Far Eastern Mathematical Journal

To content of the issue

The Eisenstein-Hecke series and their properties

Bykovskii V.A.

2016, issue 1, Ñ. 3-8

Let $\Gamma_0(N)$ be the congruence subgroup of level N. If N is not a square-free number then the Fourier coefficients of the classical Eisenstein series are not multiplicative. In the paper we construct the modified Eisenstein-Hecke series with the desired property of multiplicativity. This result is of great importance for investigating trace formulas on the space of cusp forms. Similar results were obtained earlier by S. Gelbart and H. Jacquet using the theory of adeles.

modular form, Eisenstein series, Hecke operator

Download the article (PDF-file)


[1] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, NJ, 1971.
[2] J.-M. Deshouillers, H. Ivaniec, «Kloosterman sums and Fourier coofficients of cups forms», Invent. Math., 70 (1988), 219–288.
[3] V. A. Bykovskii, N. V. Kuznetsov, A. I. Vinogradov, «Generalized summation formula for inhomogeneous convolution», In Proc. Int. Conf. Automorphic functions and their applications, Khabarovsk, 1989, 18–63.
[4] M. N. Huxley, «Scattering matrices for congruence subgroups», Modular Forms, ed. by R. Rankin, 1984, 141–156.
[5] S. Gelbart, H. Jacquet, «Forms on GL(2) from the analytic point of viev», Proc. Symp. Pure Math., V. 33, part 1, RI, Providence, 1979, 213–251.

To content of the issue