Given an automorphic representation of a unitary group, one can define an arithmetic automorphic period as the Petersson inner product of a deRham rational form. Here the deRham rational structure comes from the cohomology of Shimura varieties. When the form is holomorphic, the period can be related to special values of L-functions and is better understood. In this talk, we formulate a conjecture on relations among general arithmetic periods of representations in the same L-packet and explain a conditional proof.