Motives, Periods, and Functoriality

TL;DR

This paper develops a functorial transfer for pure motives over ield, providing criteria for period equality and implications for critical L-values, with applications to tensor, orthogonal, and twisted motives.

Contribution

It introduces a new functorial transfer for motives respecting algebraic structures and formulates criteria for period equality, advancing understanding of L-functions and motives.

Findings

Criteria for period equality in motives

Explicit examples including tensor and Rankin-Selberg motives

Implications for critical values of associated L-functions

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.