On the Shimura lift, après Selberg
type: Item
instance of: scholarly article
title: On the Shimura lift, après Selberg
zbMATH Open document ID: 0676.10019
DOI: 10.1016/0022-314X(89)90097-8
author: Barry Cipra
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Recall that in his paper ``On modular forms of half-integral weight'' [Ann. Math., II. Ser. 97, 440-481 (1973; Zbl 0266.10022)] \textit{G. Shimura} introduced certain lifting maps sending modular forms of weight \(k+\tfrac12\) \((k\in {\mathbb Z})\) to modular forms of weight \(2k\). These mappings can be defined via Fourier/Mellin series. As the author points out, a specific version of the Shimura lift was discovered earlier by Selberg (unpublished): roughly speaking, what Selberg observed was that the product of a Hecke eigenform \(f\) of weight \(k\) on \(\text{SL}_2(\mathbb Z)\) and a theta function \(\theta\) of weight \(1/2\) and level 4 ``lifts'' to the square of \(f\). The proof is elementary in the sense that it makes only use of the fact that the Fourier coefficients of \(f\) are multiplicative. In the present paper the author generalizes Selberg's observation to the extent that arbitrary levels and characters and also more general theta series are considered.
Mathematics Subject Classification ID: 11F32
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 4106952
zbMATH Keywords: Selberg's version
zbMATH Keywords: theta functions
zbMATH Keywords: arbitrary level and character
zbMATH Keywords: Shimura lift
zbMATH Keywords: theta series
reviewed by: Winfried Kohnen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4106952