Motives, Periods, and Functoriality

TL;DR

The article develops a motivic analogue of Langlands transfer by associating to a motive $M$ with a multilinear structure ${\mathsf s}$ and a representation $V$ a transferred motive $M^V$, and proves a criterion for when its Deligne periods satisfy $c^+(M^V) \approx_E c^-(M^V)$. Grounded in Betti–de Rham realizations and Tannakian formalism, the authors establish explicit determinant relations for periods, discuss how they behave under Tate twists and Yoshida invariants, and relate these to critical $L$-values via Deligne’s conjectures. The framework is illustrated through a suite of examples—tensor products, Artin motives, Asai and Rankin–Selberg L-functions—showing how period relations explain and predict rationality phenomena in automorphic $L$-values. The results unify several prior rationality statements and provide new motivic explanations for period phenomena across totally real and totally imaginary base fields, with extensions to Asai and symmetric–orthogonal contexts. The work highlights the deep interplay between motives, periods, and automorphic $L$-functions, and offers a robust toolkit for probing critical values via motivic transfer.

Abstract

Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group respecting $\mathsf{s}$, we describe a functorial transfer $M^V$. We formulate a criterion that guarantees when the two periods of $M^V$ are equal. This has an implication for the critical values of the $L$-function attached to $M^V.$ The criterion is explicated in a variety of examples such as: tensor product motives and Rankin-Selberg $L$-functions; orthogonal motives and the standard $L$-function for even orthogonal groups; twisted tensor motives and Asai $L$-functions.

Theorems & Definitions (38)

• Definition 2.2.1
• Lemma 2.4.1
• Corollary 2.4.2
• Lemma 2.5.2.1
• proof : Proof of Lemma
• Proposition 2.5.3.1
• Theorem 2.6.1
• proof
• Lemma 2.6.2
• proof