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55 #270 10D15

Asai, Tetsuya On the Fourier coefficients of automorphic forms at various cusps and some applications to Rankin’s convolution. J. Math. Soc. Japan 28 (1976), no. 1, 48–61. Let N be a square-free integer, χ a Dirichlet character (mod N). For each decomposition N = N_{1}N_{2} and the corresponding decomposition of χ as χ_{1}χ_{2} with χ_{i} a character (mod N_{i}), the Atkin-Lehner operator

f ↦ f|W_{N_{1}}, where W_{N_{1}} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with a, d ∈ N_{1}Z, b ∈ Z, c ∈ NZ, ad−

bc = N_{1}, gives a mapping from S_{k}(Γ_{0}(N), χ) to S_{k}(Γ_{0}(N), χ_{1}\overline{χ}_{2}). Let f ∈ S_{k}(Γ_{0}(N), χ) be a normalized new form in the sense of the work of A. Atkin and J. Lehner [Math. Ann. 185 (1970), 134–160; MR 42#3022] with f|T_{p} = a_{p}f for all primes p. Then it is shown that f|W_{N_{1}} = λ_{N_{1}}, where f_{N_{1}} ∈ S_{k}(Γ_{0}(N), χ_{1}\overline{χ}_{2}) is the normalized new form with pth eigenvalue \overline{χ}_{1}(p)a_{p} (if p ∤ N_{1}) or χ_{2}(p)\overline{a}_{p} (if p ∤ N_{2}) and λ is an explicitly given complex number of absolute value 1. Thus one can obtain the Fourier developments of f at all cusps of Γ_{0}(N) from its Fourier development at infinity.

The second main result, which generalizes results of R. Rankin [Proc. Cambridge Philos. Soc. 35 (1939), 351–356; Zbl 21, 392; ibid. 35 (1972), 357–372; Zbl 21, 392] for forms of level 1 and of A. Ogg [Invent. Math. 7 (1969), 297–312; MR 40#88] for forms of Haupttypus, gives an analytic continuation for the convolution of the Dirichlet series associated to two cusp forms of squarefree level and arbitrary character; this overlaps with recent work of W. Li [Math. Ann. 212 (1975), 285–315; MR 51#5498]. As an application, the author generalises the lifting given by H. Naganuma [J. Math. Soc. Japan 25 (1973), 547–555; MR 48#10987] from S_{k}(Γ_{0}(D), (D/·)) to the space of Hilbert modular forms for the real quadratic field of prime discriminant D to the case of arbitrary odd D; this was done by a different method by the reviewer [Invent. Math. 30 (1975), no. 1, 1–46; MR 52#3062]. Don Zagier