hecke operators on jacobi forms of lattice index and the relation to elliptic modular forms
TRANSCRIPT
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 22: Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic Modular Forms](https://reader033.vdocuments.mx/reader033/viewer/2022052318/5a66e10b7f8b9a91298b4607/html5/thumbnails/22.jpg)
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
![Page 23: Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic Modular Forms](https://reader033.vdocuments.mx/reader033/viewer/2022052318/5a66e10b7f8b9a91298b4607/html5/thumbnails/23.jpg)
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
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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
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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
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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 ?.
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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.
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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.
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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.
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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).
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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 .
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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 .
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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.
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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)
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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)
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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)
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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)
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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 (`)
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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 (`)
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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 (`)
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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 (`)
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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)− .
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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)− .
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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))
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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))
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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))
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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.
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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.
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Thank you!
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