product

On the symmetric domain of type IV, Borcherds has constructed automorphic forms by infinite product in his paperAutomorphic forms onOs+2,2(R)and infinite productsin 1995.

In his paper, he set the discrete groupOs+2,2(Z). And he gave an open problem: Extend the methods of this paper to level greater than 1.

To show the automorphic property is not so hard.

To show the convergence is very hard !!

In Borcherds paper, he did very complicated calculation on

asymptotic behaviors of the Fourier coefficients of modular forms to investigate the analytic continuation of the infinite product.

Abstract Jacobi forms Our main theorem Proof Application

Our problem

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Problem

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Find an easy way to show the convergence of given series that satisfies a kind of automorphic property, and apply it to Borcherds product.

The symmetric domain of type IV is a domain defined from an indefinite quadratic space of signature (s+ 2,2).

s= 0⇒H×H: very special case

s= 1⇒H2: Siegel upper half space of degree 2

Abstract Jacobi forms Our main theorem Proof Application

The simplest case

The simplest case : s= 1 (Siegel modular forms of degree 2)

Γ := Sp₂(Z)

Mk : space of Siegel modular forms of weightk Jk,m : space of Jacobi forms of weightk indexm

Fourier-Jacobi expansion

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Mk 3F(τ, z, ω) =

∑∞ m=0

ϕ_m(τ, z) exp(2π√

−1mω)

Mk 3F 7→ {ϕ_m} ∈

∏∞ m=0

Jk,m

This is not surjective. We determine the image of this map.

Abstract Jacobi forms Our main theorem Proof Application

Main theorem

The elementS :=

(_{0 1 0 0}

1 0 0 0 0 0 0 1 0 0 1 0

)

inducesF(τ, z, ω) = (−1)^kF(ω, z, τ).

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Theorem. (The image of Fourier-Jacobi expansion)

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These two maps are surjective, hence bijective.

FJ : Mk 3F 7→ {ϕm}^∞m=0∈(∏^∞

m=0

Jk,m

)sym

FJ^c : M^ck 3F 7→ {ϕ_m}^∞m=1∈(∏^∞

m=1

J^ck,m

)sym

If{ϕm} is in the image, the series

∑∞ m=0

ϕm(τ, z) exp( 2π√

−1mω) converges.

Application

Application

convergence of Maass lift

convergence of Borcherds product Reference

M. Eichler, D. Zagier,The theory of Jacobi forms, Birkh¨auser, 1985.

E. Freitag, Siegelsche Modulfunktionen, GMW 254, Springer Verlag, Berlin (1983).

R.E.Borcherds, Automorphic forms onOs+2,2(R) and infinite products, Invent. Math. 120-1(1995), 161–213.

J. Bruinier,Borcherds products onO(2, l)and Chern classes of Heegner divisors, LNM 1780. Springer-Verlag, Berlin (2002).

Abstract Jacobi forms Our main theorem Proof Application

Siegel upper half space of degree 2

We denote Siegel upper half space of degree 2 by H2:=

{

Z=^tZ= (τ z

z ω )

∈M₂(C) ImZ >0 }

.

The symplectic group G:= Sp₂(R) =

{ M=

(A B C D )

∈M₄(R)^tM J M=J:=

(O2 −E2

E2 O2

)}

acts onH2transitively by

H23Z7−→MhZi:= (AZ+B)(CZ+D)⁻¹∈H2.

Abstract Jacobi forms Our main theorem Proof Application

Siegel modular form of degree 2

For a holomorphic functionF :H2→Candk∈Z, define (F|kM)(Z) := det(CZ+D)^−kF(MhZi).

In today’s talk, we set Γ := Sp₂(Z) := Sp₂(R)∩M4(Z).

Definition. (Siegel modular form of degree 2)

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We sayF is a Siegel modular form of weightkif a holomorphic functionF onH2 satisfies the conditionF =F|kM for anyM ∈Γ.

We denote the space of all Siegel modular forms of weightkbyMk. For simplicity, we denoteF(Z) =F(τ, z, ω).

Abstract Jacobi forms Our main theorem Proof Application

Koecher principle

LetF ∈Mk. F has a Fourier expansion F(Z) = ∑

n,l,m

a(n, l, m)qⁿζ^lp^m,

whereqⁿ:=e(nτ) := exp( 2π√

−1nτ)

,ζ^l:=e(lz) andp^m:=e(mω).

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Proposition. (Koecher principle)

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a(n, l, m) = 0 if 4mn−l²<0 orm <0.

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Definition. (Siegel cusp form)

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we sayF ∈Mk is a cusp form of weightkifF satisfies the condition a(n, l, m) = 0 unless 4mn−l²>0. We denote the space of all cusp forms of weightkbyM^ck.

Abstract Jacobi forms Our main theorem Proof Application

Jacobi group

The elementT :=

(_{1 0 0 0}

0 1 0 1 0 0 1 0 0 0 0 1

)

inducesF(τ, z, ω)7→F(τ, z, ω+ 1).

DefineG^J:={M ∈G|M⁻¹T M =T}. (G= Sp₂(R) )

IfF :H2→Chas a period 1 with respect toω, thenF|kM also has a period 1 for anyM ∈G^J.

Letm∈Zandϕ(τ, z) be a holomorphic function onH×C. The image ofϕ(τ, z)p^m byM ∈G^J is a product of a holomorphic function onH×Candp^m. (p^m= exp(

2π√

−1mω) ) Proposition. (Action of Jacobi group)

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For eachm∈Z, the group G^J acts on the set of all holomorphic functions onH×C.

Abstract Jacobi forms Our main theorem Proof Application

Jacobi group invariant function

Define Γ^J :=G^J∩Γ. ( Γ = Sp₂(Z) )

Letm∈Zandϕ(τ, z) be a holomorphic function onH×C. We assume thatϕ(τ, z)p^mis Γ^J-invariant.

Namely,ϕ(τ, z) satisfies the following two equations:

ϕ(τ, z) =(cτ+d)^−ke

(−mcz² cτ+d

) ϕ

(aτ +b cτ+d, z

cτ+d ) ϕ(τ, z) =e(

m(

x²τ+ 2xz))

ϕ(τ, z+xτ +y) (e(x) = exp(

2π√

−1x) ) for any(_{a b}

c d

)∈SL2(Z) and for anyx, y∈Z. (c.f. the book by Eichler and Zagier)

Abstract Jacobi forms Our main theorem Proof Application

No negative index Jacobi forms

According to the book of Eichler and Zagier, we have the following propositions.

Proposition. (Jacobi form with negative index is zero)

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Ifm <0, aboveϕshould be the zero function.

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Proposition. (Jacobi form with zero index is constant)

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Ifm= 0, aboveϕdoes not depend onz. Namely, it is a SL₂(Z)-invariant holomorphic function onH.

Hence we may assumem≥0.

Abstract Jacobi forms Our main theorem Proof Application

Fourier expansion of Jacobi forms

Aboveϕhas a Fourier expansion ϕ(τ, z) =∑

n,l

c(n, l)qⁿζ^l. (qⁿ :=e(nτ), ζ^l:=e(lz))

Ifm= 0,c(n, l) = 0 forl6= 0.

Ifm >0,c(n, l) depends only on 4mn−l² andl (mod 2m).

Especially, ifm= 1, c(n, l) depends only on 4mn−l² and sometimes we denotec(4mn−l²) =c(n, l).

Abstract Jacobi forms Our main theorem Proof Application

Jacobi form

Definition.

(Jacobi cusp formweak Jacobi formw.h. Jacobi form)

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We say aboveJacobi cusp formweak Jacobi formw.h. Jacobi formϕis a of weight kand indexmif c(n, l) = 0 except whenn >0 and 4mn−l²>0n≥0n≥ −^∃N . We denote the space of allJacobi cusp formweak Jacobi formw.h. Jacobi formof weightkand indexm byJ^ck,mJ^wk,mJ^whk,m.

Jacobi form n≥0 and 4mn−l²≥0 Jk,m

Jacobi cusp form n >0 and 4mn−l²>0 J^ck,m

weak Jacobi form n≥0 J^wk,m

w.h. Jacobi form n≥ −^∃N J^wh_k,m

(w.h. · · ·weakly holomorphic)

Abstract Jacobi forms Our main theorem Proof Application

Property of Jacobi forms (1)

Forϕ∈J^whk,m, a positive valued function Gϕ(τ, z) :=|ϕ(τ, z)|exp

(−2πm(Imz)² Imτ

) (Imτ)^k2 is Γ^J-invariant, namelyGϕ|0,0M =Gϕ for anyM ∈Γ^J.

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Proposition. (Upper bound of a Jacobi cusp form)

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Ifϕ∈J^ck,m, Gϕ has a maximum value.

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Proposition. (Upper bound of Fourier coefficients)

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Forϕ∈J^ck,m, there exists a constant K_ϕ such that

|c(n, l)| ≤K_ϕ(

4mn−l²)^k₂ .



 ϕ(τ, z) =∑

n,l

c(n, l)qⁿζ^l, (qⁿ:=e(nτ), ζ^l:=e(lz))





Abstract Jacobi forms Our main theorem Proof Application

Property of Jacobi forms (2)

Ifm= 0,ϕis a SL2(Z)-invariant holomorphic function on H. Hence J^wk,0=Jk,0=Ak andJ^ck,0={0}, whereAk is a space of all elliptic modular forms of weightk w.r.t. SL2(Z).

J^w_∗,∗:= ⊕

k,m∈Z

J^wk,m is a graded ring ofA_∗:= ⊕

k,m∈Z

Ak.

Proposition. (Structure theorem of weak Jacobi forms)

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J^w_∗,∗ is generated byϕ0,1, ϕ₋2,1 andϕ₋1,2onA_∗. (We remark thatJ_∗,∗ is not finitely generated onA_∗).

Abstract Jacobi forms Our main theorem Proof Application

Fourier-Jacobi expansion

The Fourier Jacobi expansionis a p-expansion of F∈Mk orM^ck: F(Z) =

∑∞ m=0

ϕm(τ, z)p^m. ForF ∈Mk, we haveϕm∈Jk,m.

ForF ∈M^ck, we haveϕ0= 0 andϕm∈J^ck,m.

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Definition. (Fourier-Jacobi expansion)

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The Fourier Jacobi expansion is a map fromMk orM^ck to the infinite direct product space of Jacobi forms:

FJ : Mk 3F7→ {ϕm}^∞m=0∈ ∏^∞

m=0

Jk,m

FJ^c : M^ck 3F7→ {ϕm}^∞m=1∈

∏∞ m=1

J^ck,m

But these two maps are not surjective !!

Abstract Jacobi forms Our main theorem Proof Application

The symmetry

The elementS :=

(_{0 1 0 0}

1 0 0 0 0 0 0 1 0 0 1 0

)

inducesF(τ, z, ω)7→(−1)^kF(ω, z, τ).

S∈Γ gives the information about the image of the Fourier-Jacobi expansion. Namely, on the Fourier expansion

Mk 3F(Z) = ∑

n,l,m

a(n, l, m)qⁿζ^lp^m,

we havea(n, l, m) = (−1)^ka(m, l, n).

Proposition. (Generators of the symplectic group)

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The groupG= Sp₂(R) is generated byG^J andS.

The group Γ = Sp₂(Z) is generated by Γ^J andS.

Abstract Jacobi forms Our main theorem Proof Application

The image of the Fourier-Jacobi expansion

Let (∏^∞

m=0

Jk,m

)sym

:=

{

{ϕ_m} ∈ ∏^∞

m=0

Jk,mc_m(n, l) = (−1)^kc_n(m, l) }

(∏^∞

m=1

J^ck,m

)sym

:=

{ {ϕ_m} ∈

∏∞ m=1

J^ck,mc_m(n, l) = (−1)^kc_n(m, l) }

where we denoteϕ_m(τ, z) =∑

n,l

c_m(n, l)qⁿζ^l.

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Theorem. (The image of Fourier-Jacobi expansion)

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These two maps are surjective, hence bijective.

FJ : Mk 3F 7→ {ϕm}^∞m=0∈(∏^∞

m=0

Jk,m

)sym

FJ^c : M^ck 3F 7→ {ϕm}^∞m=1∈(∏^∞

m=1

J^ck,m

)sym

Abstract Jacobi forms Our main theorem Proof Application

Flow of our proof

Theorem. (The image of Fourier-Jacobi expansion)

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These two maps are surjective, hence bijective.

FJ : Mk 3F 7→ {ϕ_m}^∞m=0∈(∏^∞

m=0

Jk,m

)sym

FJ^c : M^ck 3F 7→ {ϕm}^∞m=1∈(∏^∞

m=1

J^ck,m

)sym

Does

∑∞ m=0

ϕ_m(τ, z)p^m converge absolutely and locally uniformly onH?

Step 1. Estimation of Jacobi cusp forms on ‘rational’ points Step 2. Convergence at ‘rational’ points

Step 3. Complete the proof of FJ^c. Step 4. Complete the proof of FJ.

Abstract Jacobi forms Our main theorem Proof Application

Step 1. (1)

First, we suppose{ϕ_m}^∞m=1∈(∏^∞

m=1

J^ck,m

)sym

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Takex, y∈Qwith same denominatorD. Then f_m(τ) :=e(mx²τ)ϕ_m(τ, xτ+y)

is an elliptic cusp form of weightkwith respect to the main congruent subgroup of levelD². The Fourier expansion off_mis

fm(τ) =∑

ν>0

am(ν)q^ν (

am(ν) = ∑

n,l∈Z mx2 +lx+n=ν

cm(n, l) )

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We remark that the number of the pair (n, l) satisfying mx²+lx+n=ν is less than 4√

mν+ 1. These (n, l) satisfies the condition 4mn−l²≤4mν.

Step 1. (2)

Because the space of the above elliptic cusp forms is finite dimensional, there existLandCsuch that

|f_m(τ)| ≤C



∑

ν≤L

|a_m(ν)|



(Imτ)^−k2.

Hence we have

Gm(τ, xτ+y)≤C







∑

ν≤L

( ∑

n,l∈Z mx2 +lx+n=ν

|cm(n, l)|)



, whereG_m meansG_ϕm.

Abstract Jacobi forms Our main theorem Proof Application

Step 2. (1)

Here we denote the Fourier coefficientc

( n l/2 l/2 m

)

=c_m(n, l).

For anyA∈SL2(Z), (_t

A O₂ O2A⁻¹

)∈Γ inducesc(T) =c(^tAT A).

Forx= ^α_β, takeA= (^∗_∗α^β)∈SL₂(Z). Then we have c

( n l/2 l/2 m

)

=c (tA

( n l/2 l/2 m

) A

)

=c

( _{∗ ∗}

∗ nβ²+lαβ+mα²

) . Becausenβ²+lαβ+mα²= (mx²+lx+n)β²≤(mx²+lx+n)D², we regard thatm in|c_m(n, l)|at Step 1 is sufficiently small.

Step 2. (2)

Namely, from

Gm(τ, xτ+y)≤C







∑

ν≤L

( ∑

n,l∈Z mx2 +lx+n=ν

|cm(n, l)|)



,

there exists a constantKsuch that

Gm(τ, xτ+y)≤CK







∑

ν≤L

( ∑

n,l∈Z mx2 +lx+n=ν

(4mn−l²)^k₂)



, namely, there exists a constantC⁰ (depends on D) such that

G_m(τ, xτ+y)≤C⁰m^k+12

Abstract Jacobi forms Our main theorem Proof Application

Step 3.

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Proposition. (Structure theorem of weak Jacobi forms)

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J^w_∗,∗ is generated byϕ0,1, ϕ₋2,1 andϕ₋1,2onA_∗.

For anyR >1 and (τ0, z0)∈H×C, there exist its neighbourhoodU andx, y∈Qsuch that G_m(τ, z)≤R^mC⁰m^k+12 for any (τ, z)∈U. Namely,

|ϕ(τ, z)| ≤R^mC⁰m^k+12 exp

(2πm(Imz)² Imτ

)

(Imτ)^−k2.

Hence we know the series

∑∞ m=1

ϕ_m(τ, z)p^mis holomorphic on H2. (

Under the assumption{ϕm}^∞m=1∈(∏^∞

m=1

J^ck,m

)sym

. )

Abstract Jacobi forms Our main theorem Proof Application

Step 4.

Let ∆₁₀∈M^c10be a unique Siegel cusp form of weight 10. We remark that ∆₁₀(τ,0, ω) = 0.

Proposition. (Freitag)

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IfF ∈Mk(k∈2Z) satisfiesF(τ,0, ω) = 0, thenF/∆₁₀∈Mk−10. Now supposek∈2Z,{ϕ_m}^∞m=0∈(∏^∞

m=0

Jk,m

)sym

and regard F:=∑_∞

m=0ϕm(τ, z)p^mas a power series ofp.

Then each coefficients ofp^m onF∆₁₀is a Jacobi cusp form, hence by Step 3,F∆10 is holomorphic.

Therefore, by above proposition,F is holomorphic.

For oddk, we have a similar proof. (use ∆₃₅ )

Abstract Jacobi forms Our main theorem Proof Application

Convergence of Maass lift

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Theorem. (Maass Lift)

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For anyϕ∈J^c_k,1, there existsF∈M^c_k such that FJ^c₁(F) =ϕ.

The Hecke operatorT₋(m) induces a map fromJ^ck,1 toJ^ck,m.

(ϕ|T₋(m)) (τ, z) := ∑

ad=m d−1

∑

b=0

a^kϕ

(aτ +b d , az

)

The series F(Z) :=

∑∞ m=1

1

m(ϕ|T₋(m)) (τ, z)p^m

=

∑∞ m=1

∑∞ n=1

∑

l



 ∑

a|(n,l,m)

a^k−1c (mn

a² , l a

) qⁿζ^lp^m

converges by our main theorem.

Lift of a weakly holomorphic Jacobi form

Now letϕ(τ, z) =∑

n,l

c(4mn−l²)qⁿζ^l∈J^wh0,1.

Calculate the Maass lift ofϕ, although it does not converge:

∑∞ m=1

∑∞ n=−N

∑

l



 ∑

a|(n,l,m)

a⁻¹c

(4mn−l² a²

) qⁿζ^lp^m

=

∑∞ m=1

∑∞ n=−N

∑

l

c(4mn−l²) (_∞

∑

a=1

a⁻¹(

qⁿζ^lp^m)a

)

=

∑∞ m=1

∑∞ n=−N

∑

l

c(4mn−l²)(

−log(

1−qⁿζ^lp^m)) .

Abstract Jacobi forms Our main theorem Proof Application

Borcherds product

Hence,formally,

∏∞ m=1

∏∞ n=−N

∏

l

(1−qⁿζ^lp^m)c(4mn−l²)

is a Γ^J-invariant function of weight 0. By slight modification, Borcherds constructs a Γ-invariant function of weightc(0)/2:

p^aζ^bq^c ∏

(m,n,l)>0

(1−qⁿζ^lp^m)c(4mn−l²)

,

wherea=1 2

∑

l>0

l²c(−l²),b=−1 2

∑

l>0

lc(−l²),c= 1 24

∑

l∈Z

c(−l²) and (m, n, l)>0 meansm >0 orm= 0, n >0 orm=n= 0, l >0.

Convergence

This infinite product (Borcherds product) is a Γ-invariant function and has a symmetry.

However, generally, this infinite product (Borcherds product) does not converge. Borcherds has investigated its analytic continuation. He has determined all zero and poles of this product and shown it to be a meromorphic modular form onH2.

By our main theorem, if we show each coefficient of thep-expansion of this infinite product, it should be a holomorphic modular form.

In fact, using this method, we can find out the Borcherds product

∆10,∆35and so on.

Abstract Jacobi forms Our main theorem Proof Application

With levels

Consider the case Γ = Γ0(N) :=

{ M=

(A B C D )

∈Sp₂(Z)C≡O2 (mod N) }

Γ₀(N) is generated by Γ₀(N)^J andS.

We can show the first part of our main theorem in a similar way.

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Theorem. (The image of Fourier-Jacobi expansion)

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The map FJ is surjective, hence bijective.

FJ : Mk 3F 7→ {ϕm}^∞m=0∈(∏^∞

m=0

Jk,m

)sym

This gives a partial answer of Borcherds open problem: Extend the methods of this paper to level greater than 1.

Automorphic forms on O(2,s+2)

Generalize our main theorem to automorphic forms on the symmetric domain of type IV:

H:=G/K (K= O(2)×O(s+ 2) : max. cpt. ) x Γ

Obstacles

Step 2: Can we makemso small ?

(Is the Fourier group sufficiently large?)

Step 3: Is the space of Jacobi forms always finitely generated?

Step 4: Does the good cusp forms like ∆10 always exist?

Abstract Jacobi forms Our main theorem Proof Application

Thank you

Thank you for your kind attention.