Ikeda’s conjecture on the period of the Duke-Imamoḡlu-Ikeda lift. (English) Zbl 1376.11038
Summary: Let \(k\) and \(n\) be positive even integers. For a cuspidal Hecke eigenform \(h\) in the Kohnen plus space of weight \(k-n/2+1/2\) for \(\varGamma_0(4),\) let \(I_n(h)\) be the Duke-Imamoḡlu-Ikeda lift of \(h\) in the space of cusp forms of weight \(k\) for \(\mathrm{Sp}_n(\mathbb{Z}),\) and \(f\) be the primitive form of weight \(2k-n\) for \(\mathrm{SL}_2(\mathbb{Z})\) corresponding to \(h\) under the Shimura correspondence. We then express the ratio \(\langle I_n(h), I_n(h) \rangle / \langle h, h \rangle\) of the period of \(I_n(h)\) to that of \(h\) in terms of special values of certain \(L\)-functions of \(f\). This proves the conjecture proposed by Ikeda concerning the period of the Duke-Imamoḡlu-Ikeda lift.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
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