The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.

中文翻译：

---

GL2n 的 L 函数：p-adic 特性和扭曲不消失

本文的主要目的是将 $p$-adic $L$-函数附加和研究到完全实域 $F$ 上 $\operatorname {GL}_{2n}$ 的上同调尖牙自守表示 $\Pi$承认 Shalika 模型。我们使用模块化符号方法，沿着 Ash 和 Ginzburg 工作的全局路线，但我们的结果更加明确，因为我们大量借鉴了所有三位作者最近和单独的作品中使用的方法。通过构造，我们的 $p$-adic $L$-函数是在 $p\infty$ 之外无分支的 $F$ 的最大阿贝尔扩展的伽罗瓦群上的分布。此外，我们在 $\Pi _p$ 上的较弱的 Panchishkine 类型条件下工作，而不是在完全普通条件下工作。最后，我们证明了所有临界点的$p$-adic $L$-函数之间的所谓Manin 关系。