Abstract

Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $s$ on $M$, and given a representation $V$ of the group respecting $s$, 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.

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