Motives, Periods, and Functoriality

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.

给定一个在\(\mathbb{Q}\)上具有\(M\)上的多重线性代数结构\(\mathsf{s}\)的纯动机\(M\)，以及给定一个尊重\(\mathsf{s}\)的群的表示\(V\)，我们描述了一个函子性转移\(M^V\)。我们制定了一个准则，该准则保证\(M^V\)的两个周期何时相等。这对与\(M^V\)相关联的\(L\)-函数的临界值有影响。该准则在各种例子中得到了阐释，例如：张量积动机和兰金 - 塞尔伯格\(L\)-函数；正交动机和偶正交群的标准\(L\)-函数；扭曲张量动机和浅井\(L\)-函数。