Definitions Application Quaternions HMF Application II Half integral weight modular forms Ariel Pacetti Universidad de Buenos Aires Explicit Methods for Modular Forms March 20, 2013 Ariel Pacetti Half integral weight modular forms The Dedekind eta function 1 πiz Y 2πinz η(z) = e 12 (1 − e ): n=1 It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η \should be" of weight 1=2. Actually η turns out to be weight 1=2 but with a character of order 24. We can consider two classical examples: Definitions Application Quaternions HMF Application II Motivation What is a half integral modular form? Ariel Pacetti Half integral weight modular forms The Dedekind eta function 1 πiz Y 2πinz η(z) = e 12 (1 − e ): n=1 It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η \should be" of weight 1=2. Actually η turns out to be weight 1=2 but with a character of order 24. Definitions Application Quaternions HMF Application II Motivation What is a half integral modular form? We can consider two classical examples: Ariel Pacetti Half integral weight modular forms It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η \should be" of weight 1=2. Actually η turns out to be weight 1=2 but with a character of order 24. Definitions Application Quaternions HMF Application II Motivation What is a half integral modular form? We can consider two classical examples: The Dedekind eta function 1 πiz Y 2πinz η(z) = e 12 (1 − e ): n=1 Ariel Pacetti Half integral weight modular forms Actually η turns out to be weight 1=2 but with a character of order 24. Definitions Application Quaternions HMF Application II Motivation What is a half integral modular form? We can consider two classical examples: The Dedekind eta function 1 πiz Y 2πinz η(z) = e 12 (1 − e ): n=1 It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η \should be" of weight 1=2. Ariel Pacetti Half integral weight modular forms Definitions Application Quaternions HMF Application II Motivation What is a half integral modular form? We can consider two classical examples: The Dedekind eta function 1 πiz Y 2πinz η(z) = e 12 (1 − e ): n=1 It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η \should be" of weight 1=2. Actually η turns out to be weight 1=2 but with a character of order 24. Ariel Pacetti Half integral weight modular forms a b It is not hard to see that if γ = c d 2 Γ0(4), then θ(γz)2 −1 = (cz + d): θ(z) d 2 So θ(z) 2 M1(Γ0(4); χ−1). Definitions Application Quaternions HMF Application II Motivation The classical theta function 1 X 2 θ(z) = e2πin z : n=−∞ Ariel Pacetti Half integral weight modular forms 2 So θ(z) 2 M1(Γ0(4); χ−1). Definitions Application Quaternions HMF Application II Motivation The classical theta function 1 X 2 θ(z) = e2πin z : n=−∞ a b It is not hard to see that if γ = c d 2 Γ0(4), then θ(γz)2 −1 = (cz + d): θ(z) d Ariel Pacetti Half integral weight modular forms Definitions Application Quaternions HMF Application II Motivation The classical theta function 1 X 2 θ(z) = e2πin z : n=−∞ a b It is not hard to see that if γ = c d 2 Γ0(4), then θ(γz)2 −1 = (cz + d): θ(z) d 2 So θ(z) 2 M1(Γ0(4); χ−1). Ariel Pacetti Half integral weight modular forms k a b f (γz) = J(γ; z) (d)f (z) 8γ = c d 2 Γ0(4N) f (z) is holomorphic at the cusps. We denote by Mk=2(4N; ) the space of such forms and Sk=2(4N; ) the subspace of cuspidal ones. Let k be an odd positive integer, N a positive integer and a character modulo N. Definition A modular form of weight k=2, level 4N and character is an holomorphic function f : H ! C such that Definitions Application Quaternions HMF Application II Definition θ(γz) We consider the factor of automorphy J(γ; z) = θ(z) . Ariel Pacetti Half integral weight modular forms k a b f (γz) = J(γ; z) (d)f (z) 8γ = c d 2 Γ0(4N) f (z) is holomorphic at the cusps. We denote by Mk=2(4N; ) the space of such forms and Sk=2(4N; ) the subspace of cuspidal ones. Definition A modular form of weight k=2, level 4N and character is an holomorphic function f : H ! C such that Definitions Application Quaternions HMF Application II Definition θ(γz) We consider the factor of automorphy J(γ; z) = θ(z) . Let k be an odd positive integer, N a positive integer and a character modulo N. Ariel Pacetti Half integral weight modular forms k a b f (γz) = J(γ; z) (d)f (z) 8γ = c d 2 Γ0(4N) f (z) is holomorphic at the cusps. We denote by Mk=2(4N; ) the space of such forms and Sk=2(4N; ) the subspace of cuspidal ones. Definitions Application Quaternions HMF Application II Definition θ(γz) We consider the factor of automorphy J(γ; z) = θ(z) . Let k be an odd positive integer, N a positive integer and a character modulo N. Definition A modular form of weight k=2, level 4N and character is an holomorphic function f : H ! C such that Ariel Pacetti Half integral weight modular forms f (z) is holomorphic at the cusps. We denote by Mk=2(4N; ) the space of such forms and Sk=2(4N; ) the subspace of cuspidal ones. Definitions Application Quaternions HMF Application II Definition θ(γz) We consider the factor of automorphy J(γ; z) = θ(z) . Let k be an odd positive integer, N a positive integer and a character modulo N. Definition A modular form of weight k=2, level 4N and character is an holomorphic function f : H ! C such that k a b f (γz) = J(γ; z) (d)f (z) 8γ = c d 2 Γ0(4N) Ariel Pacetti Half integral weight modular forms We denote by Mk=2(4N; ) the space of such forms and Sk=2(4N; ) the subspace of cuspidal ones. Definitions Application Quaternions HMF Application II Definition θ(γz) We consider the factor of automorphy J(γ; z) = θ(z) . Let k be an odd positive integer, N a positive integer and a character modulo N. Definition A modular form of weight k=2, level 4N and character is an holomorphic function f : H ! C such that k a b f (γz) = J(γ; z) (d)f (z) 8γ = c d 2 Γ0(4N) f (z) is holomorphic at the cusps. Ariel Pacetti Half integral weight modular forms Definitions Application Quaternions HMF Application II Definition θ(γz) We consider the factor of automorphy J(γ; z) = θ(z) . Let k be an odd positive integer, N a positive integer and a character modulo N. Definition A modular form of weight k=2, level 4N and character is an holomorphic function f : H ! C such that k a b f (γz) = J(γ; z) (d)f (z) 8γ = c d 2 Γ0(4N) f (z) is holomorphic at the cusps. We denote by Mk=2(4N; ) the space of such forms and Sk=2(4N; ) the subspace of cuspidal ones. Ariel Pacetti Half integral weight modular forms 1 Tn = 0 if n is not a square. 2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product. 3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ! = 2 , then Tp2 acts like ! −1 n !−1 2 k−1 a 2 + (n) p a + (p )p a 2 : p n p p n n=p Hence there exists a basis of eigenforms for the Hecke operators prime to 4N. Definitions Application Quaternions HMF Application II Hecke operators Via a double coset action, one can define Hecke operators fTngn≥1 acting on Sk=2(4N; ). They satisfy the properties: Ariel Pacetti Half integral weight modular forms 2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product. 3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ! = 2 , then Tp2 acts like ! −1 n !−1 2 k−1 a 2 + (n) p a + (p )p a 2 : p n p p n n=p Hence there exists a basis of eigenforms for the Hecke operators prime to 4N. Definitions Application Quaternions HMF Application II Hecke operators Via a double coset action, one can define Hecke operators fTngn≥1 acting on Sk=2(4N; ). They satisfy the properties: 1 Tn = 0 if n is not a square. Ariel Pacetti Half integral weight modular forms 3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ! = 2 , then Tp2 acts like ! −1 n !−1 2 k−1 a 2 + (n) p a + (p )p a 2 : p n p p n n=p Hence there exists a basis of eigenforms for the Hecke operators prime to 4N. Definitions Application Quaternions HMF Application II Hecke operators Via a double coset action, one can define Hecke operators fTngn≥1 acting on Sk=2(4N; ). They satisfy the properties: 1 Tn = 0 if n is not a square. 2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.