Lifting modular forms of half-integral weight to Siegel modular forms of even genus
Let \(f\) be a normalized Hecke eigenform in the space \(S_{2k}\) of cusp forms of even weight \(2k\) on the full modular group \(\text{SL}_2(\mathbb{Z})\), and let \(n\) be a positive integer for which \(n-k\) is even. Then there exists a Hecke eigenform \(F\) in the space \(S_{k+n}(\Gamma_{2n})\) of cusp forms of weight \(k+n\) with respect to the Siegel modular group \(\Gamma_{2n}= \text{Sp}_{2n}(\mathbb{Z})\) of even genus \(2n\), such that the standard zeta function of \(F\) is \[ \zeta(s) \prod_{j=1}^{2n} L(f,s+k+n-j), \] where \(L(f,s)\) is the Hecke \(L\)-function of \(f\). This was conjectured by W. Duke and Ö. Imamoglu, and it was proved by \textit{T. Ikeda} [Ann. Math. (2) 154, 641-681 (2001; Zbl 0998.11023)]. Ikeda also proved a formula for the coefficients \(A(T)\) in the Fourier expansion of \(F(Z)\). One of the factors in \(A(T)\) is a coefficient of the cusp form \(g\) of half-integral weight \(k+ \frac 12\) in the Kohnen plus space \(S_{k+1/2}^+\) which corresponds to \(f\) via the Shimura lift. In the paper under review, another formula for \(A(T)\) is proved. It also contains coefficients of \(g\), but its other ingredients are easier to understand than those in Ikeda's formula. The author uses his formula to define a linear map from \(S_{k+1/2}^+\) to \(S_{k+n}(\Gamma_{2n})\) which on Hecke eigenforms coincides with Ikeda's lifting map. Many interesting details of the paper cannot be reproduced here. It ends with a discussion of some open problems. One of them is a conjecture on the image of the lifting map in \(S_{k+n} (\Gamma_{2n})\) which for \(n=1\) is equal to the Maass space.
scientific article; zbMATH DE number 1753094