Elliptic Modular Forms and Their Applications - Up To

40 D. Zagierler product L(f,s) =∏ ( a p 1+p prime p s + a p 2p 2s + ···), and the second tells usthat the power series ∑ ∞ν=0 a p ν xν for p prime equals 1/(1 − a p x + p k−1 x 2 ).Combining these two statements gives Hecke’s fundamental Euler productdevelopmentL(f,s) =∏11 − a p p −s + p k−1−2s (44)p primefor the L-series of a normalized Hecke eigenform f ∈ M k (Γ 1 ),asimpleexamplebeing given byL(G k ,s) = ∏ p11 − (p k−1 +1)p −s = ζ(s) ζ(s − k +1).+ pk−1−2s For eigenforms on Γ 0 (N) there is a similar result except that the Euler factorsfor p|N have to be modified suitably.The L-series have another fundamental property, also discovered by Hecke,which is that they can be analytically continued in s and then satisfy functionalequations. We again restrict to Γ = Γ 1 and also, for convenience, tocusp forms, though not any more just to eigenforms. (The method of proof extendsto non-cusp forms but is messier there since L(f,s) then has poles, andsince M k is spanned by cusp forms and by G k ,whoseL-series is completelyknown, there is no loss in making the latter restriction.) From the estimatea n = O(n k/2 ) proved in §2.4 we know that L(f,s) converges absolutely in thehalf-plane R(s) > 1+k/2. Takes in that half-plane and consider the Eulergamma function∫ ∞Γ (s) = t s−1 e −t dt .0Replacing t by λt in this integral gives Γ (s) =λ ∫ s ∞t s−1 e −λt dt or λ −s =0Γ (s) ∫ −1 ∞t s−1 e −λt dt for any λ>0. Applying this to λ =2πn, multiplying0by a n , and summing over n, weobtain(2π) −s Γ (s) L(f,s) =∞∑∫ ∞a n t s−1 e −2πnt dt = t s−1 f(it) dt00(R(s) > k )2 +1 ,n=1∫ ∞where the interchange of integration and summation is justified by the absoluteconvergence. Now the fact that f(it) is exponentially small for t →∞(because f is a cusp form) and for t → 0 (because f(−1/z) =z k f(z) )impliesthat the integral converges absolutely for all s ∈ C and hence that thefunctionL ∗ (f,s) := (2π) −s Γ (s) L(f,s) = (2π) −s Γ (s)∞∑n=1a nn s (45)