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**On the lifting of elliptic cusp forms to Siegel cusp forms of degree \(2n\).**
*(English)*
Zbl 0998.11023

In this beautiful paper, the author generalizes the Saito-Kurokawa lifting to higher degrees. Let \(f(\tau)\in S_{2k}(SL_{2}(\mathbb{Z}))\) be a normalized Hecke eigenform. W. Duke and Ö. Imamoglu conjectured that if \(k\equiv n\) mod 2, then there exists a Hecke eigenform \(F(Z)\in S_{k+n}(Sp_{2n}(\mathbb{Z}))\) of degree \(2n\), whose standard \(L\)-function is equal to \(\zeta(S)\prod_{i=1}^{2n}L(s+k+n-i,f)\). The author solves this conjecture affirmatively.

The author’s construction is very simple and constructive. In fact, he gives an explicit form of the Fourier coefficient of \(F(Z)\). The Fourier coefficient essentially consists of two parts. The one part comes from the Fourier coefficient of a half-integral weight Hecke eigenform corresponding to \(f\) by the Shimura correspondence. The other part is expressed by a product of local factors coming from the so-called Siegel series.

In the final part of this paper, the author discusses the connection of his results with Arthur’s conjecture. It should be noted that Ibukiyama made a similar conjecture, which was formulated in terms of the Koecher-Maass series.

The author’s construction is very simple and constructive. In fact, he gives an explicit form of the Fourier coefficient of \(F(Z)\). The Fourier coefficient essentially consists of two parts. The one part comes from the Fourier coefficient of a half-integral weight Hecke eigenform corresponding to \(f\) by the Shimura correspondence. The other part is expressed by a product of local factors coming from the so-called Siegel series.

In the final part of this paper, the author discusses the connection of his results with Arthur’s conjecture. It should be noted that Ibukiyama made a similar conjecture, which was formulated in terms of the Koecher-Maass series.

Reviewer: Shoyu Nagaoka (Osaka)

### MSC:

11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |

11F37 | Forms of half-integer weight; nonholomorphic modular forms |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |