Dimensions of spaces of newforms Greg Martin University of British - PowerPoint PPT Presentation

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimensions of spaces of newforms Greg Martin University of British Columbia Canadian Number Theory Association X Meeting University of Waterloo July 17, 2008 www.math.ubc.ca/ ∼ gerg/index.shtml?slides Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Outline Cusp forms on Γ 0 ( N ) 1 Newforms on Γ 0 ( N ) 2 Consequences of the dimension formula 3 Related dimensions 4 Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Cusp forms on Γ 0 ( N ) Notation �� a b � � Γ 0 ( N ) = ∈ SL 2 ( Z ): c ≡ 0 (mod N ) c d Definition (weight- k cusp forms on Γ 0 ( N ) ) Let S k (Γ 0 ( N )) denote the C -vector space of functions f that are holomorphic on the upper half-plane ℑ z > 0 , and “holomorphic and zero at cusps”, that satisfy � az + b � = ( cz + d ) k f ( z ) f cz + d � a b � for all ∈ Γ 0 ( N ) . c d Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Cusp forms on Γ 0 ( N ) Notation �� a b � � Γ 0 ( N ) = ∈ SL 2 ( Z ): c ≡ 0 (mod N ) c d Definition (weight- k cusp forms on Γ 0 ( N ) ) Let S k (Γ 0 ( N )) denote the C -vector space of functions f that are holomorphic on the upper half-plane ℑ z > 0 , and “holomorphic and zero at cusps”, that satisfy � az + b � = ( cz + d ) k f ( z ) f cz + d � a b � for all ∈ Γ 0 ( N ) . c d Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 s 0 , ν ∞ , ν 2 , and ν 3 are certain multiplicative functions related to Γ 0 ( N ) . Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 s 0 , ν ∞ , ν 2 , and ν 3 are certain multiplicative functions related to Γ 0 ( N ) . Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 s 0 is the multiplicative function satisfying s 0 ( p α ) = 1 + 1 p for all α ≥ 1 . Ns 0 ( N ) is the index of Γ 0 ( N ) in SL 2 ( Z ) , where G denotes the quotient of the group G by its center. Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 ν ∞ is the multiplicative function satisfying: ν ∞ ( p α ) = 2 p ( α − 1 ) / 2 if α is odd; ν ∞ ( p α ) = p α/ 2 + p α/ 2 − 1 if α is even. ν ∞ ( N ) counts the number of (inequivalent) cusps of Γ 0 ( N ) . Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 ν 2 is the multiplicative function satisfying: ν 2 ( 2 ) = 1 , and ν 2 ( 2 α ) = 0 for α ≥ 2 ; if p ≡ 1 (mod 4) then ν 2 ( p α ) = 2 for α ≥ 1 ; if p ≡ 3 (mod 4) then ν 2 ( p α ) = 0 for α ≥ 1 . ν 2 ( N ) counts the number of (inequivalent) elliptic points of Γ 0 ( N ) of order 2. Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 ν 3 is the multiplicative function satisfying: ν 3 ( 3 ) = 1 , and ν 3 ( 3 α ) = 0 for α ≥ 2 ; if p ≡ 1 (mod 3) then ν 3 ( p α ) = 2 for α ≥ 1 ; if p ≡ 2 (mod 3) then ν 3 ( p α ) = 0 for α ≥ 1 . ν 3 ( N ) counts the number of (inequivalent) elliptic points of Γ 0 ( N ) of order 3. Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Dimension of space of cusp forms Notation Let g 0 ( k , N ) denote the dimension of S k (Γ 0 ( N )) . Proposition For any even integer k ≥ 2 and any integer N ≥ 1 , � k g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 � 2 ν ∞ ( N )+ c 2 ( k ) ν 2 ( N )+ c 3 ( k ) ν 3 ( N )+ δ . 2 � k c 2 ( k ) = 1 � − k � − 1 4 , 1 � 4 + 4 , so c 2 ( k ) ∈ for k even 4 4 � k c 3 ( k ) = 1 − k − 1 3 , 0 , 1 � � � 3 + 3 , so c 3 ( k ) ∈ 3 3 δ ( m ) = 1 if m = 1 , and δ ( m ) = 0 otherwise Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Where that dimension formula comes from We assume N ≥ 2 and k ≥ 4 to simplify the exposition. Notation Let g N denote the genus of the (compactified) quotient of the upper half-plane by Γ 0 ( N ) . Formula for the genus g N = Ns 0 ( N ) − ν ∞ ( N ) − ν 2 ( N ) − ν 3 ( N ) + 1 12 2 4 3 Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Where that dimension formula comes from We assume N ≥ 2 and k ≥ 4 to simplify the exposition. Notation Let g N denote the genus of the (compactified) quotient of the upper half-plane by Γ 0 ( N ) . Formula for the genus g N = Ns 0 ( N ) − ν ∞ ( N ) − ν 2 ( N ) − ν 3 ( N ) + 1 12 2 4 3 Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Where that dimension formula comes from We assume N ≥ 2 and k ≥ 4 to simplify the exposition. Notation Let g N denote the genus of the (compactified) quotient of the upper half-plane by Γ 0 ( N ) . Formula for the genus g N = Ns 0 ( N ) − ν ∞ ( N ) − ν 2 ( N ) − ν 3 ( N ) + 1 12 2 4 3 Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Where that dimension formula comes from Formula for the genus g N = Ns 0 ( N ) − ν ∞ ( N ) − ν 2 ( N ) − ν 3 ( N ) + 1 12 2 4 3 The dimension g 0 ( k , N ) of the space of weight- k cusp forms on Γ 0 ( N ) is calculated by the Riemann–Roch theorem: � k � g 0 ( k , N ) = ( k − 1 )( g N − 1 ) + 2 − 1 ν ∞ ( N ) � k � k � � + ν 2 ( N ) + ν 3 ( N ) . 4 3 Collecting the multiples of ν ∞ ( N ) , ν 2 ( N ) , and ν 3 ( N ) yields g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 2 ν ∞ ( N ) � 1 � k � 1 � k 4 − k 3 − k �� �� + 4 + ν 2 ( N ) + 3 + ν 3 ( N ) . 4 3 Dimensions of spaces of newforms Greg Martin

Cusp forms on Γ 0 ( N ) Newforms on Γ 0 ( N ) Consequences of the dimension formula Related dimensions Where that dimension formula comes from Formula for the genus g N = Ns 0 ( N ) − ν ∞ ( N ) − ν 2 ( N ) − ν 3 ( N ) + 1 12 2 4 3 The dimension g 0 ( k , N ) of the space of weight- k cusp forms on Γ 0 ( N ) is calculated by the Riemann–Roch theorem: � k � g 0 ( k , N ) = ( k − 1 )( g N − 1 ) + 2 − 1 ν ∞ ( N ) � k � k � � + ν 2 ( N ) + ν 3 ( N ) . 4 3 Collecting the multiples of ν ∞ ( N ) , ν 2 ( N ) , and ν 3 ( N ) yields g 0 ( k , N ) = k − 1 12 Ns 0 ( N ) − 1 2 ν ∞ ( N ) � 1 � k � 1 � k 4 − k 3 − k �� �� + 4 + ν 2 ( N ) + 3 + ν 3 ( N ) . 4 3 Dimensions of spaces of newforms Greg Martin