Fourier Coefficients Of Eisenstein Series Research Articles

Fourier coefficients of Eisenstein series on non-congruence subgroups

Let Γ be a subgroup of PSL2 (ℤ) of finite index. In this note we are concerned with the arithmetic nature of the Fourier coefficients of holomorphic Eisenstein series on Γ.

Fourier coefficients of Eisenstein series of degree $3$

Fourier coefficients of Eisenstein series of degree $3$

A note on local densities of quadratic forms

Let L, M be regular quadratic lattices over Zp. The local density αp(L, M) is an important invariant in the theory of representation of quadratic forms and they appear naturally in Fourier coefficients of Eisenstein series. In spite of the importance we knew little except the case when either L = M or rk L = 1 and M is unimodular. Evaluating them is a laborious task.

Table of the Fourier coefficients of Eisenstein series of degree $3$

Table of the Fourier coefficients of Eisenstein series of degree $3$

Functional Equations of Whittaker Functions on p-Adic Groups

Introduction. The Fourier coefficients of Eisenstein series on a rational tube domain have Euler product expansions, and one can view the Euler factors as functions of the weight of the Eisenstein series. Our main result is the existence of a functional equation for these functions, the local singular series. To prove this we observe that the local singular series is the value of a Whittaker function W attached to a class 1 principal series representation PS (X) induced from a character X on P, a maximal parabolic subgroup of G. That is, W transforms under left translation by elements of the unipotent radical, N, of P according to a character 4. The crucial point is to show that for suitable 4 there is only one operator, up to scalar multiple, that intertwines PS(x) with the space of Whittaker functions for 4. Functional equations then follow from the existence of intertwining operators corresponding to certain elements of the Weyl group. In contrast to [9] and [12], in dealing with Whittaker models we work with a horocycle N that is not maximal. This is necessary because we are dealing with degenerate principal series representations. In a separate paper, we apply the functional equations to the computation of local singular series for Baily's exceptional modular group. We now summarize the contents of the present paper. In Section 1 we introduce the principal series representations PS(X) on G and generic characters. By means of the Frobenius Reciprocity Law and a filtration on the space PS(x) deriving from the cellular decomposition of G with respect to a pair of parabolic subgroups, we reduce the uniqueness theorem to a study of certain integrals. In Section 2 we prove a technical lemma about p-adic fiber spaces and use it to show that the integrals mentioned above must all vanish. In Section 3 we prove the uniqueness theorem (3.2). Then we show how to obtain Whittaker maps as Cauchy principal value integrals. This gives rise to analytic families of distributions on G. As a consequence of

Eisenstein series and decomposition theory over function fields

The theory of Eisenstein series for SL2(IR) has been well-explored (cf. [11]). In [12], Langlands developed the theory of Eisenstein series in great generality, namely, for arbitrary reductive Lie groups. Harder in [5] studied the corresponding theory for simply connected Chevalley groups over function fields of one variable. The present paper is concerned with a situation which is more restricted than I"5] in the sense that only GL z is considered, and more general in the sense that the Eisenstein functions are supposed to be invariant only by an open compact subgroup. We prove that these Eisenstein series are rational functions, satisfy functional equations, etc., as expected (Sects. 2, 3, and 5). One of the main themes of this paper is to study the intertwining operators which arise from the constant Fourier coefficients of Eisenstein series (Sects. 3 and 6). These are rational operators. Since they occur in the functional equations for Eisenstein series, the degrees of their numerators are essential in computing the dimensions of the spaces of cusp forms for congruence subgroups. This will be shown in a subsequent paper [7] by Harder et al. On the other hand, the computations of these intertwining operators are inevitable if one wants to get hold of the "bad" terms occurring in the trace formula. Another main theme is discussed in Sect. 7. There we develop a decomposition theory on the space of automorphic functions (of some central quasi-character and invariant under a congruence subgroup) which are eigenfunctions of the Hecke operator T o with eigenvalue 2~ at some "good" place v. Namely, any such function is a direct sum of an Eisenstein series and a cusp form. If we interpret T~ as the "Beltrami operator" and view eigenfunctions of T~ with eigenvalue 2~ as "holomorphic functions" as explained in 1-15, 16], then this decomposition theory is an analogue of the classical decomposition theory introduced by Hecke (cf. [8, No. 24]). Consequently, the study of automorphic functions is reduced to the study of cusp forms. There is also a similar result from the representation-theoretic point

Fourier Coefficients of Eisenstein Series of One Epstein Series of One Complex Variable for the Special Linear Group

Fourier Coefficients of Eisenstein Series of One Epstein Series of One Complex Variable for the Special Linear Group

Fourier coefficients of Eisenstein series of one complex variable for the special linear group

Fourier coefficients of Eisenstein series of one complex variable for the special linear group

Some properties of Fourier coefficients of Eisenstein series of degree two.

Article Some properties of Fourier coefficients of Eisenstein series of degree two. was published on January 1, 1974 in the journal Journal für die reine und angewandte Mathematik (volume 1974, issue 265).

On Fourier coefficients of Eisenstein series

On Fourier coefficients of Eisenstein series