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
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
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
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).
Ali Ajouz (Siegen University) The history of Hecke operators 3
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).
Ali Ajouz (Siegen University) The history of Hecke operators 3
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).
Ali Ajouz (Siegen University) The history of Hecke operators 3
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).
Ali Ajouz (Siegen University) The history of Hecke operators 3
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).
Ali Ajouz (Siegen University) The history of Hecke operators 3
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).
Ali Ajouz (Siegen University) The history of Hecke operators 3
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
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.
Ali Ajouz (Siegen University) Jacobi forms of lattice index 5
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.
Ali Ajouz (Siegen University) Jacobi forms of lattice index 5
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.
Ali Ajouz (Siegen University) Jacobi forms of lattice index 5
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
r
)q
r2
8 ζr2 , (q = e2πiτ and ζ = e2πiz).
η(τ) = ∆1
24 (τ) (a modular form of weight 1/2).
Ali Ajouz (Siegen University) Jacobi forms of lattice index 6
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
r
)q
r2
8 ζr2 , (q = e2πiτ and ζ = e2πiz).
η(τ) = ∆1
24 (τ) (a modular form of weight 1/2).
Ali Ajouz (Siegen University) Jacobi forms of lattice index 6
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
r
)q
r2
8 ζr2 , (q = e2πiτ and ζ = e2πiz).
η(τ) = ∆1
24 (τ) (a modular form of weight 1/2).
Ali Ajouz (Siegen University) Jacobi forms of lattice index 6
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 .
Ali Ajouz (Siegen University) Jacobi forms of lattice index 7
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))
Ali Ajouz (Siegen University) Jacobi forms of lattice index 8
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))
Ali Ajouz (Siegen University) Jacobi forms of lattice index 8
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))
Ali Ajouz (Siegen University) Jacobi forms of lattice index 8
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))
Ali Ajouz (Siegen University) Jacobi forms of lattice index 8
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))
Ali Ajouz (Siegen University) Jacobi forms of lattice index 8
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))
Ali Ajouz (Siegen University) Jacobi forms of lattice index 8
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 a natural notion of Jacobi Eisenstein series.
There exists No Hecke theory for Jacobi forms of lattice index.
A perfect correspondence with elliptic modular forms of integerweight ?.
Ali Ajouz (Siegen University) Jacobi forms of lattice index 9
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) =
∑a
ak−drk(L)
2e−1%(D, a)Cφ
(`2
a2D, `a′r).
Here the sum is over those a | `2, a2 | `2ND, aa′ ≡ 1 (mod N), and
%(D, a) = f ·(D∆/f 2
a/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)),
where
CT (`)φ(D, r) =∑
a|gcd(`2,ND)
ak−rk(L)
2−1(
∆
a
)Cφ(`2
a2D, `a′r).
The application φ 7→ T (`)φ defines an endomorphism of Jk,L.
Ali Ajouz (Siegen University) Hecke Operators 11
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(
∆
d
)T(``′/d
)if rk(L) is even,
for all ` and `′ both prime to N.
Ali Ajouz (Siegen University) Hecke Operators 12
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).
Ali Ajouz (Siegen University) Hecke Operators 13
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 .
Ali Ajouz (Siegen University) Hecke Operators 14
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 .
Ali Ajouz (Siegen University) Hecke Operators 14
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.
Ali Ajouz (Siegen University) Hecke Operators 15
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)
2
Theta Expansion
Modular forms of weight 2k − rk(L)− 1
(Shimura Lifting)
Ali Ajouz (Siegen University) Hecke Operators 16
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)
2
Theta Expansion
Modular forms of weight 2k − rk(L)− 1
(Shimura Lifting)
Ali Ajouz (Siegen University) Hecke Operators 16
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)
2
Theta Expansion
Modular forms of weight 2k − rk(L)− 1
(Shimura Lifting)
Ali Ajouz (Siegen University) Hecke Operators 16
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)
2
Theta Expansion
Modular forms of weight 2k − rk(L)− 1
(Shimura Lifting)
Ali Ajouz (Siegen University) Hecke Operators 16
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)
2e,L2
∼=−→ Jk+d
rk(L1)
2e,L1
Theorem
The isomorphism I commutes with the Hecke operators T (`)
Ali Ajouz (Siegen University) Hecke Operators 17
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)
2e,L2
∼=−→ Jk+d
rk(L1)
2e,L1
Theorem
The isomorphism I commutes with the Hecke operators T (`)
Ali Ajouz (Siegen University) Hecke Operators 17
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)
2e,L2
∼=−→ Jk+d
rk(L1)
2e,L1
Theorem
The isomorphism I commutes with the Hecke operators T (`)
Ali Ajouz (Siegen University) Hecke Operators 17
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)
2e,L2
∼=−→ Jk+d
rk(L1)
2e,L1
Theorem
The isomorphism I commutes with the Hecke operators T (`)
Ali Ajouz (Siegen University) Hecke Operators 17
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)− .
Proof
Consider the composition of
Jk,L∼=−→ J
k−d rk(L)2e+1,(Z,(x ,y)7→2mxy)
∼=−→M2k−1−rk(L) (N/4)− .
Ali Ajouz (Siegen University) Hecke Operators 18
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)− .
Proof
Consider the composition of
Jk,L∼=−→ J
k−d rk(L)2e+1,(Z,(x ,y)7→2mxy)
∼=−→M2k−1−rk(L) (N/4)− .
Ali Ajouz (Siegen University) Hecke Operators 18
Euler products (The even rank case)
Proposition
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) =∏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)
,
where the product is over all primes p such that (p,N) = 1.
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)
2
(m, χLξ) for some ξ and suitable m
(rad(m) = rad(N))
Ali Ajouz (Siegen University) Hecke Operators 19
Euler products (The even rank case)
Proposition
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) =∏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)
,
where the product is over all primes p such that (p,N) = 1.
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)
2
(m, χLξ) for some ξ and suitable m
(rad(m) = rad(N))
Ali Ajouz (Siegen University) Hecke Operators 19
Euler products (The even rank case)
Proposition
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) =∏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)
,
where the product is over all primes p such that (p,N) = 1.
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)
2
(m, χLξ) for some ξ and suitable m
(rad(m) = rad(N))
Ali Ajouz (Siegen University) Hecke Operators 19
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.
Ali Ajouz (Siegen University) Hecke Operators 20
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.
Ali Ajouz (Siegen University) Hecke Operators 21
Thank you!
Ali Ajouz (Siegen University) Hecke Operators 22