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Sur les séries \(L\) associées aux formes modulaires. (On L-series associated to modular forms). (French) Zbl 0756.11012

The first part of this paper is concerned with theorems like Weil’s on the determination of modular forms for \(\Gamma_ 0(N)\) by functional equations for the \(L\)-series. The author formulates a result in which only a finite number of twists of the \(L\)-series of Dirichlet characters is needed. The second part of the paper is concerned with Manin’s explicit formula for the eigenvalues of Hecke operators. Here an extension of Manin’s result is formulated.
In both cases the author follows the proof given by Weil or by Manin and in both cases he overlooks the existence of Dirichlet characters which are neither primitive nor principal, so that the proofs are incomplete.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F25 Hecke-Petersson operators, differential operators (one variable)
11F11 Holomorphic modular forms of integral weight
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References:

[1] JACQUET (H.) and LANGLANDS (R.) . - Automorphic forms on GL(2) , Lecture Notes in Mathematics 114, Springer-Verlag, Berlin-Heidelberg-New York, 1970 . MR 53 #5481 | Zbl 0236.12010 · Zbl 0236.12010
[2] LI (W.-C.) Winnie . - Newforms and functional equations , Math. Ann., t. 212, 1975 , p. 285-315. MR 51 #5498 | Zbl 0278.10026 · Zbl 0278.10026
[3] LI WINNIE (W.-C.) . - Une caractérisation des représentations automorphes de GL(2) et de GL(1) , C. R. Acad. Sci. Paris, t. A, 290, 1980 , p. 681-684. MR 83h:22034b | Zbl 0437.12011 · Zbl 0437.12011
[4] LI WINNIE (W.-C.) . - Hecke-Weil-Jacquet-Langlands Theorem revisited , Lecture Notes in Mathematics 751, Springer-Verlag, Berlin-Heidelberg-New York, p. 206-220. MR 81m:10058 | Zbl 0413.10023 · Zbl 0413.10023
[5] LI WINNIE (W.-C.) . - On converse theorems for GL(2) and GL(1) , Amer. J. Math., t. 103, 1981 , p. 851-886. MR 83b:22024 | Zbl 0477.12013 · Zbl 0477.12013
[6] MANIN (Y.I.) . - Explicite formulas for the eigenvalues of Hecke operators , Acta Arith., t. 24, 1973 , p. 239-249. Article | MR 48 #3886 | Zbl 0273.10018 · Zbl 0273.10018
[7] MANIN (Y.I.) . - Cusps and the zeta function of modular curves , Izv. AN SSSR, t. 3, 36, 1972 , p. 19-66. Zbl 0243.14008 · Zbl 0243.14008
[8] OGG (A.) . - Modular forms and Dirichlet series . - Benjamin, New York, 1968 . Zbl 0191.38101 · Zbl 0191.38101
[9] PIATECKII SHAPIRO (I.I.) . - On the Weil-Jacquet-Langlands theorem. Lie groups and their Representations , Halsted, New York, 1975 , p. 583-595. Zbl 0329.22014 · Zbl 0329.22014
[10] RAZAR (M.) . - Modular forms for \Gamma 0(N) and Dirichlet series , Trans. Amer. Math. Soc., t. 231, 2, 1977 , p. 489-495. MR 56 #2926 | Zbl 0364.10015 · Zbl 0364.10015
[11] WEIL (A.) . - Uber der Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , Math. Ann., t. 168, 1967 , p. 149-156. MR 34 #7473 | Zbl 0158.08601 · Zbl 0158.08601
[12] WEIL (A.) . - Dirichlet Series and Automorphic Forms , Lecture Notes in Mathematics 189, Springer-Verlag, Berlin-Heidelberg-New York, 1971 . Zbl 0218.10046 · Zbl 0218.10046
[13] WEIL (A.) . - On the analogue of the modular group in characteristic p. Functional Analysis and Related Fields , p. 211-223, Springer-Verlag, Berlin and New York, 1970 . MR 42 #5913 | Zbl 0226.10031 · Zbl 0226.10031
[14] WEIL (A.) . - Zeta-functions and Mellin transforms , Colloque on Algebraic Geometry, Bombay, p. 409-426, Tata Institute Bombay, 1969 . MR 41 #6857 | Zbl 0193.49104 · Zbl 0193.49104
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