Ikeda's conjecture on the period of the Duke–Imamoḡlu–Ikeda lift

Abstract

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 Γ0(4), let In(h) be the Duke-Imamoḡlu-Ikeda lift of h in the space of cusp forms of weight k for Sp n (Z), and f be the primitive form of weight 2k − n for SL2(Z) corresponding to h under the Shimura correspondence. We then express the ratio In(h), In(h) / h, h of the period of In(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.be the character corresponding to Q( √ D)/Q. Here we make the convention that D * = 1 if D = 1.

Rankin-Selberg Method for Jacobi Forms of Integral Weight and of Half-Integral Weight on Symplectic Groups

“…This result is a generalization of the result for the Rankin-Selberg type Dirichlet series of Siegel cusp forms of half-integral weight obtained by Katsurada and Kawamura in [KK15, Corollary to Proposition 3.1]. It means that in [KK15] they treated Fourier coefficients of Siegel cusp forms of half-integral weight to construct the Rankin-Selberg type Dirichlet series. In this paper we shall treat Jacobi cusp forms of half-integral weight of arbitrary degree instead of Siegel cusp forms of half-integral weight and we also shall treat Fourier-Jacobi coefficients instead of Fourier coefficients.…”

“…We remark that the case t = n in Corollary 3.4 has been shown in [KZ81] (for n = 1) and in [KK15] (for n > 1).…”

On the Gross-Keating invariant of a quadratic form over a non-archimedean local field

“…The formula was also one of key ingredients in proving the conjecture on the period of the Duke-Imamoglu-Ikeda lift proposed in [12] (cf. [20]). Moreover, it was used to relate the local intersection multiplicities on certain Shimura varieties to the derivatives of certain local Whittaker functions in [30].…”

Congruence Primes of the Kim–Ramakrishnan–Shahidi Lift

“…Böecherer, Dummigan, and Schulze-Pillot [4] proved the period relation for the Yoshida lift and gave a similar result on the congruence between the Yoshida lift and non-Yoshida lift. Katsurada and Kawamura [26] proved Ikeda's conjecture on the period of the Duke-Imamoglu-Ikeda lift proposed in [23], and by using this period relation Katsurada proved Problem B for the Duke-Imamoglu-Ikeda lift in [25] (see also [8]). Based on the conjectural period relation in [23], Ibukiyama, Katsurada, Poor, and Yuen [18] proposed a conjecture on the congruence of the Ikeda-Miyawaki lift and tested it numerically.…”