Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic Modular Forms

Hecke Operators on Jacobi Forms of Lattice Index andthe Relation to Elliptic Modular Forms

Ali Ajouz (Siegen University)

July 10, 2015

Ali Ajouz (Siegen University) 1

Ramanujan tau function

∑n>0

τ(n)qn :=∆(z) = q∏n>0

(1− qn)24

=q − 24q2 + 252q3 − 1472q4 + 4830q5 − 6048q6 + · · · ,

where q = e2πiz with z ∈ H := {z ∈ C | Imz > 0}.

∆(az+bcz+d

)= (cz + d)12 ∆(z) for all

(a bc d

)∈ SL2(Z).

This function turns out to be a modular form of weight 12 for SL2(Z)(in fact, it is a cusp form).

Ramanujan’s conjectures

Ramanujan(1916) observed, but could not prove, the following properties:

1 τ(m)τ(n) =∑

d |gcd(m,n)

d11τ(mn/d2).

2∑∞

n=1 τ(n)n−s =∏

p prime

(1− p−sτ(p) + p11−2s

)−1.

Ali Ajouz (Siegen University) The history of Hecke operators 2

∑n>0

τ(n)qn :=∆(z) = q∏n>0

(1− qn)24

=q − 24q2 + 252q3 − 1472q4 + 4830q5 − 6048q6 + · · · ,

where q = e2πiz with z ∈ H := {z ∈ C | Imz > 0}.∆(az+bcz+d

)= (cz + d)12 ∆(z) for all

(a bc d

)∈ SL2(Z).

1 τ(m)τ(n) =∑

d |gcd(m,n)

d11τ(mn/d2).

2∑∞

n=1 τ(n)n−s =∏

p prime

(1− p−sτ(p) + p11−2s

)−1.

∑n>0

τ(n)qn :=∆(z) = q∏n>0

(1− qn)24

=q − 24q2 + 252q3 − 1472q4 + 4830q5 − 6048q6 + · · · ,

where q = e2πiz with z ∈ H := {z ∈ C | Imz > 0}.∆(az+bcz+d

)= (cz + d)12 ∆(z) for all

(a bc d

)∈ SL2(Z).

1 τ(m)τ(n) =∑

d |gcd(m,n)

d11τ(mn/d2).

2∑∞

n=1 τ(n)n−s =∏

p prime

(1− p−sτ(p) + p11−2s

)−1.

Hecke theory for elliptic modular forms

Mordell(1917) proved Ramanujan’s conjecture on the multiplicativenature of τ .Hecke introduced his operators to systematize Mordell’s method ofproof.

Hecke operators

Hecke defined a family of commuting operators T (n) (n ∈ N) with thefollowing properties:

T (n) : Mk → Mk ,T (n) : Sk → Sk .

T (m)T (n) =∑

d|gcd(m,n)

dk−1T (mn/d2).

∑∞`=1

T (`)`−s =∏

p prime

(1− T (p)p−s + pk−1−2s

Sk has a basis whose elements are eigenforms for all Hecke operators T (n).

Let f =∑

n≥0 af (n)qn be a Hecke eigenform. If af (1) = 1, T (n)f = λ(n)ffor some λ(n) ∈ R, then af (n) = λ(n).

Hecke operators

T (n) : Mk → Mk ,T (n) : Sk → Sk .

T (m)T (n) =∑

d|gcd(m,n)

dk−1T (mn/d2).

∑∞`=1

T (`)`−s =∏

p prime

(1− T (p)p−s + pk−1−2s

Let f =∑

Hecke operators

T (n) : Mk → Mk ,T (n) : Sk → Sk .

T (m)T (n) =∑

d|gcd(m,n)

dk−1T (mn/d2).

∑∞`=1

T (`)`−s =∏

p prime

(1− T (p)p−s + pk−1−2s

Let f =∑

Hecke operators

T (n) : Mk → Mk ,T (n) : Sk → Sk .

T (m)T (n) =∑

d|gcd(m,n)

dk−1T (mn/d2).

∑∞`=1

T (`)`−s =∏

p prime

(1− T (p)p−s + pk−1−2s

Let f =∑

Hecke operators

T (n) : Mk → Mk ,T (n) : Sk → Sk .

T (m)T (n) =∑

d|gcd(m,n)

dk−1T (mn/d2).

∑∞`=1

T (`)`−s =∏

p prime

(1− T (p)p−s + pk−1−2s

Let f =∑

Hecke operators

T (n) : Mk → Mk ,T (n) : Sk → Sk .

T (m)T (n) =∑

d|gcd(m,n)

dk−1T (mn/d2).

∑∞`=1

T (`)`−s =∏

p prime

(1− T (p)p−s + pk−1−2s

Let f =∑

Jacobi forms of lattice index

Definition

An (even positive definite) lattice over Z is pair L = (L, β), where L is afree Z-module of a finite rank, and where β : L× L→ Z is a map whichsatisfies the following properties:

1 Symmetric and Z-bilinear.

2 β(x , x) is even and strictly positive unless x = 0.

Definition

Let k ∈ Z. A Jacobi form of weight k and index L is a holomorphicfunction φ : H× L⊗Z C→ C satisfying

1 φ(aτ+bcτ+d ,

zcτ+d

)(cτ + d)−ke

(−cβ(z)cτ+d

)= φ(τ, z) for all

(a bc d

)∈ SL2(Z)

2 φ(τ, z + xτ + y)e (τβ(x) + β(x , z)) = φ(τ, z) (x , y ∈ L).

3 φ is holomorphic at infinity.

The C-vector space of all such forms is denoted by Jk,L.

Ali Ajouz (Siegen University) Jacobi forms of lattice index 4

Jacobi forms of lattice index

Definition

An (even positive definite) lattice over Z is pair L = (L, β), where L is afree Z-module of a finite rank, and where β : L× L→ Z is a map whichsatisfies the following properties:

1 Symmetric and Z-bilinear.

2 β(x , x) is even and strictly positive unless x = 0.

Definition

Let k ∈ Z. A Jacobi form of weight k and index L is a holomorphicfunction φ : H× L⊗Z C→ C satisfying

1 φ(aτ+bcτ+d ,

zcτ+d

)(cτ + d)−ke

(−cβ(z)cτ+d

)= φ(τ, z) for all

(a bc d

)∈ SL2(Z)

2 φ(τ, z + xτ + y)e (τβ(x) + β(x , z)) = φ(τ, z) (x , y ∈ L).

3 φ is holomorphic at infinity.

The C-vector space of all such forms is denoted by Jk,L.

f Ali Ajouz (Siegen University) Jacobi forms of lattice index 4

Remarks

Fourier expansion

φ is called holomorphic at infinity if it has a Fourier expansion of the form

φ(τ, z) =∑

n∈Z,r∈L#,n≥β(r)

cφ(n, r)e (nτ + β(r , z)).

L# = {y ∈ L⊗Z Q : β(x , y) ∈ Z for all x ∈ L} .

Proposition

For fixed D ≤ 0, the map Cφ(D, r) = cφ (β(r)− D, r) for D ≡ β(r)(mod Z), and Cφ(D, r) = 0 otherwise, depends only on r + L.

Definition (Jacobi cusp form)

A Jacobi form φ is called a cusp form if Cφ(0, r) = 0 for all r ∈ L# suchthat β(r) ∈ Z. By Sk,L, we denote the subspace of Jacobi forms in Jk,Lconsisting of cusp forms.

Remarks

Fourier expansion

φ(τ, z) =∑

cφ(n, r)e (nτ + β(r , z)).

Proposition

Remarks

Fourier expansion

φ(τ, z) =∑

cφ(n, r)e (nτ + β(r , z)).

Proposition

Example: Theta blocks (Gritsenko, Skoruppa, and Zagier)

Zmev =

{x = (x1, x2, . . . , xm) ∈ Zm |

m∑i=1

xi ∈ 2Z}

(m ≥ 1), with

the bilinear form (x , y) 7→ x · y =∑m

i=1 xiyi (Dot product).

Let k ∈ N such that k + m ≡ 0 mod 12. We set

Θ(τ, (z1, · · · , zm)) := ϑ(τ, z1)ϑ(τ, z2) . . . ϑ(τ, zm)η(τ)2k−m

x=(x1,··· ,xm)∈ 12Zm

( −4

2mx1···xm

)η(τ)2k−me (β(x , z) + β(x)τ) .

Θ ∈ Jk,Zmev

ϑ(τ, z) =∑

r∈Z(−4

8 ζr2 , (q = e2πiτ and ζ = e2πiz).

η(τ) = ∆1

24 (τ) (a modular form of weight 1/2).

Zmev =

{x = (x1, x2, . . . , xm) ∈ Zm |

m∑i=1

xi ∈ 2Z}

(m ≥ 1), with

x=(x1,··· ,xm)∈ 12Zm

( −4

2mx1···xm

)η(τ)2k−me (β(x , z) + β(x)τ) .

Θ ∈ Jk,Zmev

ϑ(τ, z) =∑

r∈Z(−4

η(τ) = ∆1

Zmev =

{x = (x1, x2, . . . , xm) ∈ Zm |

m∑i=1

xi ∈ 2Z}

(m ≥ 1), with

x=(x1,··· ,xm)∈ 12Zm

( −4

2mx1···xm

)η(τ)2k−me (β(x , z) + β(x)τ) .

Θ ∈ Jk,Zmev

ϑ(τ, z) =∑

r∈Z(−4

η(τ) = ∆1

Theta Decomposition

Theorem

Let φ ∈ Jk,L. Then φ can be written as a sum

φ(τ, z) =∑

x∈L#/L

hx(τ)ϑL,x(τ, z),

where the Jacobi theta series is given by

ϑL,x(τ, z) :=∑r∈L#

r≡x (mod L)

e (τβ(r) + β(r , z)),

andhx(τ) =

∑D∈Q

(D,x)∈supp(L)

Cφ (D, x) e (−Dτ) .

The hx transform as modular forms of weight k − rk(L)2 .

The features of the theory of Jacobi forms of scalar index

Definition (Jacobi forms of scalar index)

Jk,m := Jk,(Z,(x ,y) 7→2mxy).

Jk,m is finite-dimensional.

There exists a Petersson scalar product on the space of Jacobi cuspforms.

There exists a natural notion of Jacobi Eisenstein series.

There exists a Hecke theory for Jacobi forms of scalar index, i.e., foreach positive number ` relative prime to m, there exists a naturalHecke operator T (`) on Jk,m, and the space Jk,m has a basisconsisting of simultaneous eigenforms with respect to all T (`).

A perfect correspondence with elliptic modular forms of integerweight: Jk,m is isomorphic as module over the Hecke algebra to anatural subspace of the space M2k−2(Γ0(m))

Jk,m := Jk,(Z,(x ,y) 7→2mxy).

The features of the theory of Jacobi forms of lattice index

Jk,L is finite-dimensional.

There exists a Petersson scalar product < ·, · > on the space ofJacobi cusp forms.

There exists No Hecke theory for Jacobi forms of lattice index.

A perfect correspondence with elliptic modular forms of integerweight ?.

Hecke operators: odd rank

Theorem

Let L be a lattice of odd rank, level N, discriminant

∆ = (−1)rk(L)−1

2 2 det(L), and let φ be a Jacobi form in Jk,L. For a positiveinteger `, relatively prime to N, let

(T (`)φ) (τ, z) :=∑

D≤0,r∈L#

D≡β(r) mod Z

CT (`)φ(D, r) e ((β(r)− D)τ + β(r , z)),

whereCT (`)φ(D, r) =

ak−drk(L)

2e−1%(D, a)Cφ

a2D, `a′r).

Here the sum is over those a | `2, a2 | `2ND, aa′ ≡ 1 (mod N), and

%(D, a) = f ·(D∆/f 2

)if gcd(D∆, a) = f 2, and equals 0 otherwise.

The application φ 7→ T (`)φ defines an endomorphism of Jk,L.

Ali Ajouz (Siegen University) Hecke Operators 10

Hecke operators: even rank

Theorem

Let L be a lattice of even rank, level N, discriminant ∆ = (−1)rk(L)

2 det(L),and let φ be a Jacobi form in Jk,L. For a positive integer `, relatively primeto N, let

(T (`)φ) (τ, z) :=∑

D≤0,r∈L#

D≡β(r) mod Z

CT (`)φ(D, r) e ((β(r)− D)τ + β(r , z)),

CT (`)φ(D, r) =∑

a|gcd(`2,ND)

ak−rk(L)

2−1(

)Cφ(`2

a2D, `a′r).

The application φ 7→ T (`)φ defines an endomorphism of Jk,L.

The multiplicative properties

Theorem

The Hecke operators on Jk,L satisfy the following multiplicative relation:

T (`) · T (`′) =

∑d |gcd(`,`′)

d2k−rk(L)−2T(``′/d2

)if rk(L) is odd,

∑d |gcd(`2,`′2)

dk− rk(L)2−1(

)T(``′/d

)if rk(L) is even,

for all ` and `′ both prime to N.

Main Result II (Basis of simultaneous Hecke eigenforms)

Theorem

The vector space Jk,L has a basis of simultaneous eigenforms for alloperators T (`) (gcd(`,N) = 1).

Euler products (The odd rank case)

Theorem

Let φ ∈ Jk,L which is an eigenfunction of the Hecke operators T (`) for all` ∈ NL. One has the product expansion of the form

L(s, φ) :=∑

gcd(`,N)=1

λ(`)`−s =∏p

(1− p−sλ(p) + p2k−rk(L)−2−2s

)−1.

where the product is over all primes p such that (p,N) = 1.

Observation

If the Jacobi form φ lifts to an elliptic modular form f of weight2k − 1− rk(L), then L(s, φ) should be (up to a finite number of Eulerfactors) the L-series of f .

Euler products (The odd rank case)

Theorem

L(s, φ) :=∑

gcd(`,N)=1

λ(`)`−s =∏p

(1− p−sλ(p) + p2k−rk(L)−2−2s

)−1.

Observation

If the Jacobi form φ lifts to an elliptic modular form f of weight2k − 1− rk(L), then L(s, φ) should be (up to a finite number of Eulerfactors) the L-series of f .

Consequences: odd rank

Conjecture

For each L and each k with 2k − rk(L)− 1 ≥ 2, there are Heckeequivariant injections Jk,L → M2k−1−rk(L)(N/4)

Results

The conjecture is true for Jacobi Eisenstein series.

I developed some methods for obtaining such maps.

The conjecture is true for many examples.

Method 1: via Shimura correspondence I

Theorem (Shimura, Niwa)

Let N, k ∈ N, and an even Dirichlet character χ mod 4N. For eachsquare-free integer t there is a Hecke equivariant mapSt : Sk+ 1

2(4N, χ)→ M2k(N, χ2).

Jacobi Forms of weight k and Index L

Modular forms of half integral weight k − rk(L)

Theta Expansion

Modular forms of weight 2k − rk(L)− 1

(Shimura Lifting)

2(4N, χ)→ M2k(N, χ2).

Theta Expansion

(Shimura Lifting)

2(4N, χ)→ M2k(N, χ2).

Theta Expansion

(Shimura Lifting)

2(4N, χ)→ M2k(N, χ2).

Theta Expansion

(Shimura Lifting)

Method 2: via stable isomorphisms between lattices

For lattices L1 = (L1, β1), L2 = (L2, β2) over Z we define theirorthogonal sum by L1 ⊕ L2 := (L1 ⊕ L2, f ), wheref (x1 ⊕ x2, y1 ⊕ y2)= β1(x1, y1) + β2(x2, y2).

Two even lattices L1 and L2 are said to be stably isomorphic if andonly if there exists even unimodular lattices U1 and U2 such thatL1 ⊕ U1

∼= L2 ⊕ U2.

Theorem (Boylan-Skoruppa 2014)

If L1∼=st.iso L2, then there is an isomorphism between the two vector

spaces of Jacobi forms

I : Jk+d

rk(L2)

∼=−→ Jk+d

rk(L1)

Theorem

The isomorphism I commutes with the Hecke operators T (`)

∼= L2 ⊕ U2.

I : Jk+d

rk(L2)

∼=−→ Jk+d

rk(L1)

Theorem

∼= L2 ⊕ U2.

I : Jk+d

rk(L2)

∼=−→ Jk+d

rk(L1)

Theorem

∼= L2 ⊕ U2.

I : Jk+d

rk(L2)

∼=−→ Jk+d

rk(L1)

Theorem

Theorem (Skoruppa-Zagier)

For every m > 0 and k ≥ 2, there exists Hecke-equivariant isomorphisms

Jk,(Z,(x ,y)7→2mxy)

∼=−→M2k−2 (m)− ,

where M2k−2(m) is the Certain Space inside M2k−2(m) containing allnewforms whose L-series has a minus sign in its functional equation.

Theorem

If L = (L, β) ∼=st.iso

(Z, (x , y) 7→ det(L)xy

), then, for all k ∈ N with

2k − rk(L)− 1 ≥ 2, there is a Hecke-equivariant isomorphism

Jk,L∼=−→M2k−1−rk(L) (N/4)− .

Consider the composition of

Jk,L∼=−→ J

k−d rk(L)2e+1,(Z,(x ,y)7→2mxy)

∼=−→M2k−1−rk(L) (N/4)− .

Theorem (Skoruppa-Zagier)

For every m > 0 and k ≥ 2, there exists Hecke-equivariant isomorphisms

Jk,(Z,(x ,y)7→2mxy)

∼=−→M2k−2 (m)− ,

where M2k−2(m) is the Certain Space inside M2k−2(m) containing allnewforms whose L-series has a minus sign in its functional equation.

Theorem

If L = (L, β) ∼=st.iso

(Z, (x , y) 7→ det(L)xy

), then, for all k ∈ N with

2k − rk(L)− 1 ≥ 2, there is a Hecke-equivariant isomorphism

Jk,L∼=−→M2k−1−rk(L) (N/4)− .

Consider the composition of

Jk,L∼=−→ J

k−d rk(L)2e+1,(Z,(x ,y)7→2mxy)

∼=−→M2k−1−rk(L) (N/4)− .

Euler products (The even rank case)

Proposition

L(s, φ) =∏p-N

1 + pk−rk(L)

2−1−sχL(p)

1 + pk−rk(L)

2−1−sχL(p)− λ(p)p−s + p2(k− rk(L)

2−1−s)

Observation

For even rank, the shape of the L-series of a Hecke eigenform φ is like theL-series

∑gcd(`,N)=1 ξ(`)γ(`2)`−s where γ(`) are the eigenvalues of a

Hecke eigenform in Mk− rk(L)

(m, χLξ) for some ξ and suitable m

(rad(m) = rad(N))

Proposition

L(s, φ) =∏p-N

1 + pk−rk(L)

2−1−sχL(p)

1 + pk−rk(L)

2−1−sχL(p)− λ(p)p−s + p2(k− rk(L)

2−1−s)

Observation

(rad(m) = rad(N))

Proposition

L(s, φ) =∏p-N

1 + pk−rk(L)

2−1−sχL(p)

1 + pk−rk(L)

2−1−sχL(p)− λ(p)p−s + p2(k− rk(L)

2−1−s)

Observation

(rad(m) = rad(N))

Consequences: even rank

Conjecture

For each k > rk(L)2 , there are maps M

k− rk(L)2

(m, χLξ)→ Jk,L such that

T (`2) on the left corresponds to ξ(`)T (`) on the Jacobi form side.

Results

The conjecture is true if L is unimodular.

The conjecture is true if det(L) is an odd prime.

The conjecture is true for Jacobi Eisenstein series.

The conjecture is true for many examples.

Summery: my contribution to knowledge

I developed a systematic Hecke theory for Jacobi forms of latticeindex along the lines of Hecke’s theory of modular forms.

Thank you!

Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic Modular Forms

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Transcript of Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic Modular Forms