On the Shimura lift, après Selberg
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 As the author points out, a specific version of the Shimura lift was discovered earlier by Selberg (unpublished): roughly speaking, what Selberg observed was that the product of a Hecke eigenform \(f\) of weight \(k\) on \(\text{SL}_2(\mathbb Z)\) and a theta function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.
scientific article; zbMATH DE number 4106952
scientific article