Weil representations over abelian varieties Luca Candelori - PowerPoint PPT Presentation

Weil representations over abelian varieties Luca Candelori Louisiana State University LSU, April 7th, 2015 Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 1 / 31

Weil representations They are finite-dimensional complex representations of the form ρ : Mp 2 g ( Z ) − → GL ( V ) Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 2 / 31

Weil representations They are finite-dimensional complex representations of the form ρ : Mp 2 g ( Z ) − → GL ( V ) 1 → {± 1 } → Mp 2 g ( Z ) → Sp 2 g ( Z ) → 1 Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 2 / 31

Weil representations They are finite-dimensional complex representations of the form ρ : Mp 2 g ( Z ) − → GL ( V ) 1 → {± 1 } → Mp 2 g ( Z ) → Sp 2 g ( Z ) → 1 They ‘encode’ the transformation laws of theta functions. Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 2 / 31

Example: one-variable theta functions of rank 1 lattices Let q = e 2 π i τ , τ ∈ h , m ∈ 2 Z > 0 . � m 2 n 2 θ m , 0 ( q ) = q n ∈ Z Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 3 / 31

Example: one-variable theta functions of rank 1 lattices Let q = e 2 π i τ , τ ∈ h , m ∈ 2 Z > 0 . � m 2 n 2 θ m , 0 ( q ) = q n ∈ Z   �   q n 2 / 2 m θ null , m ( q ) =   n ≡ ν mod m n ∈ Z ν ∈ Z / m Z Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 3 / 31

Example: one-variable theta functions of rank 1 lattices �� a � � b φ 2 = c τ + d . Let γ = , φ ∈ Mp 2 ( Z ), c d Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 4 / 31

Example: one-variable theta functions of rank 1 lattices �� a � � b φ 2 = c τ + d . Let γ = , φ ∈ Mp 2 ( Z ), c d � a τ + b � θ null , m = φ ρ m ( γ ) θ null , m ( τ ) , c τ + d Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 4 / 31

Example: one-variable theta functions of rank 1 lattices �� a � � b φ 2 = c τ + d . Let γ = , φ ∈ Mp 2 ( Z ), c d � a τ + b � θ null , m = φ ρ m ( γ ) θ null , m ( τ ) , c τ + d where ρ m : Mp 2 ( Z ) → GL ( C [ Z / m Z ]) is the Weil representation attached to the quadratic form x �→ mx 2 / 2. Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 4 / 31

Example: one-variable theta functions of rank 1 lattices ρ m : Mp 2 ( Z ) → GL ( C [ Z / m Z ]) Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 5 / 31

Example: one-variable theta functions of rank 1 lattices ρ m : Mp 2 ( Z ) → GL ( C [ Z / m Z ]) �� 1 � � �� 0 � � , √ τ 1 − 1 T = , 1 , S = 0 1 1 0 Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 5 / 31

Example: one-variable theta functions of rank 1 lattices ρ m : Mp 2 ( Z ) → GL ( C [ Z / m Z ]) �� 1 � � �� 0 � � , √ τ 1 − 1 T = , 1 , S = 0 1 1 0 { δ ν } ⊆ C [ Z / m Z ] Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 5 / 31

Example: one-variable theta functions of rank 1 lattices ρ m : Mp 2 ( Z ) → GL ( C [ Z / m Z ]) �� 1 � � �� 0 � � , √ τ 1 − 1 T = , 1 , S = 0 1 1 0 { δ ν } ⊆ C [ Z / m Z ] ρ m ( T )( δ ν ) = e − π i ν 2 / m δ ν √ � i e 2 π i νµ/ m δ µ ρ m ( S )( δ ν ) = √ m µ ∈ Z / m Z Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 5 / 31

Example: one-variable theta functions of rank r lattices Let q = e 2 π i τ , τ ∈ h , ( L , Q ) a positive-definite rank r (even) lattice. � q Q ( λ ) θ L , 0 ( q ) = λ ∈ L Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 6 / 31

Example: one-variable theta functions of rank r lattices Let q = e 2 π i τ , τ ∈ h , ( L , Q ) a positive-definite rank r (even) lattice. � q Q ( λ ) θ L , 0 ( q ) = λ ∈ L �� � q Q ( λ + ν ) θ null , L ( q ) = λ ∈ L ν ∈ L ′ / L Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 6 / 31

Example: one-variable theta functions of rank r lattices �� a � � b φ 2 = c τ + d . Let γ = , φ ∈ Mp 2 ( Z ), c d Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 7 / 31

Example: one-variable theta functions of rank r lattices �� a � � b φ 2 = c τ + d . Let γ = , φ ∈ Mp 2 ( Z ), c d � a τ + b � = φ r ρ L ( γ ) θ null , L ( τ ) , θ null , L c τ + d Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 7 / 31

Example: one-variable theta functions of rank r lattices �� a � � b φ 2 = c τ + d . Let γ = , φ ∈ Mp 2 ( Z ), c d � a τ + b � = φ r ρ L ( γ ) θ null , L ( τ ) , θ null , L c τ + d where ρ L : Mp 2 ( Z ) → GL ( C [ L ′ / L ]) is the Weil representation attached to the lattice ( L , Q ). Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 7 / 31

Example: one-variable theta functions of rank r lattices ρ L : Mp 2 ( Z ) → GL ( C [ L ′ / L ]) Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 8 / 31

Example: one-variable theta functions of rank r lattices ρ L : Mp 2 ( Z ) → GL ( C [ L ′ / L ]) �� 1 � � �� 0 � � , √ τ 1 − 1 T = , 1 , S = 0 1 1 0 Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 8 / 31

Example: one-variable theta functions of rank r lattices ρ L : Mp 2 ( Z ) → GL ( C [ L ′ / L ]) �� 1 � � �� 0 � � , √ τ 1 − 1 T = , 1 , S = 0 1 1 0 { δ ν } ⊆ C [ L ′ / L ] Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 8 / 31

Example: one-variable theta functions of rank r lattices ρ L : Mp 2 ( Z ) → GL ( C [ L ′ / L ]) �� 1 � � �� 0 � � , √ τ 1 − 1 T = , 1 , S = 0 1 1 0 { δ ν } ⊆ C [ L ′ / L ] ρ m ( T )( δ ν ) = e − 2 π iQ ( ν ) δ ν √ r � i e 2 π iB ( ν,µ ) δ µ ρ m ( S )( δ ν ) = � | L ′ / L | µ ∈ L ′ / L Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 8 / 31

Further examples Let ( C g / Λ , H ) be a complex torus with a symmetric principal polarization. Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 9 / 31

Further examples Let ( C g / Λ , H ) be a complex torus with a symmetric principal polarization. Let � e 2 π i � λ, T λ � θ H , 0 = λ ∈ Z g where T ∈ h g . Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 9 / 31

Further examples Let ( C g / Λ , H ) be a complex torus with a symmetric principal polarization. Let � e 2 π i � λ, T λ � θ H , 0 = λ ∈ Z g where T ∈ h g . For k ∈ 2 Z > 0 , let � � � e 2 π i � λ + c 1 , T ( λ + c 1 ) � θ null , H k = λ ∈ Z g c 1 ∈ 1 k Z g / Z g Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 9 / 31

Geometric interpretations Andr´ e Weil, sur certains groupes d’op´ erateurs unitaires (1964): A force d’habitude, le fait que les s´ eries thˆ eta d´ efinissent des fonctions modulaires a presque cess´ e de nous ´ etonner. Mais l’apparition du groupe symplectique comme un deus ex machina dans les c´ el` ebres travaux de Siegel sur les formes quadratiques n’a rien perdu encore de son caract` ere myst´ erieux. Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 10 / 31

Geometric interpretations Andr´ e Weil, sur certains groupes d’op´ erateurs unitaires (1964): A force d’habitude, le fait que les s´ eries thˆ eta d´ efinissent des fonctions modulaires a presque cess´ e de nous ´ etonner. Mais l’apparition du groupe symplectique comme un deus ex machina dans les c´ el` ebres travaux de Siegel sur les formes quadratiques n’a rien perdu encore de son caract` ere myst´ erieux. Question Can we construct Weil representations geometrically ? Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 10 / 31

Heisenberg groups Let S be a noetherian scheme and let H → S be a commutative finite flat group scheme. Luca Candelori (Louisiana State University) Weil representations over abelian varieties LSU, April 7th, 2015 11 / 31