Elliptic Modular Forms and Their Applications - Up To

38 D. Zagierbecause the transformation law (2) of f implies that the summand associatedto a matrix M = ( abcd)∈Mm is indeed unchanged if M is replaced by γMwith γ ∈ Γ 1 , and from (40) one also easily sees that T m f is holomorphicin H and satisfies the same transformation law and growth properties as f,so T m indeed maps M k (Γ 1 ) to M k (Γ 1 ). Finally, to calculate the effect of T mon Fourier developments, we note that a set of representatives of Γ 1 \M m isgiven by the upper triangular matrices ( )ab0 d with ad = m and 0 ≤ b01d k∑b (mod d)( ) az + bf. (41)dIf f(z) has the Fourier development (3), then a further calculation with (41),again left to the reader, shows that the function T m f(z) has the Fourier expansionT m f(z) = ∑ (m/d) ∑ k−1 a n q mn/d2d|mn≥0d>0d|n= ∑ n≥0( ∑r|(m,n)r>0r k−1 a mn/r 2)q n .(42)An easy but important consequence of this formula is that the operators T m(m ∈ N) all commute.Let us consider some examples. The expansion (42) begins σ k−1 (m)a 0 +a m q + ···,soiff is a cusp form (i.e., a 0 =0), then so is T m f.Inparticular,since the space S 12 (Γ 1 ) of cusp forms of weight 12 is 1-dimensional, spannedby Δ(z), it follows that T m Δ is a multiple of Δ for every m ≥ 1. SincetheFourier expansion of Δ begins q + ··· and that of T m Δ begins τ(m)q + ···,the eigenvalue is necessarily τ(m), soT m Δ=τ(m)Δ and (42) givesτ(m) τ(n) =∑r|(m,n)r 11 τ( mnr 2 )for all m, n ≥ 1 ,proving Ramanujan’s multiplicativity observations mentioned in §2.4. By thesame argument, if f ∈ M k (Γ 1 ) is any simultaneous eigenfunction of all ofthe T m , with eigenvalues λ m ,thena m = λ m a 1 for all m. We therefore havea 1 ≠0if f is not identically 0, and if we normalize f by a 1 =1(such anf is called a normalized Hecke eigenform, orHecke form for short) then wehaveT m f = a m f, a m a n = ∑r k−1 a mn/r 2 (m, n ≥ 1) . (43)r|(m,n)Examples of this besides Δ(z) are the unique normalized cusp forms f(z) =Δ(z)E k−12 (z) in the five further weights where dim S k (Γ 1 )=1(viz. k =16,18, 20, 22 and 26) and the function G k (z) for all k ≥ 4, forwhichwehaveT m G k = σ k−1 (m)G k , σ k−1 (m)σ k−1 (n) = ∑ r|(m,n) rk−1 σ k−1 (mn/r 2 ).(This