On the rationality of period integrals and special value formulas in the compact case

Abstract

We study rationality properties of period integrals that appear in the Gan–Gross–Prasad conjectures in the compact case using Gross’ theory of algebraic modular forms. In situations where the refined Gan–Gross–Prasad are known, our rationality result for period can be interpreted as a special value formula for automorphic $L$-functions which proves automorphic versions of Deligne’s conjecture on rationality of periods. Moreover, this special value formula is well suited to $p$-adic interpolation, as illustrated in [10].