Ikeda's conjecture on the period of the Duke-Imamoglu-Ikeda lift

TL;DR

This paper proves Ikeda's conjecture relating the period of the Duke-Imamoglu-Ikeda lift to special L-values, establishing a precise formula connecting automorphic forms and their periods.

Contribution

The paper confirms Ikeda's conjecture by expressing the period ratio of the lift and the original form in terms of special L-function values, advancing understanding of automorphic periods.

Findings

Established explicit formula for period ratio in terms of L-values

Proved Ikeda's conjecture on the period of the Duke-Imamoglu-Ikeda lift

Connected automorphic periods with special values of L-functions

Abstract

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Sp(n,Z), and f the primitive form of weight 2k-n for SL(2,Z) corresponding to h under the Shimura correspondence. We then express the ratio <I(h),I(h)>/< h, h > of the period of I(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-Imamoglu-Ikeda lift.