abstract:

We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions, $L^S(s,\Pi\times\Pi')$, over arbitrary CM-fields $F$, relating critical values to explicit powers of $(2\pi i)$. Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of $(2\pi i)$, it is one of the advantages of our approach, that it applies to very general non-cuspidal isobaric automorphic representations $\Pi'$ of ${\rm GL}_n(\Bbb{A}_F)$. As an application, this enables us to establish a certain algebraic version of the Gan-Gross-Prasad conjecture, as refined by N. Harris, for totally definite unitary groups. As another application we obtain a generalization of a result of Harder-Raghuram on quotients of consecutive critical values, proved by them for totally real fields, and achieved here for arbitrary CM-fields $F$ and pairs $(\Pi,\Pi')$ of relative rank one.

摘要：

我们证明了 Asai-$L$-函数 $L^S(s,\Pi',{\rm As}^\pm)$ 和 Rankin-Selberg $L$-函数 $L^ S(s,\Pi\times\Pi')$，在任意 CM 字段 $F$ 上，将临界值与 $(2\pi i)$ 的显式幂相关联。除了将阿基米德 zeta 积分对我们公式的贡献确定为 $(2\pi i)$ 的具体幂之外，我们方法的优点之一是它适用于非常普遍的非尖峰同构自守表示 $\Pi '$ 的 ${\rm GL}_n(\Bbb{A}_F)$。作为一个应用，这使我们能够为完全确定的酉群建立由 N. Harris 改进的 Gan-Gross-Prasad 猜想的某个代数版本。作为另一个应用，我们获得了对连续临界值商的 Harder-Raghuram 结果的推广，