Jacobi Forms of Lattice Index Andreea Mocanu The University of - PowerPoint PPT Presentation

Jacobi Forms of Lattice Index Andreea Mocanu The University of Nottingham 7th of December, 2016 Andreea Mocanu Jacobi Forms of Lattice Index

Aim of the talk 1 Definition of Jacobi forms 2 Examples and parallels 3 Some results Andreea Mocanu Jacobi Forms of Lattice Index

I. What are Jacobi forms? Some notation: As usual, e m ( x ) = e 2 π ix / m and write e ( x ) when m = 1. � a b � Γ = SL 2 ( Z ) , with elements A = . c d Upper-half plane: H = { τ ∈ C : ℑ ( τ ) > 0 } . The weight of a Jacobi form will be k ∈ Z + . Apart from the weight, Jacobi forms also have an index . Some prerequisites: Denote by L = ( L , β ) , where: L is a finite rank Z -module. β : L × L → Z is symmetric, positive-definite, even Z -bilinear form. Andreea Mocanu Jacobi Forms of Lattice Index

Remark 1. Even means β ( x , x ) ∈ 2 Z , ∀ x ∈ L . 2. We denote β ( x ) := 1 2 β ( x , x ) . The rank of L is rk ( L ) (note: L ≃ Z rk ( L ) ). The determinant of L is det( L ) := det( G ) , where G is the gram matrix of L with respect to β : pick { e i } i = 1 , rk ( L ) a Z -basis for L = ⇒ G = ( β ( e i , e j )) i , j . Note this also gives β ( x , y ) = x t Gy . The dual of L is L # := { y ∈ L ⊗ Q : β ( x , y ) ∈ Z , ∀ x ∈ L } . The level of L is lev ( L ) := min N + { N : N · β ( y ) ∈ Z , ∀ y ∈ L # } Andreea Mocanu Jacobi Forms of Lattice Index

What is the modular group? Andreea Mocanu Jacobi Forms of Lattice Index

The Jacobi Group More prerequisites: the Heisenberg group associated to L is H L ( Z ) := { h = ( x , y , 1 ) : x , y ∈ L } , with hh ′ = ( x + x ′ , y + y ′ , 1 ) . Remark Γ acts on H L ( Z ) from the right via ( x , y , 1 ) A = (( x , y ) A , 1 ) . Combine action of Γ and H L ( Z ) to get Definition (The Jacobi group associated to L ) We define J L ( Z ) to be the semi-direct product Γ ⋉ H L ( Z ) , with composition law: ( A , h )( A ′ , h ′ ) = ( AA ′ , h A ′ h ′ ) . Andreea Mocanu Jacobi Forms of Lattice Index

More actions � � A τ, z + x τ + y J L ( Z ) acts on H × ( L ⊗ C ) via ( A , h )( τ, z ) = . c τ + d We have a modular variable and an elliptic variable. J L ( Z ) acts on Hol ( H × ( L ⊗ C )) . If φ ∈ Hol ( H × ( L ⊗ C )) , then � � φ | k , L ( A , h ) := φ | k , L A | k , L h , where � z � � − c β ( z ) � ( c τ + d ) − k e φ | k , L A ( τ, z ) := φ A τ, c τ + d c τ + d and φ | k , L h ( τ, z ) := φ ( τ, z + x τ + y ) e ( τβ ( x ) + β ( x , z )) . Andreea Mocanu Jacobi Forms of Lattice Index

Jacobi forms Definition (Jacobi forms of lattice index) The space J k , L of Jacobi forms of weight k and index L consists of all φ ∈ Hol ( H × ( L ⊗ C )) that satisfy 1 φ | k , L ( A , h ) = φ, ∀ ( A , h ) ∈ J L ( Z ) . 2 φ has a Fourier expansion of the form: � c ( n , r ) e ( n τ + β ( r , z )) . n ∈ Z , r ∈ L # n ≥ β ( r ) Andreea Mocanu Jacobi Forms of Lattice Index

II. Examples and intuition 1) Modular interpretation We have a ‘modular interpretation’: elliptic modular forms f ∈ M k (Γ) are in 1 : 1 correspondence with functions F (Λ τ ) ( Λ τ = Z τ ⊕ Z ) satisfying F ( λ Λ τ ) = λ − k F (Λ τ ) , for all λ ∈ C × (Koblitz). Consider the following: H L ( Z ) acts on H × ( L ⊗ C ) via h ( τ, z ) = ( τ, z + x τ + y ) . This is properly discontinuous and fixed point free, so H L ( Z ) \ ( H × ( L ⊗ C )) is an rk ( L ) − dimensional complex manifold E L . Andreea Mocanu Jacobi Forms of Lattice Index

The projection H × ( L ⊗ C ) → H induces a projection E L → H whose fiber over τ is ( L ⊗ C ) / L τ ⊕ L ≃ ( C / Λ τ ) rk ( L ) =: T τ, L . � � � � Any A ∈ Γ gives an isomorphism of tori T τ, L , 0 ≃ T A τ, L , 0 , z induced by the map z �→ c τ + d . � � z Consider the action of Γ on H × ( L ⊗ C ) : ( τ, z ) �→ A τ, . c τ + d Combine the actions of Γ and H L ( Z ) and set A L = J L ( Z ) \ H × ( L ⊗ Z C ) . There is a projection � � A L → Γ \ H , whose fiber over τ is T τ, L / Aut T τ, L , 0 . Andreea Mocanu Jacobi Forms of Lattice Index

Conclusion Jacobi forms φ ∈ J k , L become functions Φ( L τ , z ) , with L τ := L τ ⊕ L and z ∈ T τ, L = ( L ⊗ C ) / L τ , which satisfy: Φ( L τ , z + ω ) = e ( − τβ ( x ) − β ( x , z ))Φ( L τ , z ) , Φ( λ L τ , λ z ) = λ − k e ( λ c β ( z ))Φ( L τ , z ) , for λ ∈ C × and ω = x τ + y ∈ L τ . Elliptic modular forms can be interpreted as global sections of line bundles on the modular curve Γ \ H ∪ { cusps } . Jacobi forms play a similar role for J L ( Z ) \ H × ( L ⊗ Z C ) ∪ { cusps } . This also gives: J k , 0 ≃ M k (Γ) , for 0 = ( L , 0 ) . Andreea Mocanu Jacobi Forms of Lattice Index

2) Theta functions Example (Jacobi theta functions associated to L ) Fix x ∈ L # . Define: � ϑ L , x ( τ, z ) := e ( τβ ( r ) + β ( r , x )) . r ∈ L # r ≡ x mod L These transform ‘nicely’ with weight rk ( L ) / 2 and index L . They give isomorphism between spaces of Jacobi forms and spaces of vector-valued Hilbert modular forms (Boylan). Every Jacobi form has a theta-expansion (Ajouz). When rk ( L ) is odd, this gives a connection to half-integral elliptic modular forms. Andreea Mocanu Jacobi Forms of Lattice Index

3) Jacobi forms of scalar index Example Fix L = ( Z , ( x , y ) �→ 2 mxy ) , for m ≥ 0. We get J k , L = J k , m , studied extensively by Eichler and Zagier. We get a connection to Siegel modular forms, because: Let Γ J denote J L ( Z ) for this particular choice of L . → Γ J ֒ Γ ֒ → Sp 2 ( Z ) . Every SMF of degree 2 has a Jacobi-Fourier expansion (Piatetski–Shapiro). This also holds for degree g > 2, where now the expansion is in terms of Jacobi forms of matrix index ( g − 1 ) (Bringmann). Andreea Mocanu Jacobi Forms of Lattice Index

Jacobi forms and Algebraic Geometry From the work of Gritsenko: Modular forms of orthogonal type can be obtained as liftings of Jacobi forms. The former determine Lorentzian Kac–Moody Lie (super) algebra of Borcherds type. Jacobi forms are solutions to the mirror symmetry problem for K 3 surfaces. For a compact complex manifold, one defines its elliptic genus, which can be a weak Jacobi form ( n ≥ 0). And much more... Andreea Mocanu Jacobi Forms of Lattice Index

III. Work done 1) Poincaré and Eisenstein series Definition Let r ∈ L # / L and D ∈ Q ≤ 0 be such that β ( r ) ≡ D mod Z . We define g L , r , D := e ( τ ( β ( r ) − D ) + β ( r , z )) . When D < 0, we define � P k , L , r , D := g L , r , D | k , L γ γ ∈ J L ( Z ) ∞ \ J L ( Z ) and, when β ( r ) ∈ Z , let E k , L , r := 1 � g L , r , 0 | k , L γ. 2 γ ∈ J L ( Z ) ∞ \ J L ( Z ) Andreea Mocanu Jacobi Forms of Lattice Index

Why the interest? Both are elements of J k , L . Eisenstein series: Perpendicular to Jacobi cusp forms ( n > β ( r ) ) with respect to a suitably defined Petersson scalar product . We get a decomposition: J k , L = S k , L ⊕ J Eis k , L . Their twists by Dirichlet characters modulo N x (level of x ) form a basis of eigenforms of J Eis k , L with respect to (again) suitably defined Hecke operators . Andreea Mocanu Jacobi Forms of Lattice Index

Poincaré series (previously undefined in this setting): They are cusp forms. They reproduce Fourier coefficients of other cusp forms via the Petersson scalar product. Furthermore, they generate S k (Γ) . Our main interest is in reproducing kernels of linear operators defined between spaces of Jacobi cusp forms and elliptic modular forms. Andreea Mocanu Jacobi Forms of Lattice Index

Proposition For any φ ∈ S k , L , � φ, P k , L , r , D � = λ k , L , D c ( n , r ) , where � k − rk ( L ) � λ k , L , D := 2 − 2 k + rk ( L ) 2 ( π | D | ) − k + rk ( L ) det( L ) − 1 + 2 Γ + 1 − 1 2 2 2 and c ( n , r ) is the Fourier coefficient of φ corresponding to e ( τ ( β ( r ) − D ) + β ( r , z )) . Andreea Mocanu Jacobi Forms of Lattice Index

Theorem For k > rk ( L ) + 2 , P k , L , r , D is a cusp form. It has the following Fourier expansion: � G k , L , D , r ( n ′ , r ′ ) e n ′ τ + β ( r ′ , z ) � � P k , L , r , D ( τ, z ) = , n ′ ∈ Z , r ′ ∈ L # n ′ >β ( r ′ ) where G k , L , D , r ( n ′ , r ′ ) := δ L ( D , r , D ′ , r ′ ) + ( − 1 ) k δ L ( D , r , D ′ , − r ′ ) + 2 π i k � D ′ 2 − rk ( L ) � k − 1 4 2 × det( L ) − 1 � H L , c ( n , r , n ′ , r ′ ) � 2 · D c ≥ 1 1 � � 4 π ( DD ′ ) 2 + ( − 1 ) k H L , c ( n , r , n ′ , − r ′ ) � · J k − rk ( L ) , − 1 c 2 where D ′ = β ( r ′ ) − n ′ , we use J k − rk ( L ) − 1 ( · ) for the Bessel function and 2 H L , c ( n , r , n ′ , r ′ ) := c − rk ( L ) − 1 � e c ( β ( r ′ , λ + r )) K ( n ′ , β ( λ ) + β ( r + λ ) + n ; c ) . 2 λ ( c ) In the last equation, λ runs through a complete set of representatives of L / cL and K ( n ′ , β ( λ ) + β ( r + λ + n ); c ) is a Kloosterman sum. Andreea Mocanu Jacobi Forms of Lattice Index