On Period Relations for Automorphic L-functions I

This paper is the first in a series of two dedicated to the study of period relations of the type $$L(\frac{1}{2}+k,\Pi)\;\in\;(2\pi i)^{d\cdot k}\Omega_{(-1)^k}{\mathbb Q}(\Pi),\quad \frac{1}{2}+k\;\text{critical},$$ for certain automorphic representations $\Pi$ of a reductive group $G.$ In this paper we discuss the case $G={\mathrm{GL}}(n+1)\times{\mathrm{GL}}(n).$ The case $G={\mathrm{GL}}(2n)$ is discussed in part two. Our method is representation-theoretic and relies on the author's recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation $\Pi$ under consideration. The new period relations we prove are in accordance with Deligne's Conjecture on special values of $L$-functions and we expect our method to apply to other cases as well.