Diagonalizing modular forms
type: Item
instance of: scholarly article
title: Diagonalizing modular forms
zbMATH Open document ID: 0594.10018
DOI: 10.1016/0021-8693(86)90064-5
author: Wen-Ching Winnie Li
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let S(N,k,\(\epsilon)\) denote the space of cusp forms of weight k, level N, and character \(\epsilon\). The theory of newforms describes the structure of S(N,k,\(\epsilon)\). This paper gives a sharper result in this direction. For each prime q dividing N, an operator \(C_ q\), arising from the familiar \(U_ q\) and the Atkin-Lehner \(W_ q\) operator, is introduced as follows: \(C_ q=U_ q\) if \(\epsilon\) is not a character mod N/q; \(C_ q=U_ q+W_ q^{-1} U_ q W_ q+q^{k/2-1} W_ q^{-1}\) if \(\epsilon\) is a character mod N/q and \(ord_ q N=1\); \(C_ q=U_ q+W_ q^{-1} U_ q W_ q\) if \(\epsilon\) is a character mod N/q and \(ord_ q N>1\). Then the author proves that the space S(N,k,\(\epsilon)\) can be decomposed as a direct sum of common eigenspaces of all Hecke operators \(T_ p\), p \(| N\), and \(C_ q\), q \(\nmid N\), each of dimension one, containing a nonzero form whose eigenvalue is the pth Fourier coefficient with respect to \(T_ p\) for p \(| N\) and the qth Fourier coefficient with respect to \(C_ q\) for q \(\nmid N\). This result was first obtained by \textit{A. Pizer} [J. Algebra 83, 39-64 (1983; Zbl 0515.10020)] for the case of trivial character. The result of this paper can be extended to the space of Eisenstein series. It is also proved that the absolute eigenvalue of \(C_ q\) is equal to \(q^{(k-1)/2}\) if \(\epsilon\) is not a character mod N/q; and less than \(2q^{k/2}\) if \(\epsilon\) is a character mod N/q except possibly for the case \(ord_ q N=1\) and \(q=2\) or 3.
reviewed by: Dingyi Pei
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3955013
zbMATH Keywords: decomposition
zbMATH Keywords: diagonalization
zbMATH Keywords: space of cusp forms
zbMATH Keywords: newforms
zbMATH Keywords: Hecke operators
zbMATH Keywords: Fourier coefficient
zbMATH Keywords: space of Eisenstein series
zbMATH Keywords: absolute eigenvalue
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90064-5
OpenAlex ID: W1980443712
cites work: Hecke operators on \(\Gamma_0(m)\)
cites work: Twists of newforms and pseudo-eigenvalues of \(W\)-operators
cites work: Q4187223
cites work: La conjecture de Weil. I
cites work: Newforms and functional equations
cites work: Q5181799
cites work: Hecke operators for \(\Gamma_ 0(N)\)
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description: scientific article; zbMATH DE number 3955013