Hecke operators on \(\Gamma_0(m)\). (English) Zbl 0177.34901

The paper is concerned with the vector space of cusp forms of positive integral weight (even integral negative dimension) on the subgroup \(\Gamma_0(m)\) of the modular group. For every proper divisor \(m'\) of \(m\), and every divisor \(d\) of \(m/m'\), the form \(F(d\tau)\) is on \(\Gamma_0(m)\) if \(F(\tau)\) is on \(\Gamma_0(m')\); the space spanned by all such forms we call “oldforms”. The “newforms” are then the orthogonal complement of the oldforms with respect to the Petersson scalar product. the newforms have a basis of forms \(F_i(\tau)\) which are eigenfunctions of all Hecke operators \(T_p\) for \((p,m)=1\); it is proved here that each \(F_i(\tau)\) is also an eigenfunction of the corresponding operators \(U_q\), and of certain non-modular involutory operators \(W_q\) in the normalizer of \(\Gamma_0(m)\), for each prime \(q\) dividing \(m\). The eigenvalues of \(U_q\) and \(W_q\) are determined, and it is shown that any two distinct newforms have an infinity of different eigenvalues for the operators \(T_p\). Various additional results are given for the case \(m=m'q^2\), where \(q\) is an odd prime, \(2,4\), or \(8\).
Reviewer: A. O. L. Atkin


11F25 Hecke-Petersson operators, differential operators (one variable)
11F12 Automorphic forms, one variable
Full Text: DOI EuDML


[1] Eichler, M.: Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktionen. Arch. der Math.5, 355-366 (1954). · Zbl 0059.03804
[2] Gantmacher, F. R.: The theory of matrices. New York: Chelsea 1959. · Zbl 0085.01001
[3] Gunning, R. C.: Lectures on modular forms. Princeton: University Press 1962. · Zbl 0178.42901
[4] Hecke, E.: Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, I, II. Math. Ann.114, 1-28, 316-351 (1937). · Zbl 0015.40202
[5] – Analytische Arithmetik der positiven quadratischen Formen. Kgl. Danske Videnskabernes Selskab. XIII,12 (1940).
[6] Igusa, J.-I.: Kroneckerian model of fields of elliptic modular functions. Amer. J. Math.81, 561-577 (1959). · Zbl 0093.04502
[7] Lehner, J.: Discontinuous groups and automorphic functions. Providence, 1964. · Zbl 0178.42902
[8] ?? Newman, M.: Weierstrass points of ?0(n). Ann. of Math.79, 360-368 (1964). · Zbl 0124.29203
[9] Newman, M.: The normalizer of certain modular subgroups. Canad. J. Math.8, 29-31 (1956). · Zbl 0071.02501
[10] Petersson, H.: Konstruktion der sämtlichen Lösungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwicklung, I, II, III. Math. Ann.116, 401-412 (1939),117, 39-64 (1939),117, 277-300 (1940). · JFM 65.0355.02
[11] Rankin, R. A.: Hecke operators on congruence subgroups of the modular group. Math. Ann.168, 40-58 (1967). · Zbl 0145.32001
[12] Selberg, A.: On the estimation of Fourier coefficients of modular forms. Proc. Symposium Pure Math. (Amer. Math. Soc.) VIII, 1-15 (1965). · Zbl 0142.33903
[13] Shimura, G.: Sur les intégrales attachées aux formes automorphes. J. Math. Soc. Japan11, 291-311 (1959). · Zbl 0090.05503
[14] Weil, A.: On some exponential sums. Proc. Acad. Sci. USA.34, 204-207 (1948). · Zbl 0032.26102
[15] ?? Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.168, 149-156 (1967). · Zbl 0158.08601
[16] Wohlfahrt, K.: Über Operatoren Heckescher Art bei Modulformen reeller Dimension. Math. Nachr.16, 233-256 (1957). · Zbl 0080.06101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.