A Brief Introduction to Modular Forms Catherine M. Hsu Department - PowerPoint PPT Presentation

A Brief Introduction to Modular Forms Catherine M. Hsu Department of Mathematics University of Oregon Coding Theory, Cryptography, and Number Theory Seminar Clemson University September 18, 2017 Catherine M. Hsu University of Oregon September 18, 2017 1 / 23

Congruence Subgroups for SL ( 2 , Z ) Let N > 1 be an integer. � 1 0 � Γ( N ) = { γ ∈ SL ( 2 , Z ) | γ ≡ ( mod N ) } 0 1 � 1 ∗ � Γ 1 ( N ) = { γ ∈ SL ( 2 , Z ) | γ ≡ ( mod N ) } 0 1 Γ 0 ( N ) = { γ ∈ SL ( 2 , Z ) | γ ≡ ( ∗ ∗ 0 ∗ ) ( mod N ) } SL ( 2 , Z ) acts on h via Möbius transformations: F � a b � z ∈ h , γ = ∈ SL ( 2 , Z ) , c d i ζ 3 ζ 6 γ z := az + b cz + d . − 1 − 1 0 1 1 2 2 Catherine M. Hsu University of Oregon September 18, 2017 2 / 23

Modular forms: Definition A modular form of weight k and level N is a complex function f : h → C satisfying the following properties: f is holomorphic on h ; 1 � a b � f ( γ z ) = ( cz + d ) k f ( z ) , ∀ γ = ∈ Γ 0 ( N ) ; 2 c d f is holomorphic at the cusps. 3 — f vanishes at the cusps. 4 Catherine M. Hsu University of Oregon September 18, 2017 3 / 23

Modular forms with Nebentypus Consider the following spaces of modular forms: M k (Γ 1 ( N )) , S k (Γ 1 ( N )) M k (Γ 0 ( N )) , S k (Γ 0 ( N )) For a Dirichlet character ε : ( Z / N Z ) × → C × , we say a modular form f ∈ M k (Γ 1 ( N )) has Nebentypus ε if f ( γ z ) = ε ( d )( cz + d ) k f ( z ) , ∀ γ ∈ Γ 0 ( N ) . We denote this space of modular forms by M k ( N , ε ) . Catherine M. Hsu University of Oregon September 18, 2017 4 / 23

Fourier expansions of modular forms For f ∈ M k (Γ 0 ( N )) , we have f ( z ) = f ( z + 1 ) , and hence, there is a Fourier expansion for f at ∞ : ∞ � a n q n , q = e 2 π iz . f ( z ) = n = 0 The coefficients { a n } are called the Fourier coefficients of f . Catherine M. Hsu University of Oregon September 18, 2017 5 / 23

Decomposition and Dimension of M k (Γ 1 ( N )) For each N > 1 , we have a decomposition � M k (Γ 1 ( N )) = M k ( N , ε ) , ε where ε runs over all Dirichlet characters mod N such that ε ( − 1 ) = ( − 1 ) k . We can also compute the dimension of M k ( SL ( 2 , Z ))) :  0 , if k < 0 or k is odd ,   dim M k ( SL ( 2 , Z )) = ⌊ k / 12 ⌋ + 1 , if k �≡ 2 ( mod 12 ) ,   ⌊ k / 12 ⌋ , if k ≡ 2 ( mod 12 ) . Catherine M. Hsu University of Oregon September 18, 2017 6 / 23

First Example: Eisenstein Series Let k > 2 be an even integer and define for each z ∈ h 1 � G k ( z ) = ( mz + n ) k . ( m , n ) � =( 0 , 0 ) Then, G k ( z ) ∈ M k ( SL ( 2 , Z )) with Fourier expansion ∞ � � 1 − 2 k � σ k − 1 ( n ) q n G k ( z ) = 2 ζ ( k ) . B k n = 1 � �� � E k ( z ) Catherine M. Hsu University of Oregon September 18, 2017 7 / 23

Identities involving sums of powers of divisors For k = 4 , 6 , 8 , 10 , and 14 , the dimension of M k ( SL ( 2 , Z )) is 1. Each of these spaces is spanned by the Eisenstein series E k ( z ) , and so, we have the following equalities: E 4 ( z ) 2 = E 8 ( z ) E 4 ( z ) E 6 ( z ) = E 10 ( z ) E 6 ( z ) E 8 ( z ) = E 4 ( z ) E 10 ( z ) = E 14 ( z ) Catherine M. Hsu University of Oregon September 18, 2017 8 / 23

Identities involving sums of powers of divisors (cont.) Comparing Fourier coefficients then yields identities such as n − 1 σ 3 ( m ) σ 3 ( n − m ) = σ 7 ( n ) − σ 3 ( n ) � 120 m = 1 n − 1 σ 3 ( m ) σ 9 ( n − m ) = σ 13 ( n ) − 11 σ 9 ( n ) + 10 σ 3 ( n ) � 2640 m = 1 Catherine M. Hsu University of Oregon September 18, 2017 9 / 23

Proof of identity with E 4 ( z ) 2 = E 8 ( z ) E 4 ( z ) = 1 + 240 q + 2160 q 2 + · · · = 1 + 240 � ∞ n = 1 σ 3 ( n ) q n E 8 ( z ) = 1 + 480 q + 61920 q 2 + · · · = 1 + 480 � ∞ n = 1 σ 7 ( n ) q n 2 Since E 4 ( z ) 2 = E 8 ( z ) , for each n ≥ 1 , we have n − 1 � 480 · σ 3 ( n ) + 240 2 σ 3 ( m ) σ 3 ( n − m ) 2 = 480 · σ 7 ( n ) m = 1 n − 1 σ 3 ( m ) σ 3 ( n − m ) = σ 7 ( n ) − σ 3 ( n ) � ⇒ 120 m = 1 Catherine M. Hsu University of Oregon September 18, 2017 10 / 23

Congruences between modular forms For a prime p ∈ Z , we say that two modular forms ∞ ∞ � � a n q n , b n q n f 1 = f 2 = n = 0 n = 0 are congruent mod p if a n ≡ b n ( mod p ) , ∀ n ≥ 0 , where p ⊆ Q is a prime ideal lying over p . Catherine M. Hsu University of Oregon September 18, 2017 11 / 23

Congruences between Eisenstein series Let p ∈ Z be prime. If k , k ′ are two even integers satisfying k ≡ k ′ ( mod p − 1 ) , then Fermat’s Little Theorem implies σ k − 1 ( n ) ≡ σ k ′ − 1 ( n ) ( mod p ) , ∀ n ≥ 1 . Thus, a n ( E k ) ≡ a n ( E k ′ ) ( mod p ) , ∀ n ≥ 1 . We also have a congruence between a 0 ( E k ) and a 0 ( E k ′ ) so that E k ≡ E k ′ ( mod p ) . Catherine M. Hsu University of Oregon September 18, 2017 12 / 23

The Discriminant Function For z ∈ h , define 1 � E 4 ( z ) 3 − E 6 ( z ) 2 � ∆( z ) = . 1728 Since ∆ vanishes at ∞ , we have ∆ ∈ S 12 ( SL ( 2 , Z )) . Moreover, ∞ ∞ � � ( 1 − q n ) 24 = τ ( n ) q n , ∆( z ) = q n = 1 n = 1 where τ ( n ) is the Ramanujan tau function. Catherine M. Hsu University of Oregon September 18, 2017 13 / 23

A Congruence of Ramanujan The first few values of τ ( n ) are given below: n 1 2 3 4 5 6 7 · · · τ ( n ) 1 − 24 252 − 1472 4830 − 6048 − 16744 · · · In particular, we note that τ ( n ) is multiplicative and satisfies τ ( n ) ≡ σ 11 ( n ) ( mod 691 ) , ∀ n ≥ 1 . Catherine M. Hsu University of Oregon September 18, 2017 14 / 23

Hecke theory: Definitions For each integer m ≥ 1 , there is a linear operator T m , called the m th Hecke operator, acting on M k ( SL ( 2 , Z )) . If M m denotes the set of 2 × 2 integral matrices with determinant m , then for a modular form f ( z ) ∈ M k ( SL ( 2 , Z )) and z ∈ h , � az + b � � T m f ( z ) = m k − 1 ( cz + d ) − k f . cz + d ( a b c d ) ∈ SL ( 2 , Z ) \M m Catherine M. Hsu University of Oregon September 18, 2017 15 / 23

Hecke theory: Equivalent definitions Hecke operators also arise in the context of: abstract Hecke rings such as R (Γ 0 ( N ) , ∆ 0 ( N )) modular correspondences on (Γ 0 ( N ) \ h ) × (Γ 0 ( N ) \ h ) certain moduli spaces such as S 1 ( N ) Catherine M. Hsu University of Oregon September 18, 2017 16 / 23

Hecke theory: Fourier expansions Let f ( z ) have Fourier expansion f ( z ) = � ∞ n = 0 a n q n . Then � � � � r k − 1 a mn / r 2 q n . T m f ( z ) = n ≥ 0 r | ( m , n ) r > 0 Important observation: The Hecke operators T m all commute! Catherine M. Hsu University of Oregon September 18, 2017 17 / 23

Hecke action on the discriminant function Consider the action of T m on ∆ ∈ S 12 ( SL ( 2 , Z )) . Since dim ( S 12 ( SL ( 2 , Z ))) = 1 , T m ∆ must be a multiple of ∆ for each m ≥ 1. In particular, since T m ∆ = τ ( m ) q + · · · , ∆ = q + · · · , we must have T m ∆ = τ ( m )∆ , ∀ m ≥ 1 . Catherine M. Hsu University of Oregon September 18, 2017 18 / 23

Hecke action on eigenforms More generally, if f ( z ) is a normalized Hecke eigenform, then T m f ( z ) = λ m a 0 + λ m q + · · · , = σ k − 1 ( m ) a 0 + a m q + · · · . Hence, for each m ≥ 1 , we have an equality λ m = a m . Applying this with the formula for the action of T m on f yields � r k − 1 a mn / r 2 . a m a n = r | ( m , n ) r > 0 Catherine M. Hsu University of Oregon September 18, 2017 19 / 23

Old and new spaces of S k (Γ 0 ( N )) Let d , M , N > 0 be integers such that dM | N , and define ι ∗ d , M , N : S k (Γ 0 ( M )) → S k (Γ 0 ( N )) , f ( z ) �→ d k − 1 f ( dz ) . For a fixed N , we define the old subspace of S k (Γ 0 ( N )) by � S k (Γ 0 ( N )) old = ι ∗ d , M , N ( S k (Γ 0 ( M )) , where the sum is taken over all d , M with dM | N and M � = N . Catherine M. Hsu University of Oregon September 18, 2017 20 / 23

Old and new spaces of S k (à 0 ( N )) (cont.) Moreover, there is a Hecke-equivariant decomposition � S k (à 0 ( N )) = S k (à 0 ( N )) old S k (à 0 ( N )) new . � �� � � �� � images of spanned by level-raising newforms Catherine M. Hsu University of Oregon September 18, 2017 21 / 23

Old and new spaces of S 2 (Γ 0 ( 33 )) Using various dimension formulas, we find that dim ( S 2 (Γ 0 ( 33 ))) = 3 . Since S 2 (Γ 0 ( 3 )) = 0 , we have a decomposition S 2 (Γ 0 ( 33 )) old = ι ∗ 1 , 11 , 33 ( S 2 (Γ 0 ( 11 ))) ⊕ ι ∗ 3 , 11 , 33 ( S 2 (Γ 0 ( 11 ))) . Thus, � S 2 (Γ 0 ( 33 )) = S 2 (Γ 0 ( 33 )) old S 2 (Γ 0 ( 33 )) new . � �� � � �� � dim 2 dim 1 Catherine M. Hsu University of Oregon September 18, 2017 22 / 23

Importance of Hecke theory There are many deep connections between Hecke theory and the theory of modular forms including: The strong multiplicity one theorem Duality between spaces of cusp forms and Hecke algebras Galois representations attached to Hecke eigenforms Catherine M. Hsu University of Oregon September 18, 2017 23 / 23